Water is leaking out of an inverted conical tank at a rate

Differentiate:(a)f!x"!1 x (b)y !s 3 x 2

Find equations of the tangent line and normal line to the curve at the point . Illustrate by graphing the curve and these lines

Find the points on the curve y !x 4 "6x V 2 !4 where the tangent line is horizonta

The equation of motion of a particle is ,s !2t s 3 "5t 2 !3t !4 where is measured in centimeters and in seconds. Find the acceleration as a function of time. What is the acceleration after 2 seconds?

If , find and . Compare the graphs of and

At what point on the curve is the tangent line parallel to the line ?

(a) How is the number e defined? (b) Use a calculator to estimate the values of the limits lim hl0 2.7h "1 h and lim hl0 2.8h "1 h correct to two decimal places. What can you conclude about the value of e?

(a) Sketch, by hand, the graph of the function , paying particular attention to how the graph crosses the y-axis. What fact allows you to do this?b) What types of functions are and ? Compare the differentiation formulas for and t. (c) Which of the two functions in part (b) grows more rapidly when x is large?

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Find an equation of the tangent line to the curve at the given poin

Find an equation of the tangent line to the curve at the given poin

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen y !x "sx !1, 0"

Find f#!x" Compare the graphs of and and use them to explain why your answer is reasonable

Find f#!x" Compare the graphs of and and use them to explain why your answer is reasonable

Find f#!x" Compare the graphs of and and use them to explain why your answer is reasonable

Find f#!x" Compare the graphs of and and use them to explain why your answer is reasonable

(a) Use a graphing calculator or computer to graph the function in the viewing rectangle ("3, 5)by ("10, 50). f !x"!x 4 "3x 3 "6x 2 !7x !30 (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.8.) (c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b)

(a) Use a graphing calculator or computer to graph the function in the viewing rectangle by . (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.8.) (c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).

Find the first and second derivatives of the function

Find the first and second derivatives of the function

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of , , and .

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of , , and .

The equation of motion of a particle is , where is in meters and is in seconds. Find (a) the velocity and acceleration as functions of , (b) the acceleration after 2 s, and (c) the acceleration when the velocity is 0.

The equation of motion of a particle is s !2t s t 3 "7t 2 !4t !1, where is in meters and is in seconds. (a) Find the velocity and acceleration as functions of . (b) Find the acceleration after 1 s. ; (c) Graph the position, velocity, and acceleration functions on the same screen

Find the points on the curve y !2x 3 !3x 51. 2 "12x !1 where the tangent is horizonta

For what values of does the graph of f !x"!x 3 !3x 2 !x !3 have a horizontal tangent?

Show that the curve y !6x 3 !5x "3 has no tangent line with slope 4

Find an equation of the tangent line to the curve that is parallel to the line

Find equations of both lines that are tangent to the curve and are parallel to the line 12x "y !1

At what point on the curve y !1 !2e x "3x is the tangent line parallel to the line ? Illustrate by graphing the curve and both lines

Find an equation of the normal line to the parabola y !x that is parallel to the line

Where does the normal line to the parabola at the point (1, 0) intersect the parabola a second time? Illustrate with a sketch.

Draw a diagram to show that there are two tangent lines to the parabola that pass through the point . Find the coordinates of the points where these tangent lines intersect the parabola

(a) Find equations of both lines through the point that are tangent to the parabola . (b) Show that there is no line through the point that is tangent to the parabola. Then draw a diagram to see why.

Use the definition of a derivative to show that if , then . (This proves the Power Rule for the case

Find the derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

Find a second-degree polynomial such tha

The equation is called a differential equation because it involves an unknown function and its derivatives and . Find constants such that the function satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)

Find a cubic function whose graph has horizontal tangents at the points an

Find a parabola with equation that has slope 4 at , slope at , and passes through the point

Let f !x"!# 2 !x x 2 !2x "2 if x #1 if x $1 Is differentiable at 1? Sketch the graphs of and

At what numbers is the following function differentiable? Give a formula for t%and sketch the graphs of t and t%. t!x"!# !1 !2x x 2 x if x &!1 if !1 #x #1 if x $1

(a) For what values of is the function differentiable? Find a formula for . (b) Sketch the graphs of an

Where is the function differentiable? Give a formula for and sketch the graphs of and

Find the parabola with equation whose tangent line at (1, 1) has equation

Suppose the curve has a tangent line when with equation and a tangent line when with equation . Find the values of , , , and

For what values of and is the line tangent to the parabola when

Find the value of such that the line is tangent to the curve

Let f !x"!# x 2 mx "b if x #2 if x $2 Find the values of and that make differentiable everywher

A tangent line is drawn to the hyperbola at a point . (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is . (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where is located on the hyperbola

Evaluate lim xl1 x 1000 !1 x !1

Draw a diagram showing two perpendicular lines that intersect on the -axis and are both tangent to the parabola . Where do these lines intersect

If , how many lines through the point are normal lines to the parabola ? What if

Sketch the parabolas and . Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

(a) If , find . (b) Find the derivative

Differentiate the function f!t"!st !a "bt

If f!x"!sx t!x, where t!4"!2 and t%!4"!3 , find f%!4

Let y !x 2 "x !2 x 3 "6 then y%! !x 3 "6"d dx !x 2 "x !2"!!x 2 "x !2"d dx !x 3 "6" !x 3 "6" 2 !!x 3 "6"!2x "1"!!x 2 "x !2"!3x 2 " !x 3 "6" 2 !!2x 4 "x 3 "12x "6"!!3x 4 "3x 3 !6x 2 " !x 3 "6" 2 !!x 4 !2x 3 "6x 2 "12x "6 !x 3 "6" 2

Find an equation of the tangent line to the curve y !ex %!1 "x 2 at the point .

Find the derivative of y !!x 2 "1"!x 3 "1" in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?

Find the derivative of the function F!x"!x !3xsx sx in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?

Differentiate f!x"!!x 3 "2x"ex

Differentiate t!x"!sx ex

Differentiate y !ex x 2

Differentiate y !e x 1 "x

Differentiate t!x"!3x !1 2x "1

Differentiate f!t"!2t 4 "t 2

Differentiate V!x"!!2x3 "3"!x4 !2x"

Differentiate Y!u"!!u!2 "u!3 "!u5 !2u2 "

Differentiate F!y"!(1 y2 !3 y4 )!y "5y3 . "

Differentiate R!t"!!t "et "(3 !st

Differentiate y !x 3 1 !x 2

Differentiate y !x "1 x 3 "x !2

Differentiate y !t 2 "2 t 4 !3t 2 "1

Differentiate y !t !t !1"

Differentiate y !!r 2 !2r"e

Differentiate y !1 s "ke

Differentiate y !"v3 !2vsv v

Differentiate z !w3%2 !w "ce

Differentiate f!t"!2t 2 "st

Differentiate t!t"!t !st t 1%3

Differentiate f!x"!A B "Ce

Differentiate f!x"!1 !xex x "e

Differentiate f!x"!x x " c x

Differentiate f!x"!ax "b cx "d

Find f'(x)and f''(x) f!x"!x 4 e x

Find f'(x)and f''(x) f!x"!x 5%2 ex

Find f'(x)and f''(x) f!x"!x 2 1 "2x

Find f'(x)and f''(x) f!x"!x 3 "ex

Find an equation of the tangent line to the given curve at the specified point

Find an equation of the tangent line to the given curve at the specified point

Find equations of the tangent line and normal line to the given curve at the specified point.

Find equations of the tangent line and normal line to the given curve at the specified point.

(a) The curve is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen

(a) The curve is called a serpentine. Find an equation of the tangent line to this curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

(a) If , find . ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and

a) If , find . ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and

(a) If , find and . ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of

(a) If , find and . ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and

If , find

If , find

Suppose that , , , and . Find the following values.

Suppose that , , , and . Find

If , where and , fin

If h!2"!4 and h%!2"!!3 , find d dx ( h!x" x ),x!2

If and are the functions whose graphs are shown, let and

Let and , where and are the functions whose graphs are shown

If is a differentiable function, find an expression for the deriv- q !f !p" ative of each of the following functions

If is a differentiable function, find an expression for the derivative of each of the following functions.

How many tangent lines to the curve ) pass through the point ? At which points do these tangent lines touch the curve?

Find equations of the tangent lines to the curve that are parallel to the line

In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule

A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write . Then the total revenue earned with selling price p is . (a) What does it mean to say that and ? (b) Assuming the values in part (a), find and interpret your answer

(a) Use the Product Rule twice to prove that if , , and are differentiable, then . (b) Taking in part (a), show that (c) Use part (b) to differentiate

(a) If , where and have derivatives of all orders, show that . (b) Find similar formulas for and . (c) Guess a formula for

Find expressions for the first five derivatives of . Do you see a pattern in these expressions? Guess a formula for and prove it using mathematical induction

(a) If t is differentiable, the Reciprocal Rule says that Use the Quotient Rule to prove the Reciprocal Rule. (b) Use the Reciprocal Rule to differentiate the function in Exercise 18. (c) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is, for all positive integers n. d dx !x!n "!!n

Differentiate y !x 2 sin x

Differentiate . For what values of x does the graph of have a horizontal tangent?

An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time . (See Figure 5 and note that the downward direction is positive.) Its position at time t is

Find the 27th derivative of cos x

Find lim xl 0 sin 7x 4x

Calculate lim xl 0 x cot x

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Differentiate

Prove that d dx !csc x"!"csc x cot x

Prove that d dx !sec x"!sec x tan x

Prove that d dx !cot x"!"csc2 x

Prove, using the definition of derivative, that if f!x"!cos x then f#!x"!"sin x.

Find an equation of the tangent line to the curve at the given point.y !sec x, !%%3, 2"

Find an equation of the tangent line to the curve at the given point.

Find an equation of the tangent line to the curve at the given point.y !x !cos x !0, 1"

Find an equation of the tangent line to the curve at the given point.y !!0, 1"1 sin x !cos x

(a) Find an equation of the tangent line to the curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen

(a) Find an equation of the tangent line to the curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen

(a) If , find . ; (b) Check to see that your answer to part (a) is reasonable by graphing both and for

(a) If , find and . ; (b) Check to see that your answers to part (a) are reasonable by graphing , , and

If H!$"!$sin $,find H#!$"and H'!$"

If f !x"!sec x, find f '!%%4".

(a) Use the Quotient Rule to differentiate the function 37. $ (b) Simplify the expression for by writing it in terms of and , and then find . (c) Show that your answers to parts (a) and (b) are equivalent.

Suppose and , and let and Find (a) and (b)

For what values of does the graph of have a horizontal tangent?

Find the points on the curve at which the tangent is horizontal.

A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is , where is in seconds and in centimeters. (a) Find the velocity and acceleration at time . (b) Find the position, velocity, and acceleration of the mass at time . In what direction is it moving at that time?

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is , , where is measured in centimeters and in seconds. (Take the positive direction to be downward.) (a) Find the velocity and acceleration at time . (b) Graph the velocity and acceleration functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest?

A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does change with respect to when ?

An object with weight is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is where is a constant called the coefficient of friction. (a) Find the rate of change of with respect to . (b) When is this rate of change equal to 0? ; (c) If lb and , draw the graph of as a function of and use it to locate the value of for which . Is the value consistent with your answer to part (b

Find the limit.lim xl0 sin 3x x

Find the limit.lim xl0 sin 4x sin 6x

Find the limit.lim tl0 tan 6t sin 2t

Find the limit.lim $l0 cos $"1 sin $

Find the limit.lim $l0 sin!cos $" sec $

Find the limit.lim tl0 sin2 3t t 2

Find the limit.lim xl0 sin!x 2 " x

Find the limit.lim %l%%4 1 "tan x sin x "cos x

Find the limit.lim xl1 sin!x "1" x 2 !x "2

Differentiate each trigonometric identity to obtain a new (or familiar) identity. (a) (b) (c)

A semicircle with diameter sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice cream cone, as shown in the figure. If is the area of the 51. s semicircle and is the area of the triangle, find

The figure shows a circular arc of length and a chord of length , both subtended by a central angle . Find

Find F!x"!sx F#!x"2 !1

Differentiate (a)y=sin (x2) and (b) y=sin2x

Differentiate y=(x3-1)

Find f'(x) if f (x)=1 s 3 x 2 !x !1

Find the derivative of the function t!t"!#t "2 2t !1 $

Differentiate y !!2x !1" 5 !x 3 "x !1" 4

Differentiate y !esin x

If f"x#!sin"cos"tan x## then f$"x#!cos"cos"tan x##d dx cos"tan x# !cos"cos"tan x##%#sin"tan x#&d dx "tan x# !#cos"cos"tan x## sin"tan x# sec2 x

Differentiate y !esec 3&

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Write the composite function in the form . [Identify the 3 inner function and the outer function .] Then find the derivative

Find the derivative of the function y !e#5x y !! cos 3x

Find the derivative of the function y !e#5x y !! cos 3x

Find the derivative of the function

Find the derivative of the function fx1 x 4 23 Fxs

Find the derivative of the function

Find the derivative of the function fts 3 tt1 tan t 1 t

Find the derivative of the function y cosa x 3 x 3 f

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function tx1 4x 5 3 x x2 8 y xe

Find the derivative of the function htt 4 1 3 t 3 1 4 tx1

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function y !e#5x y !! cos 3x

Find the derivative of the function

Find the derivative of the function G"y#!"y #1# 4 "y2 %2y#F 5

Find the derivative of the function

Find the derivative of the function y !e u #e#u e u %e#u

Find the derivative of the function

Find the derivative of the function y !x sin 1 x

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function

Find the derivative of the function y !x sin 1 x

Find the derivative of the function

Find the derivative of the function f"t#!*t t 2 %4

Find the derivative of the function

Find the derivative of the function y !ek tan sx

Find the derivative of the function

Find the derivative of the function y !sin"sin"sin x##

Find the derivative of the function

Find the derivative of the function y !sx %sx %sx

Find the derivative of the function t"x#!"2rarx %n# p

Find the derivative of the function y !23x2

Find the derivative of the function y !cosssin"tan (x#

Find the derivative of the function y !%x %"x %sin2 x# 3 &4

Find the first and second derivatives of the function h"x#!sx 2 %1

Find the first and second derivatives of the function y !xe cx

Find the first and second derivatives of the function !e )x sin *x

Find the first and second derivatives of the function y !ee x

Find an equation of the tangent line to the curve at the given point y !"1 %2x#"0

Find an equation of the tangent line to the curve at the given point y !sin x %sin "0, 0#

Find an equation of the tangent line to the curve at the given point y !sin"sin x#"(, 0#

Find an equation of the tangent line to the curve at the given point y !x "1, 1(e#

(a) Find an equation of the tangent line to the curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen

(a) The curve is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen

(a) If , find . ; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f$

The function , , arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of produced by a graphing device to make a rough sketch of the graph of . (b) Calculate and use this expression, with a graphing

Find all points on the graph of the function at which the tangent line is horizontal.

Find the -coordinates of all points on the curve at which the tangent line is horizontal

If , where , , , , and , find

If , where and , find

A table of values for , , , and is given. (a) If , find . (b) If , find

Let and be the functions in Exercise 63. (a) If , find . (b) If , find

If and are the functions whose graphs are shown, let , , and . Find each derivative, if it exists. If it does not exist, explain why

If is the function whose graph is shown, let and . Use the graph of to estimate the value of each derivative

If is the function whose graph is shown, let and . Use the graph of to estimate the value of each derivative

Suppose is differentiable on and is a real number. Let and . Find expressions for (a) and

Let , where , , , , and . Find

If is a twice differentiable function and , find in terms of , , and

If , where and , find

If , where , , , , and , find

Show that the function satisfies the differential equation

For what values of does the function satisfy the equation

Find the 50th derivative of

Find the 1000th derivative of

The displacement of a particle on a vibrating string is given by the equation where is measured in centimeters and in seconds. Find the velocity of the particle after seconds

If the equation of motion of a particle is given by , the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time . (b) When is the velocity 0

A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by . In view of these data, the brightness of Delta Cephei at time , where is measured in days, has been modeled by the function (a) Find the rate of change of the brightness after days. (b) Find, correct to two decimal places, the rate of increase after one day

In Example 4 in Section 1.3 we arrived at a model for the length of daylight (in hours) in Philadelphia on the th day of the year: L"t#!12 %2.8 sin+2( 365 "t #80#,Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21

The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is where is measured in centimeters and in seconds. Find the velocity after seconds and graph both the position and velocity functions for

Under certain circumstances a rumor spreads according to the equation where is the proportion of the population that knows the rumor at time and and are positive constants. [In Section 9.4 we will see that this is a reasonable equation for .] (a) Find . (b) Find the rate of spread of the rumor. ; (c) Graph for the case , with measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor

A particle moves along a straight line with displacement velocity , and acceleration . Show that Explain the difference between the meanings of the derivatives

Air is being pumped into a spherical weather balloon. At any time , the volume of the balloon is and its radius is . (a) What do the derivatives and represent? (b) Express in terms of .

The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge remaining on the capacitor (measured in microcoulombs, +C) at time (measured in seconds). (a) Use a graphing calculator or computer to find an exponential model for the charge. (b) The derivative represents the electric current (measured in microamperes, +A) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when s. Compare with the result of Example 2 in Section 2.1.

The table gives the US population from 1790 to 1860. (a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines. (c) Use the exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b). (d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,558,000. Can you explain the discrepancy?

Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to find the derivative in Example 5 and compare with the answer in that example. Then use the simplify command and compare again. (b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents?

(a) Use a CAS to differentiate the function and to simplify the result. (b) Where does the graph of have horizontal tangents? (c) Graph and on the same screen. Are the graphs consistent with your answer to part (b)?

Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.

Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule

(a) If is a positive integer, prove that (b) Find a formula for the derivative of that is similar to the one in part (a

Suppose is a curve that always lies above the -axis and never has a horizontal tangent, where is differentiable everywhere. For what value of is the rate of change of with respect to eighty times the rate of change of with respect to ?

Use the Chain Rule to show that if is measured in degrees, then (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would not be as simple if we used degree measure.)

(a) Write and use the Chain Rule to show that (b) If , find and sketch the graphs of and . Where is not differentiable? (c) If , find and sketch the graphs of and . Where is not differentiable?

If and , where and are twice differentiable functions, show that d2 y dx 2 !d2 y du2 !du dx $ 2 % dy du d2 u dx 2

f and , where and possess third derivatives, find a formula for similar to the one given in Exercise 95.

(a) Find by implicit differentiation. (b) Solve the equation explicitly for and differentiate to get in terms of . (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part (a)

(a) Find by implicit differentiation. (b) Solve the equation explicitly for and differentiate to get in terms of . (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part (a)

(a) Find by implicit differentiation. (b) Solve the equation explicitly for and differentiate to get in terms of . (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part (a) 1 x ! 1 y !1

(a) Find by implicit differentiation. (b) Solve the equation explicitly for and differentiate to get in terms of . (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part (a) cos x !sy !5

Find by implicit differentiation

Find by implicit differentiation 2sx !sy !3

Find by implicit differentiation

Find by implicit differentiation 2x 3 !x 2 y #xy x 3 !2

Find by implicit differentiation

Find by implicit differentiation 5 !x 2 y 3 !1 !yex

Find by implicit differentiation x "2 y2 !x sin y !4

Find by implicit differentiation

Find by implicit differentiation

Find by implicit differentiation

Find by implicit differentiation

Find by implicit differentiation sx !y !1 !x2 y2

Find by implicit differentiation

Find by implicit differentiation tan!x #y"!y 1 !x s 2

Find by implicit differentiation

Find by implicit differentiation sin x !cos y !sin x cos y

If and , find

If , find

Regard as the independent variable and as the dependent variable and use implicit differentiation to find .

Regard as the independent variable and as the dependent variable and use implicit differentiation to find .

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

Use implicit differentiation to find an equation of the tangent line to the curve at the given poin

(a) The curve with equation is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point . ; (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)

(a) The curve with equation is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point . (b) At what points does this curve have horizontal tangents? ; (c) Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen

Find by implicit differentiation

Find by implicit differentiation sx !sy !1

Find by implicit differentiation x 3 !y 3 !1

Find by implicit differentiation x 4 !y4 !a4

Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a) Graph the curve with equation At how many points does this curve have horizontal tangents? Estimate the -coordinates of these points. (b) Find equations of the tangent lines at the points (0, 1) and (0, 2). (c) Find the exact -coordinates of the points in part (a). (d) Create even more fanciful curves by modifying the equation in part (a

(a) The curve with equation has been likened to a bouncing wagon. Use a computer algebra system to graph this curve and discover why. (b) At how many points does this curve have horizontal tangent lines? Find the -coordinates of these points.

Find the points on the lemniscate in Exercise 29 where the tangent is horizontal.

Show by implicit differentiation that the tangent to the ellipse at the point is

Find an equation of the tangent line to the hyperbola at the point

Show that the sum of the - and -intercepts of any tangent line to the curve is equal to

Show, using implicit differentiation, that any tangent line at a point to a circle with center is perpendicular to the radius OP

The Power Rule can be proved using implicit differentiation for the case where is a rational number, , and is assumed beforehand to be a differentiable function. If , then . Use implicit differentiation to show that y"!p q x!p#q"#1

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible

Find the derivative of the function. Simplify where possible F!*"!arcsin ssin *

Find the derivative of the function. Simplify where possible y !cos#1 !e2x "

Find the derivative of the function. Simplify where possible y !arctan'1 #x 1 !x

Find . Check that your answer is reasonable by comparing the graphs of and

Find . Check that your answer is reasonable by comparing the graphs of and

Prove the formula for by the same method as for .

(a) One way of defining is to say that and or . Show that, with this definition, (b) Another way of defining that is sometimes used is to say that and , . Show that, with this definition,

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other, that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.x ax !by !0 2 !y 2 !r 2 ,

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other, that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.x 2 !by 2 !y 2 !ax

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other, that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.2 !2y y !cx 2 !k

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other, that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes.x 2 !3y y !ax 2 !b

The equation represents a rotated ellipse, that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses

(a) Where does the normal line to the ellipse at the point intersect the ellipse a second time? ; (b) Illustrate part (a) by graphing the ellipse and the normal line

Find all points on the curve where the slope of the tangent line is

Find equations of both the tangent lines to the ellipse that pass through the point

(a) Suppose is a one-to-one differentiable function and its inverse function is also differentiable. Use implicit differentiation to show that provided that the denominator is not 0.

(a) Show that is one-to-one. (b) What is the value of ? (c) Use the formula from Exercise 67(a) to find

The figure shows a lamp located three units to the right of the -axis and a shadow created by the elliptical region . If the point is on the edge of the shadow, how far above the -axis is the lamp located?

Differentiate y !ln!x V 3 !1"

Finf d/dxIn(sin x)

Differentiate f!x"!sln x

Differentiate f!x"!log10!2 !sin x

Find d dx ln x !1 sx #2

Find f'(x) if f (x)= In /x/

Differentiate y !x 3#4 sx 2 !1 !3x !2"5

Differentiate y !x sx

Explain why the natural logarithmic function is used much more frequently in calculus than the other logarithmic functions

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function f!x"!sin x ln!5x"

Differentiate the function f!t"!1 !ln t 1 #ln

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function y !1 ln x

Differentiate the function

Differentiate the function H!z"!ln'a2 #z 2 a2 !z 2

Differentiate the function

Differentiate the function

Differentiate the function

Differentiate the function

Find y' and y'' y !x 2 ln!2x"

Find y' and y'' y !ln x x 2

Find y' and y''

Find y' and y'' y !ln!sec x !tan x"

Differentiate and find the domain of .f!x"!x 1 #ln!x #1"

Differentiate and find the domain of .f!x"!1 1 !ln x

Differentiate and find the domain of .

Differentiate and find the domain of .f!x"!ln ln ln x

If , find

If f!x"!ln!1 !e , find f "!0". 2x

Find an equation of the tangent line to the curve at the given point

Find an equation of the tangent line to the curve at the given point

If , find . Check that your answer is reasonable by comparing the graphs of and

Find equations of the tangent lines to the curve at the points and . Illustrate by graphing the curve and its tangent lines

Use logarithmic differentiation to find the derivative of the function y !!2x !1" 5 !x 4 #3" 6

Use logarithmic differentiation to find the derivative of the function y !sx e x 2 !x 2 !1" 10

Use logarithmic differentiation to find the derivative of the function y !sin2 x tan4 x !x 2 !1" 2

Use logarithmic differentiation to find the derivative of the function y !'4 x 2 !1 x 2 #1

Use logarithmic differentiation to find the derivative of the function y !x x

Use logarithmic differentiation to find the derivative of the function y !xcos x

Use logarithmic differentiation to find the derivative of the function y !xsin x

Use logarithmic differentiation to find the derivative of the function y !sx x

Use logarithmic differentiation to find the derivative of the function y !!cos x" x

Use logarithmic differentiation to find the derivative of the function y !!sin x" ln x

Use logarithmic differentiation to find the derivative of the function y !!tan x" 1#x

Use logarithmic differentiation to find the derivative of the function y !!ln x" cos x

Find y' if y=In(x2 + y2)

Find y' if xy=yx

Find a formula for if

Find d9 dx 9 !x 8 ln x"

Use the definition of derivative to prove that lim xl0 ln!1 !x" x !1

Show that lim for any x -0. n l ()1 ! x n * n !e

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time and after 3 s. ; (h) Graph the position, velocity, and acceleration functions for . (i) When is the particle speeding up? When is it slowing down?f!t"!t 3 !12t 1. 2 "36

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time and after 3 s. ; (h) Graph the position, velocity, and acceleration functions for . (i) When is the particle speeding up? When is it slowing down?f!t"!0.01t 4 !0.04t 3

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time and after 3 s. ; (h) Graph the position, velocity, and acceleration functions for . (i) When is the particle speeding up? When is it slowing down?

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time and after 3 s. ; (h) Graph the position, velocity, and acceleration functions for . (i) When is the particle speeding up? When is it slowing down?f!t"!te!t#2

Graphs of the velocity functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

Graphs of the position functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

The position function of a particle is given by . (a) When does the particle reach a velocity of 5 m#s?(b) When is the acceleration 0? What is the significance of 15. this value of ?

If a ball is given a push so that it has an initial velocity of down a certain inclined plane, then the distance it has rolled after seconds is . (a) Find the velocity after 2 s. (b) How long does it take for the velocity to reach ?

If a stone is thrown vertically upward from the surface of the moon with a velocity of , its height (in meters) after seconds is . (a) What is the velocity of the stone after 3 s? (b) What is the velocity of the stone after it has risen 25 m?

If a ball is thrown vertically upward with a velocity of 80 ft#s, then its height after seconds is . (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?

If a ball is thrown vertically upward with a velocity of 80 ft#s, then its height after seconds is . (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?

If a ball is thrown vertically upward with a velocity of 80 ft#s, then its height after seconds is . (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?

(a) Find the average rate of change of the area of a circle with respect to its radius as changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when . (c) Show that the rate of change of the area of a circle with respect to its radius (at any ) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount . How can you approximate the resulting change in area if is small?

(a) Find the average rate of change of the area of a circle with respect to its radius as changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when . (c) Show that the rate of change of the area of a circle with respect to its radius (at any ) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount . How can you approximate the resulting change in area if is small?

(a) Find the average rate of change of the area of a circle with respect to its radius as changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when . (c) Show that the rate of change of the area of a circle with respect to its radius (at any ) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount . How can you approximate the resulting change in area if is small?

(a) The volume of a growing spherical cell is , where the radius is measured in micrometers (1 ,m ). Find the average rate of change of with respect to when changes from (i) 5 to 8 ,m (ii) 5 to 6 ,m (iii) 5 to 5.1 ,m (b) Find the instantaneous rate of change of with respect to when ,m. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c

The mass of the part of a metal rod that lies between its left end and a point meters to the right is kg. Find the linear density (see Example 2) when is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest?

If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricellis Law gives the volume of water remaining in the tank after minutes as Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings

The quantity of charge in coulombs (C) that has passed through a point in a wire up to time (measured in seconds) is given by . Find the current when (a) s and (b) s. [See Example 3. The unit of current is an ampere ( A C#s).] At what time is the current lowest?

Newtons Law of Gravitation says that the magnitude of the force exerted by a body of mass on a body of mass is where is the gravitational constant and is the distance between the bodies. (a) Find and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 N#km when r !20,000 km. How fast does this force change when r !10,000 km?

Boyles Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: . (a) Find the rate of change of volume with respect to pressure.(b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given b

If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value , then where is a constant. (a) Find the rate of reaction at time . (b) Show that if C , then (c) What happens to the concentration as ? (d) What happens to the rate of reaction as ? (e) What do the results of parts (c) and (d) mean in practical terms

In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number of bacteria after hours and use it to estimate the rate of growth of the bacteria population after 2.5 hours

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function where is measured in hours. At time the population is 20 cells and is increasing at a rate of . Find the values of and . According to this model, what happens to the yeast population in the long run?

The table gives the population of the world in the 20th century. (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data.(c) Use your model in part (b) to find a model for the rate of population growth in the 20th century. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) Estimate the rate of growth in 1985.

The table shows how the average age of first marriage of Japanese women varied in the last half of the 20th century. (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for . (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models fo

Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference , and viscosity . (a) Find the velocity of the blood along the centerline , at radius cm, and at the wall . (b) Find the velocity gradient at , , and . (c) Where is the velocity the greatest? Where is the velocity changing most?

The frequency of vibrations of a vibrating violin string is given by where is the length of the string, is its tension, and is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3d ed. (Pacific Grove, CA: Brooks/Cole, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when and are constant), (ii) the tension (when and are constant), and (iii) the linear density (when and are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency . (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string

The cost, in dollars, of producing yards of a certain fabric is T (a) Find the marginal cost function. (b) Find and explain its meaning. What does it predict? (c) Compare with the cost of manufacturing the 201st yard of fabric

The cost function for production of a commodity is (a) Find and interpret . (b) Compare with the cost of producing the 101st item.

If is the total value of the production when there are workers in a plant, then the average productivity of the workforce at the plant is (a) Find . Why does the company want to hire more workers if ? (b) Show that if is greater than the average productivity

If denotes the reaction of the body to some stimulus of strength , the sensitivity is defined to be the rate of change of the reaction with respect to . A particular example is that when the brightness of a light source is increased, the eye reacts by decreasing the area of the pupil. The experimental formula has been used to model the dependence of on when is measured in square millimeters and is measured in appropriate units of brightness. (a) Find the sensitivity. ; (b) Illustrate part (a) by graphing both and as functions of . Comment on the values of and at low levels of brightness. Is this what you would expect?

The gas law for an ideal gas at absolute temperature (in kelvins), pressure (in atmospheres), and volume (in liters) is , where is the number of moles of the gas and is the gas constant. Suppose that, at a certain instant, atm and is increasing at a rate of 0.10 atm"min and and is decreasing at a rate of 0.15 L"min. Find the rate of change of with respect to time at that instant if mo

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation where is the birth rate of the fish, is the maximum population that the pond can sustain (called the carrying capacity), and is the percentage of the population that is harvested. (a) What value of corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if is raised to 5%?

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by , and caribou, given by , in northern Canada. The interaction has been modeled by the equations (a) What values of and correspond to stable populations? (b) How would the statement The caribou go extinct be represented mathematically? (c) Suppose that , , , and . Find all population pairs that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

Use the fact that the world population was 2560 million in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 20th century. (Assume that the growth rate is proportional to the population size.) What is the relative growth rate? Use the model to estimate the world population in 1993 and to predict the population in the year 2020.

The half-life of radium-226 is 1590 years. (a) A sample of radium-226 has a mass of 100 mg. Find a formula for the mass of the sample that remains after years. (b) Find the mass after 1000 years correct to the nearest milligram. (c) When will the mass be reduced to 30 mg?

A bottle of soda pop at room temperature ( F) is placed in a refrigerator where the temperature is F. After half an hour the soda pop has cooled to F. (a) What is the temperature of the soda pop after another half hour? (b) How long does it take for the soda pop to cool to F?

If $1000 is invested at 6% interest, compounded annually, then after 1 year the investment is worth , after 2 years its worth , and after years its worth . In general, if an amount is invested at an interest rate in this example), then after years its worth . Usually, however, interest is compounded more frequently, say, times a year. Then in each compounding period the interest rate is and there are compounding periods in years, so the value of the investment is

A population of protozoa develops with a constant relative 3. growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells. (a) Find the relative growth rate. (b) Find an expression for the number of cells after hours. (c) Find the number of cells after 8 hours. (d) Find the rate of growth after 8 hours. (e) When will the population reach 20,000 cells?

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a) Find an expression for the number of bacteria after hours. (b) Find the number of bacteria after 3 hours. (c) Find the rate of growth after 3 hours. (d) When will the population reach 10,000?

A bacteria culture grows with constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000. (a) Find the initial population. (b) Find an expression for the population after t hours (c) Find the number of cells after 5 hours. (d) Find the rate of growth after 5 hours. (e) When will the population reach 200,000?

The table gives estimates of the world population, in millions, from 1750 to 2000: (a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. (c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.

The table gives the population of the United States, in millions, for the years 19002000. (a) Use the exponential model and the census figures for 1900 and 1910 to predict the population in 2000. Compare with the actual figure and try to explain the discrepancy. (b) Use the exponential model and the census figures for 1980 and 1990 to predict the population in 2000. Compare with the actual population. Then use this model to predict the population in the years 2010 and 2020. ; (c) Graph both of the exponential functions in parts (a) and (b) together with a plot of the actual population. Are these models reasonable ones?

Experiments show that if the chemical reaction takes place at , the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: (a) Find an expression for the concentration N O after t seconds if the initial concentration is C.(b) How long will the reaction take to reduce the concentration of N O to 90% of its original value?

Bismuth-210 has a half-life of 5.0 days. (a) A sample originally has a mass of 800 mg. Find a formula for the mass remaining after days. (b) Find the mass remaining after 30 days. (c) When is the mass reduced to 1 mg? (d) Sketch the graph of the mass function

The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after years. (b) How much of the sample remains after 100 years? (c) After how long will only 1 mg remain?

A sample of tritium-3 decayed to 94.5% of its original amount after a year. (a) What is the half-life of tritium-3? (b) How long would it take the sample to decay to 20% of its original amount?

Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of C begins to decrease through radioactive decay. Therefore the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74% as much C radioactivity as does plant material on the earth today. Estimate the age of the parchment.

A curve passes through the point and has the property that the slope of the curve at every point is twice the -coordinate of . What is the equation of the curve?

A roast turkey is taken from an oven when its temperature has reached and is placed on a table in a room where the temperature is . (a) If the temperature of the turkey is after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to ?

A thermometer is taken from a room where the temperature is C to the outdoors, where the temperature is . After one minute the thermometer reads C. (a) What will the reading on the thermometer be after one more minute? (b) When will the thermometer read C?

When a cold drink is taken from a refrigerator, its temperature is C. After 25 minutes in a C room its temperature has increased to C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature be 15+C?

A freshly brewed cup of coffee has temperature C in a C room. When its temperature is C, it is cooling at a rate of C per minute. When does this occur?

The rate of change of atmospheric pressure with respect to altitude is proportional to , provided that the temperature is constant. At C the pressure is kPa at sea level and kPa at m. (a) What is the pressure at an altitude of 3000 m? (b) What is the pressure at the top of Mount McKinley, at an altitude of 6187 m?

(a) If $1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (viii) continuously.(b) Suppose $1000 is borrowed and the interest is compounded continuously. If is the amount due after years, where , graph for each of the interest rates 6%, 8%, and 10% on a common screen.

(a) If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) If is the amount of the investment at time for the case of continuous compounding, write a differential equation and an initial condition satisfied by

(a) How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b) What is the equivalent annual interest rate?

Air is being pumped into a spherical balloon so that its volume increases at a rate of . How fast is the radius of the balloon increasing when the diameter is 50 cm?

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft!s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m !min, find the rate at which the water level is rising when the water is 3 m deep.

Car A is traveling west at and car B is traveling north at . Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection?

A man walks along a straight path at a speed of 4 ft!s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?

If is the volume of a cube with edge length and the cube expands as time passes, find in terms of

a) If is the area of a circle with radius and the circle expands as time passes, find in terms of . (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of , how fast is the area of the spill increasing when the radius is 30 m?

Each side of a square is increasing at a rate of . At what rate is the area of the square increasing when the area of the square is 16 cm

The length of a rectangle is increasing at a rate of and its width is increasing at a rate of . When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

A cylindrical tank with radius 5 m is being filled with water at a rate of . How fast is the height of the water increasing?

The radius of a sphere is increasing at a rate of . How fast is the volume increasing when the diameter is 80 mm?

If and , find when

If and , find when

If and , find when

A particle moves along the curve . As it reaches the point , the -coordinate is increasing at a rate of . How fast is the -coordinate of the point changing at that instant?

a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem A plane flying horizontally at an altitude of 1 mi and a speed of 500 m!h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station

a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem If a snowball melts so that its surface area decreases at a rate of 1 cm !min, find the rate at which the diameter decreases when the diameter is 10 cm

a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft!s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?

a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km!h and ship B is sailing north at 25 km!h. How fast is the distance between the ships changing at 4:00 PM?

Two cars start moving from the same point. One travels south at 60 mi!h and the other travels west at 25 mi!h. At what rate is the distance between the cars increasing two hours later?

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m!s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

A man starts walking north at 4 ft!s from a point . Five minutes later a woman starts walking south at 5 ft!s from a point 500 ft due east of . At what rate are the people moving apart 15 min after the woman starts walking?

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft!s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment?

The altitude of a triangle is increasing at a rate of 1 cm!min while the area of the triangle is increasing at a rate of 2 cm !min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is ?

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m!s, how fast is the boat approaching the dock when it is 8 m from the dock?

At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km!h and ship B is sailing north at 25 km!h. How fast is the distance between the ships changing at 4:00 PM?

A particle is moving along the curve . As the particle passes through the point , its -coordinate increases at a rate of . How fast is the distance from the particle to the origin changing at this instant?

Water is leaking out of an inverted conical tank at a rate of 10,000 cm !min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm!min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft !min, how fast is the water level rising when the water is 6 inches deep?

A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 , how fast is the water level rising when the water is 30 cm deep?

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 , how fast is the water level rising when the depth at the deepest point is 5 ft?

Gravel is being dumped from a conveyor belt at a rate of 30 , and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

A kite 100 ft above the ground moves horizontally at a speed of 8 ft!s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad!s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is

How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

Boyles Law states that when a sample of gas is compressed at a constant temperature, the pressure and volume satisfy the equation , where is a constant. Suppose that at a certain instant the volume is 600 cm , the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa!min. At what rate is the volume decreasing at this instant?

When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation , where is a constant. Suppose that at a certain instant the volume is 400 cm and the pressure is 80 kPa and is decreasing at a rate of 10 kPa!min. At what rate is the volume increasing at this instant?

If two resistors with resistances and are connected in parallel, as in the figure, then the total resistance , measured in ohms ( ), is given by If and are increasing at rates of and , respectively, how fast is changing when and

Brain weight as a function of body weight in fish has been modeled by the power function , where B and W are measured in grams. A model for body weight as a function of body length (measured in centimeters) is . If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species brain growing when the average length was 18 cm?

Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of . How fast is the length of the third side increasing when the angle between the sides of fixed length is 60 ?

Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley (see the figure). The point is on the floor 12 ft directly beneath and between the carts. Cart A is being pulled away from at a speed of 2 ft!s. How fast is cart B moving toward at the instant when cart A is 5 ft from ?

A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Lets assume the rocket rises vertically and its speed is 600 ft!s when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment?(b) If the television camera is always kept aimed at the rocket, how fast is the cameras angle of elevation changing at that same moment?

A lighthouse is located on a small island 3 km away from the nearest point on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from ?

A plane flies horizontally at an altitude of and passes directly over a tracking telescope on the ground. When the angle of elevation is , this angle is decreasing at a rate of . How fast is the plane traveling at that time?

A Ferris wheel with a radius of is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?

A plane flying with a constant speed of 300 km!h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30 . At what rate is the distance from the plane to the radar station increasing a minute later?

Two people start from the same point. One walks east at 3 mi!h and the other walks northeast at 2 mi!h. How fast is the distance between the people changing after 15 minutes?

A runner sprints around a circular track of radius 100 m at a constant speed of 7 m!s. The runners friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one oclock?

Find the linearization of the function at and use it to approximate the numbers and . Are these approximations overestimates or underestimates?

For what values of is the linear approximation

Compare the values of and if and changes (a) from 2 to 2.05 and (b) from 2 to 2.01

The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?

Find the linearization of the function at a f !x"!x a !%1

Find the linearization of the function at a f !x"!ln x a !1

Find the linearization of the function at a f !x"!cos x a !&$2

Find the linearization of the function at a f !x"!x a !16

Find the linear approximation of the function at and use it to approximate the numbers and . Illustrate by graphing and the tangent lin

Find the linear approximation of the function at and use it to approximate the numbers and . Illustrate by graphing and the tangent line

Verify the given linear approximation at . Then determine the values of for which the linear approximation is accurate to within 0.1

Verify the given linear approximation at . Then determine the values of for which the linear approximation is accurate to within 0.1 tan x #x

Verify the given linear approximation at . Then determine the values of for which the linear approximation is accurate to within 0.1

Verify the given linear approximation at . Then determine the values of for which the linear approximation is accurate to within 0.1

Find the differential of each function.

Find the differential of each function.

Find the differential of each function.

Find the differential of each function.

(a) Find the differential and (b) evaluate for the given values of and y !e x !0 dx !0.1

(a) Find the differential and (b) evaluate for the given values of and y !1$!x $1"x !1 dx !%0.01

(a) Find the differential and (b) evaluate for the given values of and y !tan x x !&$4 dx !%0.1

(a) Find the differential and (b) evaluate for the given values of and y !cos x x !&$3 dx !0.0

Compute and for the given values of and . Then sketch a diagram like Figure 5 showing the line segments with lengths , , and y !2x %x x !2 #x !%0.4

Compute and for the given values of and . Then sketch a diagram like Figure 5 showing the line segments with lengths , , and y !sx x !1 #x !1

Compute and for the given values of and . Then sketch a diagram like Figure 5 showing the line segments with lengths , , and y !2$x x !4 #x !1

Compute and for the given values of and . Then sketch a diagram like Figure 5 showing the line segments with lengths , , and y !e x !0 #x !0.5

Use a linear approximation (or differentials) to estimate the given number !2.001" 5

Use a linear approximation (or differentials) to estimate the given number e%0.015

Use a linear approximation (or differentials) to estimate the given number

Use a linear approximation (or differentials) to estimate the given number 1$1002

Use a linear approximation (or differentials) to estimate the given number tan 44"

Use a linear approximation (or differentials) to estimate the given number s99.8

Explain, in terms of linear approximations or differentials, why the approximation is reasonable

Explain, in terms of linear approximations or differentials, why the approximation is reasonable

Explain, in terms of linear approximations or differentials, why the approximation is reasonable n 1.05 #0.05

Let and (a) Find the linearizations of , , and at . What do you notice? How do you explain what happened? ; (b) Graph , , and and their linear approximations. For which function is the linear approximation best? For which is it worst? Explain.

The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube

The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error?

The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error?

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.

a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height , inner radius , and thickness . (b) What is the error involved in using the formula from part (a)?

One side of a right triangle is known to be 20 cm long and the opposite angle is measured as , with a possible error of . (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error?

If a current passes through a resistor with resistance , ( Ohms Law states that the voltage drop is . If is constant and is measured with a certain error, use differentials to show that the relative error in calculating is approximately the same (in magnitude) as the relative error in .

When blood flows along a blood vessel, the flux (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius of the blood vessel: (This is known as Poiseuilles Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in is about four times the relative change in . How will a 5% increase in the radius affect the flow of blood?

Establish the following rules for working with differentials (where denotes a constant and and are functions of ). (a) (b) (c) (d) (e) (f)

On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula for the period of a pendulum of length L, the author obtains the equation aT !%t sin (for the tangential acceleration of the bob of the pendulum. He then says, for small angles, the value of in radians is very nearly the value of ; they differ by less than 2% out to about 20. (a) Verify the linear approximation at 0 for the sine function: ; (b) Use a graphing device to determine the values of for which and differ by less than 2%. Then verify Hechts statement by converting from radians to degrees

Suppose that the only information we have about a function is that and the graph of its derivative is as shown. (a) Use a linear approximation to estimate and . (b) Are your estimates in part (a) too large or too small? Explain

Suppose that we dont have a formula for but we know that and for all . (a) Use a linear approximation to estimate and . (b) Are your estimates in part (a) too large or too small? Explain.

Find the numerical value of each expression.

Find the numerical value of each expression.

Find the numerical value of each expression.

Find the numerical value of each expression.

Find the numerical value of each expression.

Find the numerical value of each expression.

Prove the identity. sinhxsinh x si

Prove the identity. coshxcosh x s

Prove the identity. cosh x sinh x ex 9

Prove the identity.cosh x sinh x ex c

Prove the identity.sinhx ysinh x cosh y cosh x sinh y co

Prove the identity.coshx ycosh x cosh y sinh x sinh y si

Prove the identity.

Prove the identity.tanhx ytanh x tanh y 1 tanh x tanh y cot

Prove the identity.sinh 2x 2 sinh x cosh x

Prove the identity.cosh 2x cosh2 x sinh2 x

Prove the identity.tanhln xx 2 1 x 2 1 17

Prove the identity.tanh x 1 tanh x e 2x t

Prove the identity.

. If , find the values of the other hyperbolic functions at x

If and , find the values of the other hyperbolic functions at x

a) Use the graphs of , , and in Figures 13 to draw the graphs of , , and .(b) Check the graphs that you sketched in part (a) by using a x y graphing device to produce them.

Use the definitions of the hyperbolic functions to find each of the following limits.

Prove the formulas given in Table 1 for the derivatives of the functions (a) , (b) , (c) , (d) , and (e) .

Give an alternative solution to Example 3 by letting and then using Exercise 9 and Example 1(a) with replaced by

Prove Equation 4

Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with replaced by y

For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. (a) (b) (c

Prove the formulas given in Table 6 for the derivatives of the following functions. (a) (b) (c) (d) (e

7 Find the derivative. Simplify where possible

7 Find the derivative. Simplify where possible f!x"!x sinh x %cosh x

7 Find the derivative. Simplify where possible

7 Find the derivative. Simplify where possible

7 Find the derivative. Simplify where possible y !x coth!1 $x 2

7 Find the derivative. Simplify where possible y !e cosh 3x

7 Find the derivative. Simplify where possible f!t"!csch t!1 %ln csch t

7 Find the derivative. Simplify where possible f!t"!sech2 !e

7 Find the derivative. Simplify where possible y !sinh!cosh x"

7 Find the derivative. Simplify where possible y !arctan!tanh x"

7 Find the derivative. Simplify where possible y !'1 $tanh x 1 %tanh x

7 Find the derivative. Simplify where possible G!x"!1 %cosh x 1 $cosh x

7 Find the derivative. Simplify where possible y !x sx 2 sinh%1 !2x

7 Find the derivative. Simplify where possible y !tanh%1 y !x sx

7 Find the derivative. Simplify where possible y !x tanh%1 x $ln s1 %x 2

7 Find the derivative. Simplify where possible y !x sinh%1 !x$3"%s9 $x 2

7 Find the derivative. Simplify where possible y !sech%1 s1 %x 2 , x .0

7 Find the derivative. Simplify where possible y !coth%1 sx 2 $1

The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y !211.49 %20.96 cosh 0.03291765x for the central curve of the arch, where and are measured in meters and . ; (a) Graph the central curve. (b) What is the height of the arch at its center? (c) At what points is the height 100 m? (d) What is the slope of the arch at the points in part (c)?

If a water wave with length moves with velocity in a body of water with depth , then where is the acceleration due to gravity. (See Figure 5.) Explain why the approximation is appropriate in deep water

A flexible cable always hangs in the shape of a catenary , where and are constants and (see Figure 4 and Exercise 52). Graph several members of the family of functions . How does the graph change as varies?

A telephone line hangs between two poles 14 m apart in the shape of the catenary , where and are measured in meters. (a) Find the slope of this curve where it meets the right pole. (b) Find the angle between the line and the pole.

Using principles from physics it can be shown that when a cable is hung between two poles, it takes the shape of a curve that satisfies the differential equation where is the linear density of the cable, is the acceleration due to gravity, and is the tension in the cable at its lowest point, and the coordinate system is chosen appropriately. Verify that the function is a solution of this differential equation

(a) Show that any function of the form y !cosh x satisfies the differential equation . (b) Find such that , , and

Evaluate lim . x l + sinh x ex

At what point of the curve does the tangent have slope 1?

If , show that

Show that if and , then there exist numbers and such that equals either or . In other words, almost every function of the form is a shifted and stretched hyperbolic sine or cosine function.

State each differentiation rule both in symbols and in words. e (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule

State the derivative of each function. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) y !tanh%1 y !cosh x %1 x y !sinh%1 y !cosh x y !tanh x x y !tan y !sinh x %1 y !cos x %1 x y !sin%1 y !sec x y !cot x x y !cos x y !tan x y !csc x y !ln x y !loga x y !sin x y !ax y !ex y !x n

(a) How is the number defined? (b) Express as a limit. (c) Why is the natural exponential function used more often in calculus than the other exponential functions ? (d) Why is the natural logarithmic function used more often in calculus than the other logarithmic functions ?

a) Explain how implicit differentiation works. (b) Explain how logarithmic differentiation works

a) Write an expression for the linearization of at . (b) If , write an expression for the differential . (c) If , draw a picture showing the geometric meanings of #y and dy.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If and are differentiable, then d dx (f !x"$t!x"*!f'!x"$t'!x

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If and are differentiable, then d dx (f!x"t!x"*!f'!x"t'!x"

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If and are differentiable, then d dx (f!t!x""*!f'!t!x""t'!x"

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If and are differentiable, then d dx sf!x"!f'!x" 2sf !x"

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If and are differentiable, then d dx f(sx ) !f '!x" 2sx

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.An equation of the tangent line to the parabola at !%2, 4"is y %4 !2x!x $2"

Calculate y'.

Calculate y'.y !cos!tan x

Calculate y'.y !sx " 1 s 3 x 4

Calculate y'.y !3x $2 s2x "1

Calculate y'.y !2xsx 2 "1

Calculate y'.y !ex 1 "x 2

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Calculate y'.y !ssin sx

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Calculate y'.y !!cos x" x

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Calculate y'.y !!x 2 "1" 4 !2x "1" 3 !3x $1" 5

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Calculate y'.y !10tan #&

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Calculate y'.y !arctan(arcsin sx

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Calculate y'.y !!x "%" 4 x 4 "%4

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If f!t"!s4t "1, find f !!2".

If t!&"!&sin &, find t !!#%6"

Find y'' if x6 + y6 =1

Find...if f!x"!1%!2 $x"

Use mathematical induction (page 77) to show that if , then im t l 0 t 3 tan3 !2t"

Find an equation of the tangent to the curve at the given point

Find an equation of the tangent to the curve at the given point

Find an equation of the tangent to the curve at the given point

Find equations of the tangent line and normal line to the curve at the given point.

Find equations of the tangent line and normal line to the curve at the given point.

If , find . Graph and on the same screen and comment.

(a) If , find . (b) Find equations of the tangent lines to the curve at the points and . ; (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. ; (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of and

(a) If , , find and . ; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and

At what points on the curve , , is the tangent line horizontal?

Find the points on the ellipse where the tangent line has slope 1

If , show that f'!x" f !x"!1 x $a " 1 x $b " 1 x $c

(a) By differentiating the double-angle formula obtain the double-angle formula for the sine function. (b) By differentiating the addition formula obtain the addition formula for the cosine function. sin!x "a"!sin x cos a "cos x sin a

Suppose that and , where C'!t" , , , , and . Find (a) and

If and are the functions whose graphs are shown, let , , and . Find (a) , (b) , and

Find in terms of f !x"!x "2 t!x

Find in terms of f !x"!t!x 2

Find in terms of

Find in terms of f !x"!t!t!x"

Find in terms of

Find in terms of f !x"!et!x"

Find in terms of

Find in terms of f !x"!t!ln x"

Find in terms of and h!x"!f !x"t!x" f !x""t!x"

Find in terms of and h!x"!(f !x" t!x"

Find in terms of and h!x"!f !t!sin 4x"

a) Graph the function in the viewing rectangle by . (b) On which interval is the average rate of change larger: or ? (c) At which value of is the instantaneous rate of change larger: or ? (d) Check your visual estimates in part (c) by computing and comparing the numerical values of and

At what point on the curve is the tangent horizontal?

(a) Find an equation of the tangent to the curve that is parallel to the line . (b) Find an equation of the tangent to the curve that passes through the origin.

Find a parabola that passes through the point and whose tangent lines at and have slopes 6 and , respectively

The function , where a, b, and K are positive constants and , is used to model the concentration at time t of a drug injected into the bloodstream. (a) Show that limtl-C!t"!0.(b) Find , the rate at which the drug is cleared from circulation. (c) When is this rate equal to 0?

An equation of motion of the form represents damped oscillation of an object. Find the velocity and acceleration of the object

A particle moves along a horizontal line so that its coordinate at time is , , where and are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction.

A particle moves on a vertical line so that its coordinate at time is , . (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval . ; (d) Graph the position, velocity, and acceleration functions for . (e) When is the particle speeding up? When is it slowing down?

The volume of a right circular cone is , where is the radius of the base and is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant

The mass of part of a wire is kilograms, where is measured in meters from one end of the wire. Find the linear density of the wire when m.

The cost, in dollars, of producing units of a certain commodity is (a) Find the marginal cost function. (b) Find and explain its meaning. (c) Compare with the cost of producing the 101st item.

A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000?

Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?

Let be the concentration of a drug in the bloodstream. 2 As the body eliminates the drug, decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus , where is a positive number called the elimination constant of the drug. (a) If is the concentration at time , find the concentration at time . (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug?

A cup of hot chocolate has temperature in a room kept at . After half an hour the hot chocolate cools to . (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to ?

The volume of a cube is increasing at a rate of 10 . How fast is the surface area increasing when the length of an edge is 30 cm?

A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of , how fast is the water level rising when the water is 5 cm deep?

A balloon is rising at a constant speed of . A boy is cycling along a straight road at a speed of . When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?

A waterskier skis over the ramp shown in the figure at a speed of . How fast is she rising as she leaves the ramp?

The angle of elevation of the sun is decreasing at a rate of . How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is #%6

a) Find the linear approximation to near 3. (b) Illustrate part (a) by graphing and the linear approximation. (c) For what values of is the linear approximation accurate to within 0.1?

a) Find the linearization of at . State the corresponding linear approximation and use it to give an approximate value for . ; (b) Determine the values of for which the linear approximation given in part (a) is accurate to within 0.1

Evaluate if , , and

A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window

Express the limit as a derivative and evaluate im x l1 x 17 $1 x $1

Express the limit as a derivative and evaluate im hl0 s 4 16 "h $2 h

Express the limit as a derivative and evaluate lim &l #%3 cos &$0.5 &$#%3

Evaluate lim xl0 s1 "tan x $s1 "sin x x 3

Suppose is a differentiable function such that and . Show that

Find if it is known that d dx &f !2x"'!x

Show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant.

How many lines are tangent to both of the parabolas and ? Find the coordinates of the points at which these tangents touch the parabolas.

For what values of does the equation have exactly one solution?

Find points and on the parabola so that the triangle formed by the -axis and the tangent lines at and is an equilateral triangle.

Find the point where the curves and are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent

Show that the tangent lines to the parabola at any two points with -coordinates and must intersect at a point whose -coordinate is halfway between and .

Show that d dx )sin2 x 1 "cot x " cos2 x 1 "tan x *!$cos 2x

Show that sin$1 !tanh x"!tan$1 !sinh x"

A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m east and 50 m north of the origin. At what point on the highway will the cars headlights illuminate the statue?

Prove that dn dx n !sin4 x "cos4 x"!4n$1 cos!4x "n#%2

Let be a point on the parabola with focus . Let be the angle between the parabola and the line segment , and let be the angle between the horizontal line and the parabola as in the figure. Prove that . (Thus, by a principle of geometrical optics, light from a source placed at will be reflected along a line parallel to the -axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.)

Suppose that we replace the parabolic mirror of Problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, is a semicircle with center . A ray of light coming in toward the mirror parallel to the axis along the line will be reflected to the point on the axis so that (the angle of incidence is equal to the angle of reflection). What happens to the point as is taken closer and closer to the axis

If and are differentiable functions with and , show that

Evaluate lim x l 0 sin!a "2x"$2 sin!a "x""sin a x 2

(a) The cubic function has three distinct zeros: 0, 2, and 6. Graph and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function has three distinct zeros: , , and . Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros intersects the graph of at the third zero.

For what value of does the equation have exactly one solution?

For which positive numbers is it true that for all ?

If y !x sa2 $1 $2 sa2 $1 arctan sin x a "sa2 $1 "cos x show that !1 a "cos x

Given an ellipse , where , find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals

Find the two points on the curve that have a common tangent line

Suppose that three points on the parabola have the property that their normal lines intersect at a common point. Show that the sum of their -coordinates is 0

A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius are drawn using all lattice points as centers. Find the smallest value of such that any line with slope intersects some of these circles

A cone of radius centimeters and height centimeters is lowered point first at a rate of 1 cm%s into a tall cylinder of radius centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged?

A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is , where is the radius and is the slant height.) If we pour the liquid into the container at a rate of , then the height of the liquid decreases at a rate of 0.3 cm%min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container?

Find the th derivative of the function f !x"!x n n %!1 $x"

The figure shows a circle with radius 1 inscribed in the parabola . Find the center of the circle.

If is differentiable at , where , evaluate the following limit in terms of im xla f!x"$f !a" sx $sa

The figure shows a rotating wheel with radius 40 cm and a connecting rod with length 1.2 m. The pin slides back and forth along the -axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod, , in radians per second, when . (b) Express the distance in terms of . (c) Find an expression for the velocity of the pin in term

Tangent lines and are drawn at two points and on the parabola and they intersect at a point . Another tangent line is drawn at a point between and ; it intersects at and at . Show that PQ1 $ $PP1 $"$PQ2 $ $PP2 $!1

Show that dn dx n !e ax sin bx"!r n e ax sin!bx "n& where and are positive numbers

Evaluate lim . xl# esin x $1 x $#

Let and be the tangent and normal lines to the ellipse at any point on the ellipse in the first quadrant. Let and be the - and -intercepts of and and be the intercepts of . As moves along the ellipse in the first quadrant (but not on the axes), what values can , , , and take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is.

Evaluate lim x l 0 sin!3 "x" 2 $sin 9 x

(a) Use the identity for (see Equation 14b in Appendix D) to show that if two lines and intersect at an angle , then where and are the slopes of and , respectively. (b) The angle between the curves and at a point of intersection is defined to be the angle between the tangent lines to and at (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) and (ii) and