Solved: Produce graphs of that reveal all the important aspects of the curve. Estimate

Explain the difference between an absolute minimum and a local minimum.

Suppose is a continuous function defined on a closed interval . (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for ? (b) What steps would you take to find those maximum and minimum values?

For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

Use the graph to state the absolute and local maximum and minimum values of the function.

Use the graph to state the absolute and local maximum and minimum values of the function.

Sketch the graph of a function that is continuous on [1, 5] and has the given properties Absolute minimum at 2, absolute maximum at 3, local minimum at 4

Sketch the graph of a function that is continuous on [1, 5] and has the given properties Absolute minimum at 1, absolute maximum at 5, local maximum at 2, local minimum at 4

Sketch the graph of a function that is continuous on [1, 5] and has the given properties Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4

Sketch the graph of a function that is continuous on [1, 5] and has the given properties . f has no local maximum or minimum, but 2 and 4 are critical numbers

(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.

(a) Sketch the graph of a function on [$1, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [$1, 2] that has a local maximum but no absolute maximum.

(a) Sketch the graph of a function on [$1, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [$1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.

(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! 8 $ 3x x " 1

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! 3 $ 2x x % 5

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) !x" ! x 0 ' x ' 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) !x" ! x 0 ' x % 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) !x" ! x 0 % x ' 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! x 0 % x % 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! x 0 % x % 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! 1 ! !x ! 1" $2 % x ' 5

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f!x" ! ln x 0 ' x % 2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f!t" ! cos t $3#%2 % t % 3#%2

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f!x" ! 1 $ sx

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! e x

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f !x" ! ( 1 $ x 2x $ 4 if 0 % x ' 2 if 2 % x % 3

Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) f!x" ! ( 4 $ x 2 2x $ 1 if $2 % x ' 0 if 0 % x % 2

Find the critical numbers of the function. f !x" ! 5x 2 $ x 2 ! 4x

Find the critical numbers of the function. f !x" ! x 3 ! x f !x" ! 5x 2 $ x

Find the critical numbers of the function. f!x" ! x 2 ! x 3 ! 3x 2 $ 24x

Find the critical numbers of the function. f !x" ! x 3 ! x f!x" ! x 2 ! x

Find the critical numbers of the function. s!t" ! 3t 4 ! 4t 3 $ 6t 2

Find the critical numbers of the function. t!t" ! & 3t $ 4 &

Find the critical numbers of the function. t!y" ! y $ 1 y 2 $ y ! 1

Find the critical numbers of the function. h!p" ! p $ 1 p2 ! 4

Find the critical numbers of the function. h!t" ! t 3%4 $ 2t 1%4

Find the critical numbers of the function. t!x" ! s1 $ x 2

Find the critical numbers of the function. F!x" ! x 4%5 !x $ 4" 2

Find the critical numbers of the function. t!x" ! x 1%3 $ x$2%3

Find the critical numbers of the function. f!*" ! 2 cos * ! sin t!*" ! 4* $ tan * 2 41. *

Find the critical numbers of the function. !*" ! 4* $ tan *

Find the critical numbers of the function. f!x" ! x ln x 2 e$3x

Find the critical numbers of the function. f!x" ! x $2 f!x" ! x ln x

A formula for the derivative of a function is given. How many critical numbers does have? f&!x" ! 5e 2 $ 1 $0.1 & x & sinx $ 1

A formula for the derivative of a function is given. How many critical numbers does have? f&!x" ! 5e 2 $ 1 $0.1 & x & sinx $ 1

Find the absolute maximum and absolute minimum values of on the given interval. f !x" ! 3x #0, 3$ 2 $ 12x ! 5

Find the absolute maximum and absolute minimum values of on the given interval. f !x" ! x #0, 3$ 3 $ 3x ! 1

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

Find the absolute maximum and absolute minimum values of on the given interval.

If a and b are positive numbers, find the maximum value of f!x" ! x 0 % x % 1 a !1 $ x"

Use a graph to estimate the critical numbers of f !x" ! & x 3 $ 3x 2 ! 2 & correct to one decimal place.

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values

Between and , the volume (in cubic centimeters) of 1 kg of water at a temperature is given approximately by the formula V ! 999.87 $ 0.06426T ! 0.0085043T 2 $ 0.0000679T 3 Find the temperature at which water has its maximum density.

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 F ! +W + sin * ! cos * where is a positive constant called the coefficient of friction and where . Show that is minimized when

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function where is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 19932003.

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval . Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.

When a foreign object lodged in the trachea (windpipe) forces v a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity of the airstream is related to the radius of the trachea by the equation where is a constant and is the normal radius of the trachea. The restriction on is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than is prevented (otherwise the person would suffocate). (a) Determine the value of in the interval at which has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of on the interval? (c) Sketch the graph of on the interval

Show that 5 is a critical number of the function t!x" ! 2 ! !x $ 5" 3 but does not have a local extreme value at 5.

Prove that the function f!x" ! x 101 ! x 51 ! x ! 1 has neither a local maximum nor a local minimum

If has a minimum value at , show that the function has a maximum value at .

Prove Fermats Theorem for the case in which has a local minimum at

A cubic function is a polynomial of degree 3; that is, it has the form , where . (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?

Verify that the function satisfies the three hypotheses of f c Rolles Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolles Theorem. f"x# ! 5 # 12x % 3x %1, 3&

Verify that the function satisfies the three hypotheses of f c Rolles Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolles Theorem. f"x# ! x %0, 3& 3 # x 2 # 6x % 2

Verify that the function satisfies the three hypotheses of f c Rolles Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolles Theorem. f"x# ! sx # %0, 9& 1 3 x

Verify that the function satisfies the three hypotheses of f c Rolles Theorem on the given interval. Then find all numbers that satisfy the conclusion of Rolles Theorem. f"x# ! cos 2x, %&'8, 7&'8&

Let f "x# ! 1 . Show that f"#1# ! f"1# but there is no number in such that . Why does this not contradict Rolles Theorem?

Let f"x# ! tan x . Show that f"0# ! f"& but there is no number in "0, &# such that f '"c# ! 0. Why does this not contradict Rolles Theorem?

Use the graph of to estimate the values of that satisfy the conclusion of the Mean Value Theorem for the interval .

Use the graph of given in Exercise 7 to estimate the values of that satisfy the conclusion of the Mean Value Theorem for the interval %

(a) Graph the function in the viewing rect- f ! angle by . (b) Graph the secant line that passes through the points and on the same screen with . (c) Find the number that satisfies the conclusion of the Mean Value Theorem for this function and the interval . Then graph the tangent line at the point and notice that it is parallel to the secant line

(a) In the viewing rectangle by , graph the function and its secant line through the points and . Use the graph to estimate the -coordinates of the points where the tangent line is parallel to the secant line. (b) Find the exact values of the numbers that satisfy the conclusion of the Mean Value Theorem for the interval and compare with your answers to part (a).

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem. f "x# ! 3x %#1, 1& 11. 2 % 2x % 5

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem. f "x# ! x %0, 2& 3 % x # 1

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem. f "x# ! e#2x , %0, 3&

Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem. f"x# ! %1, 4& x x % 2

Let f"x# ! "x # 3# c #2 . Show that there is no value of c in "1, 4# such that f "4# # f "1# ! f '"c#"4 # 1# . Why does this not contradict the Mean Value Theorem?

Let f"x# ! 2 # c )2x # 1 . Show that there is no value of such that . Why does this not contradict the Mean Value Theorem?

Show that the equation has exactly one real root.

Show that the equation has exactly one real root.

Show that the equation has at most one root in the interval %#2, 2&

Show that the equation has at most two real roots.

(a) Show that a polynomial of degree 3 has at most three real roots. (b) Show that a polynomial of degree has at most real roots

(a) Suppose that is differentiable on and has two roots. Show that f ' has at least one root. (b) Suppose is twice differentiable on and has three roots. Show that has at least one real root. (c) Can you generalize parts (a) and (b)?

If f "1# ! 10 and f'"x# ( 2 for 1 , x , 4 , how small can f "4# possibly be?

Suppose that 3 , f'"x# , 5 for all values of x . Show that 18 , f"8# # f"2# , 30

Does there exist a function f such that f "0# ! #1 f "2# ! 4 , and f '"x# , 2

Suppose that and are continuous on and differentiable on . Suppose also that and for . Prove that . [Hint: Apply the Mean Value Theorem to the function .]

Show that s1 % x * 1 % x ) 0

Suppose is an odd function and is differentiable everywhere. Prove that for every positive number , there exists a number in such that f '"c# ! f"b#'b

Use the Mean Value Theorem to prove the inequality ) sin a # sin b ) , ) a # b for all a and b

If (c a constant) for all , use Corollary 7 to show that for some constant .

Let f "x# ! 1'x and t"x# ! 1 x 1 % 1 x if if x ) 0 x * 0 Show that for all in their domains. Can we conclude from Corollary 7 that is constant?

Use the method of Example 6 to prove the identity 2 sin x ( 0 #1 x ! cos#1 "1 # 2x 2 #

Prove the identity arcsin x # 1 x % 1 ! 2 arctan sx # & 2

At 2:00 PM a cars speedometer reads 30 mi'h. At 2:10 PM it reads 50 mi'h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi'h .

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider , where and are the position functions of the two runners.]

A number a is called a fixed point of a function if . Prove that if for all real numbers x, then f has at most one fixed point.

Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.

Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.

Suppose you are given a formula for a function . (a) How do you determine where is increasing or decreasing? (b) How do you determine where the graph of is concave upward or concave downward? (c) How do you locate inflection points?

(a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?

The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?

The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?

The graph of the second derivative of a function is shown. State the -coordinates of the inflection points of . Give reasons for your answers.

The graph of the first derivative of a function is shown. (a) On what intervals is increasing? Explain. (b) At what values of does have a local maximum or minimum? Explain. c) On what intervals is concave upward or concave downward? Explain. (d) What are the -coordinates of the inflection points of ? Why?

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! 2x 3 ' 3x 2 % 36x

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! 4x 3 ' 3x 2 % 6x ' 1

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! x4 % 2x 11. 2 ' 3

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! x 2 x 2 ' 3

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! sin x ' cos x, 0 * x * 2+

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f !x" ! cos 0 * x * 2+ 2 x % 2 sin x

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! e ln x 2x ' e%x

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! x 2 f!x" ! e ln x

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! !ln x"#sx

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points. f!x" ! sx e%x

Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer? f !x" ! x 5 % 5x ' 3

Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer? f!x" ! x x 2 ' 4

Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer? f!x" ! x ' s1 % x

(a) Find the critical numbers of . (b) What does the Second Derivative Test tell you about the behavior of at these critical numbers? (c) What does the First Derivative Test tell you?

Suppose is continuous on . (a) If and , what can you say about ? (b) If and , what can you say about ?

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions.

Suppose f!3" ! 2, f"!3" ! f "!x" $ 0 f !!x" # 0 1 , and and for all . (a) Sketch a possible graph for . (b) How many solutions does the equation have? Why? (c) Is it possible that ? Why?

The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum? (c) On what intervals is concave upward or downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f

The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum? (c) On what intervals is concave upward or downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph

Suppose the derivative of a function is f "!x" ! !x ' 1" f 2 !x % 3" 5 !x % 6" 4 . On what interval is f increasing?

Use the methods of this section to sketch the curve y ! x a 3 % 3a2 x ' 2a3 , where is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

(a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value. f !x" ! x ' 1 sx 2 ' 1

(a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value. f!x" ! x 2 e%x

(a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates. . f !x" ! cos x ' , 0 * x * 2+ 1 2 cos 2x

(a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates. f!x" ! x 3 !x % 2" 4

Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f f!x" ! x 4 ' x 3 ' 1 sx 2 ' x ' 1

Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f f!x" ! x 2 tan%1 x 1 ' x 3

A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.

Let be the temperature at time where you live and suppose that at time you feel uncomfortably hot. How do you feel about the given data in each case?

Let be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, or ? Is the graph of K concave upward or concave downward? Why?

Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, , and is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.

The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, lets scale the function so as to remove the factor and lets analyze the special case where . So we study the function (a) Find the asymptote, maximum value, and inflection points of . (b) What role does play in the shape of the curve? ; (c) Illustrate by graphing four members of this family on the same screen.

Find a cubic function that has a local maximum value of at and a local minimum value of 0 at 1.

For what values of the numbers and does the function f !x" ! axe bx have the maximum value ?

Show that the curve y ! !1 ' x"#!1 ' x 2 " has three points of inflection and they all lie on one straight line.

how that the curves and touch the curve at its inflection points.

Suppose is differentiable on an interval and for all numbers in except for a single number . Prove that is increasing on the entire interval .

Assume that all of the functions are twice differentiable and the second derivatives are never

Assume that all of the functions are twice differentiable and the second derivatives are never

Assume that all of the functions are twice differentiable and the second derivatives are never

Show that for . [Hint: Show that is increasing on .]

(a) Show that for . (b) Deduce that for . (c) Use mathematical induction to prove that for and any positive integer ,

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three -intercepts , and , show that the -coordinate of the inflection point is !x1 ' x2 ' x3 "#3

For what values of does the polynomial have two inflection points? One inflection point? None? Illustrate by graphing for several values of c. How does the graph change as c decreases?

Prove that if is a point of inflection of the graph of and exists in an open interval that contains , then . [Hint: Apply the First Derivative Test and Fermats Theorem to the function

Show that if , then , but is not an inflection point of the graph of

Show that the function has an inflection point at but does not exist.

Suppose that is continuous and , but . Does have a local maximum or minimum at ? Does have a point of inflection

The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions whose values at 0 are all 0 and, for (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that has neither a local maximum nor a local minimum at 0, has a local minimum, and has a local maximum.

Given that im xla f!x" ! 0 lim xla t!x" ! 0 lim xla h!x" ! 1 lim xla p!x" ! ! lim xla q!x" ! ! which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.

Given that im xla f!x" ! 0 lim xla t!x" ! 0 lim xla h!x" ! 1 lim xla p!x" ! ! lim xla q!x" ! ! which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.

Given that im xla f!x" ! 0 lim xla t!x" ! 0 lim xla h!x" ! 1 lim xla p!x" ! ! lim xla q!x" ! ! which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.

Given that im xla f!x" ! 0 lim xla t!x" ! 0 lim xla h!x" ! 1 lim xla p!x" ! ! lim xla q!x" ! ! which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Find the limit. Use lHospitals Rule where appropriate. If there is a more elementary method, consider using it. If lHospitals Rule doesnt apply, explain why.

Use a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. lim xl! &1 $ 2 x '

Use a graph to estimate the value of the limit. Then use lHospitals Rule to find the exact value. lim xl0 5x " 4x 3x " 2x

Illustrate lHospitals Rule by graphing both and near to see that these ratios have the same limit as . Also calculate the exact value of the limit.

Illustrate lHospitals Rule by graphing both and near to see that these ratios have the same limit as . Also calculate the exact value of the limit.

Prove that lim xl! ex x n ! ! for any positive integer . This shows that the exponential function approaches infinity faster than any power of x

Prove that lim xl! ln x x p ! 0 for any number . This shows that the logarithmic function approaches more slowly than any power of x

What happens if you try to use lHospitals Rule to evaluate lim xl! x sx 2 $ 1 Evaluate the limit using another method

If an object with mass is dropped from rest, one model for its speed after seconds, taking air resistance into account, is v ! mt c !1 " e "ct#m " where is the acceleration due to gravity and is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; is the proportionality constant.) (a) Calculate . What is the meaning of this limit? (b) For fixed , use lHospitals Rule to calculate . What can you conclude about the velocity of a falling object in a vacuum?

If an initial amount of money is invested at an interest rate compounded times a year, the value of the investment after years is A ! A0&1 $ r n ' nt If we let , we refer to the continuous compounding of interest. Use lHospitals Rule to show that if interest is compounded continuously, then the amount after years is A ! A0ert

If a metal ball with mass is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the ball travels in time is s!t" ! m c ln cosh*tc mt where is a positive constant. Find lim cl 0$ s!t"

If an electrostatic field acts on a liquid or a gaseous polar dielectric, the net dipole moment per unit volume is P!E" ! e E $ e"E e E " e"E " 1 E Show that m E l 0$ P!E" ! 0

A metal cable has radius and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is . The velocity of an electrical impulse in the cable is v ! "c& r R' 2 ln& r R' where is a positive constant. Find the following limits and interpret your answers.

The first appearance in print of lHospitals Rule was in the book Analyse des Infiniment Petits published by the Marquis de lHospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function y ! s2a3 x " x 4 " as 3 aax a " s 4 ax 3 as approaches , where . (At that time it was common to write instead of .) Solve this problem.

The figure shows a sector of a circle with central angle . Let A!(" be the area of the segment between the chord PR and the arc . Let be the area of the triangle . Find im( l 0$ *!("#+!("

If f #is continuous, f!2" ! 0 and f#!2" ! 7 evaluate im xl0 f !2 $ 3x" $ f!2 $ 5x" x

For what values of and is the following equation true? im xl0 & sin 2x x3 $ a $ b x2 ' ! 0

If f is continuous, use lHospitals Rule to show that lim hl0 f!x $ h" " f !x " h" 2h ! f #!x" Explain the meaning of this equation with the aid of a diagram.

If f is continuous, show that lim h l 0 f!x $ h" " 2f !x" $ f !x " h" h2 ! f )!x"

Let f !x" ! ( e"1#x 2 0 if x " 0 if x ! 0 (a) Use the definition of derivative to compute . (b) Show that has derivatives of all orders that are defined on . [Hint: First show by induction that there is a polynomial and a nonnegative integer such that f x " 0 !n" !x" ! pn!x"f !x"#xk n

Let f!x" ! () x ) x 1 if x " 0 if x ! 0 (a) Show that is continuous at . (b) Investigate graphically whether is differentiable at by zooming in several times toward the point on the graph of . (c) Show that is not differentiable at . How can you reconcile this fact with the appearance of the graphs in part (b)?

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

Use the guidelines of this section to sketch the curve.

In the theory of relativity, the mass of a particle is m m0 s1 v2 c2 y where is the rest mass of the particle, is the mass when the particle moves with speed relative to the observer, and is the speed of light. Sketch the graph of as a function of v

In the theory of relativity, the energy of a particle is E sm0 2 c4 h2 c 2 2 m where is the rest mass of the particle, is its wave length, and is Plancks constant. Sketch the graph of as a function of . What does the graph say about the energy?

The figure shows a beam of length embedded in concrete walls. If a constant load is distributed evenly along its length, the beam takes the shape of the deflection curve where and are positive constants. ( is Youngs modulus of elasticity and is the moment of inertia of a cross-section of the beam.) Sketch the graph of the deflection curve

Coulombs Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge at a position between them. It follows from Coulombs Law that the net force acting on the middle particle is Fx k x 2 k x 2 2 0  x  2 1 x y where is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?

Find an equation of the slant asymptote. Do not sketch the 2 % 4x curve.

Find an equation of the slant asymptote. Do not sketch the 2 % 4x curve.

Find an equation of the slant asymptote. Do not sketch the 2 % 4x curve.

Find an equation of the slant asymptote. Do not sketch the 2 % 4x curve.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

Show that the curve has two slant asymptotes: and . Use this fact to help sketch the curve. y ! x % (#2 y

Show that the curve has two slant asymptotes: and . Use this fact to help sketch the curve.

Show that the lines and are slant asymptotes of the hyperbol !x 2 #a2 " ! !y 2 #b2 " ! 1

Let f !x" ! !x 3 % 1"#x . Show that This shows that the graph of approaches the graph of , and we say that the curve is asymptotic to the parabola . Use this fact to help sketch the graph of f

Discuss the asymptotic behavior of f!x" ! !x 4 % 1"#xin the same manner as in Exercise 70. Then use your results to help sketch the graph of f

Use the asymptotic behavior of f!x" ! cos x % 1#x 2 to sketch its graph without going through the curve-sketching procedure of this section

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! 4x 4 ! 32x 3 $ 89x 2 ! 95x $ 29

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! x6 ! 15x 5 $ 75x 4 ! 125x 3 ! x

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! x 6 ! 10x 5 ! 400x 4 $ 2500x 3

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! x 2 ! 1 40x 3 $ x $ 1

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! x x 3 ! x 2 ! 4x $ 1

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! tan x $ 5 cos x

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! x !4 & x & 4 2 ! 4x $ 7 cos x

Produce graphs of that reveal all the important aspects of f the curve. In particular, you should use graphs of and to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f!x" ! e x x 2 ! 9

Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly f !x" ! 1 $ 1 x $ 8 x 2 $ 1 x 3

Produce graphs of that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly Single Variable Calculus: Early Transcendentals (Stewart), 6th ed. 349 / 913

(a) Graph the function. (b) Use lHospitals Rule to explain the behavior as x l 0 . (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f!x" ! x 2 11. ln x

(a) Graph the function. (b) Use lHospitals Rule to explain the behavior as x l 0 . (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f !x" ! xe1#x

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values f !x" ! !x $ 4"!x ! 3" 2 x 4 !x ! 1"

Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values f!x" ! !2x $ 3" 2 !x ! 2" 5 x 3 !x ! 5" 2

If is the function considered in Example 3, use a computer algebra system to calculate and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate and use it to estimate the intervals of concavity and inflection points.

f is the function of Exercise 14, find and and use their graphs to estimate the intervals of increase and decrease and concavity of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

Use a computer algebra system to graph and to find and . Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of f

(a) Graph the function. (b) Explain the shape of the graph by computing the limit as or as . (c) Estimate the maximum and minimum values and then use calculus to find the exact values. (d) Use a graph of to estimate the x-coordinates of the inflection points.

(a) Graph the function. (b) Explain the shape of the graph by computing the limit as or as . (c) Estimate the maximum and minimum values and then use calculus to find the exact values. (d) Use a graph of to estimate the x-coordinates of the inflection points.

In Example 4 we considered a member of the family of functions that occur in FM synthesis. Here we investigate the function with . Start by graphing in the viewing rectangle by . How many local maximum points do you see? The graph has more than are visible to the naked eye. To discover the hidden maximum and minimum points you will need to examine the graph of f * very carefully. In fact, it helps to look at the graph of f # at the same time. Find all the maximum and minimum values and inflection points. Then graph in the viewing rectangle by and comment on symmetry.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

Describe how the graph of varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when changes. You should also identify any transitional values of at which the basic shape of the curve changes.

The family of functions , where , , and are positive numbers and , has been used to model the concentration of a drug injected into the bloodstream at time . Graph several members of this family. What do they have in common? For fixed values of and , discover graphically what happens as increases. Then use calculus to prove what you have discovered.

Investigate the family of curves given by , where is a real number. Start by computing the limits as . Identify any transitional values of where the basic shape changes. What happens to the maximum or minimum points and inflection points as changes? Illustrate by graphing several members of the family.

Investigate the family of curves given by the equation . Start by determining the transitional value of at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.

(a) Investigate the family of polynomials given by the equation . For what values of does the curve have minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the parabola . Illustrate by graphing this parabola and several members of the family.

(a) Investigate the family of polynomials given by the equation . For what values of does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve . Illustrate by graphing this curve and several members of the family

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

Find two numbers whose difference is 100 and whose product is a minimum.

Find two positive numbers whose product is 100 and whose sum is a minimum.

Find a positive number such that the sum of the number and its reciprocal is as small as possible.

Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

Find the dimensions of a rectangle with area whose perimeter is as small as possible.

A model used for the yield of an agricultural crop as a function of the nitrogen level in the soil (measured in appropriate units) is Y kN 1 N2 N where is a positive constant. What nitrogen level gives the best yield?

The rate at which photosynthesis takes place for a species of phytoplankton is modeled by the function P 100I I 2 I 4 in where is the light intensity (measured in thousands of footcandles). For what light intensity is a maximum?

Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

A box with a square base and open top must have a volume of 32,000 cm . Find the dimensions of the box that minimize the amount of material used

If 1200 cm of material is available to make a box with a square base and an open top, find the largest possible volume of the box

A rectangular storage container with an open top is to have a volume of 10 m . The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container

Do Exercise 14 assuming the container has a lid that is made from the same material as the sides.

(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.

Find the point on the line that is closest to the origin.

Find the point on the line 6x y 9 1 that is closest to the point .

Find the points on the ellipse 4x 2 y 19. 2 4 3 that are farthest away from the point .

Find, correct to two decimal places, the coordinates of the point on the curve y ! tan x that is closest to the point !1, 1"

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r

Find the area of the largest rectangle that can be inscribed in the ellipse x 2 #a2 $ y 2 #b2 ! 1

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side if one side of the rectangle lies on the base of the triangle.

Find the dimensions of the rectangle of largest area that has its base on the -axis and its other two vertices above the -axis and lying on the parabola y ! 8 ! x 2

Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.

A right circular cylinder is inscribed in a sphere of radius r . Find the largest possible volume of such a cylinder.

A right circular cylinder is inscribed in a cone with height and base radius . Find the largest possible volume of such a cylinder.

A right circular cylinder is inscribed in a sphere of radius . Find the largest possible surface area of such a cylinder.

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 56 on page 23.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm , find the dimensions of the poster with the smallest area.

A poster is to have an area of 180 in with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?

A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?

Answer Exercise 33 if one piece is bent into a square and the other into a circle.

A cylindrical can without a top is made to contain of liquid. Find the dimensions that will minimize the cost of the metal to make the can.

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest lad der that will reach from the ground over the fence to the wall of the building?

A cone-shaped drinking cup is made from a circular piece of paper of radius by cutting out a sector and joining the edges and . Find the maximum capacity of such a cup

A cone-shaped paper drinking cup is to be made to hold of water. Find the height and radius of the cup that will use the smallest amount of paper

A cone with height is inscribed in a larger cone with height so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h ! 1 3 H

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 a plane, then the magnitude of the force is F ! .W . sin - $ cos - where is a constant called the coefficient of friction. For what value of is smallest?

If a resistor of ohms is connected across a battery of volts with internal resistance ohms, then the power (in watts) in the external resistor is P ! E2 R !R $ r" 2 If and are fixed but varies, what is the maximum value of the power?

For a fish swimming at a speed relative to the water, the energy expenditure per unit time is proportional to . It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current , then the time required to swim a distance is and the total energy required to swim the distance is given by E!v" ! av 3 ! L v ! u

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area is given by where , the length of the sides of the hexagon, and , the height, are constants. (a) Calculate . (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of and ). Note: Actual measurements of the angle in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than .

A boat leaves a dock at 2:00 PM and travels due south at a speed of 20 kmh. Another boat has been heading due east at 15 kmh and reaches the same dock at 3:00 PM. At what time were the two boats closest together?

Solve the problem in Example 4 if the river is 5 km wide and point is only 5 km downstream from .

A woman at a point on the shore of a circular lake with radius 2 mi wants to arrive at the point diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of 4 mih and row a boat at 2 mih. How should she proceed?

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is over land to a point on the north bank and under the river to the tanks. To minimize the cost of the pipeline, where should be located?

Suppose the refinery in Exercise 47 is located 1 km north of the river. Where should be located?

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?

Find an equation of the line through the point that cuts off the least area from the first quadrant.

Let and be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point

At which points on the curve does the tangent line have the largest slope?

(a) If is the cost of producing units of a commodity, then the average cost per unit is . Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If , in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

(a) Show that if the profit is a maximum, then the marginal revenue equals the marginal cost. (b) If is the cost function and is the demand function, find the production level that will maximize profit.

A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at , the average attendance had been 27,000. When ticket prices were lowered to , the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue?

During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for each and his sales averaged 20 per day. When he increased the price by , he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry , what should the selling price be to maximize his profit?

A manufacturer has been selling 1000 television sets a week G at each. A market survey indicates that for each rebate offered to the buyer, the number of sets sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is , how should the manufacturer set the size of the rebate in order to maximize its profit?

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is per month. A market survey suggests that, on average, one additional unit will remain vacant for each increase in rent. What rent should the manager charge to maximize revenue?

Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.

The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be?

A point needs to be located somewhere on the line so that the total length of cables linking to the points , , and is minimized (see the figure). Express as a function of and use the graphs of and to estimate the minimum value.

The graph shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of the car. At very low speeds the engine runs inefficiently, so initially decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that is minimized for this car when mi#h. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Lets call this consumption . Using the graph, estimate the speed at which has its minimum value.

Let be the velocity of light in air and the velocity of light in water. According to Fermats Principle, a ray of light will travel from a point in the air to a point in the water by a path that minimizes the time taken. Show that where (the angle of incidence) and (the angle of refraction) are as shown. This equation is known as Snells Law.

Two vertical poles and are secured by a rope going from the top of the first pole to a point on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when $1 ! $ 2

The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose to minimize ?

A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

An observer stands at a point , one unit away from a track. Two runners start at the point in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observers angle of sight between the runners. [Hint: Maximize .]

A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle . How should be chosen so that the gutter will carry the maximum amount of water?

Where should the point be chosen on the line segment so as to maximize the angle ?

A painting in an art gallery has height and is hung so that its lower edge is a distance above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle subtended at his eye by the painting?)

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length and width . [Hint: Express the area as a function of an angle .]

The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuilles Laws gives the resistance of the blood as where is the length of the blood vessel, is the radius, and is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius branching at an angle into a smaller vessel with radius Manfred Cage / Peter Arnold b A B r r C a vascular branching r2

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point on a straight shoreline, flies to a point on the shoreline, and then flies along the shoreline to its nesting area . Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points and are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as land, to what point should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio W#L mean in terms of the birds flight? What would a small value mean? Determine the ratio corresponding to the minimum expenditure of energy. (c) What should the value of be in order for the bird to fly directly to its nesting area ? What should the value of be for the bird to fly to and then along the shore to D?

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point on a line ! parallel to the line joining the light sources and at a distance meters from it (see the figure). We want to locate on ! so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity at the point . (b) If m, use graphs of and to show that the intensity is minimized when m, that is, when is at the midpoint of !. (c) If m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere between m and m there is a transitional value of at which the point of minimal illumination abruptly changes. Estimate this value of by graphical methods. Then find the exact value of .

The figure shows the graph of a function . Suppose that Newtons method is used to approximate the root of the equation with initial approximation . (a) Draw the tangent lines that are used to find and , and estimate the numerical values of and . (b) Would be a better first approximation? Explain.

Follow the instructions for Exercise 1(a) but use as the starting approximation for finding the root

Suppose the line is tangent to the curve when . If Newtons method is used to locate a root of the equation and the initial approximation is , find the second approximation

For each initial approximation, determine graphically what happens if Newtons method is used for the function whose graph is shown. (a) (b) (c) (d) (e) 3 y 0 1 5 x x1 ! 4 x1 ! 5 x1 ! 0 x1 ! 1 x1 ! 3

Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x x1 ! 1 3 " 2x ! 4 ! 0

Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) 3 x 3 " 1 2 x 2 " 3 ! 0

Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x x1 ! 1 5 ! x ! 1 ! 0

Use Newtons method with the specified initial approximation to find , the third approximation to the root of the given equation. (Give your answer to four decimal places.) x x1 ! !1 5 " 2 ! 0

Use Newtons method with initial approximation to find , the second approximation to the root of the equation . Explain how the method works by first graphing the function and its tangent line at

Use Newtons method with initial approximation to find , the second approximation to the root of the equation . Explain how the method works by first graphing the function and its tangent line at

Use Newtons method to approximate the given number correct to eight decimal places. s 5 20

Use Newtons method to approximate the given number correct to eight decimal places. 100s100

Use Newtons method to approximate the indicated root of the equation correct to six decimal places. Use Newtons method to approximate the indicated root of the equation correct to six decimal places. The root x )1, 2* 4 ! 2x 3 " 5x 2 ! 6 ! 0 of in the interval )1, 2*

Use Newtons method to approximate the indicated root of the equation correct to six decimal places.

Use Newtons method to approximate the indicated root of 2.2x 5 ! 4.4x 3 " 1.3x 2 ! 0.9x ! 4.0 ! 0 the equation correct to six decimal places.

Use Newtons method to approximate the indicated root of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all roots of the equation correct to six decimal places.

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x 6 ! x 5 ! 6x 4 ! x 2 " x " 10 ! 0

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x 2 !4 ! x 2 " ! 4 x 2 " 1

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x " ! 2x 2 s2 ! x ! x 2 ! 1

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 3 sin!x 2 x " ! 2x

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 4e 3 " 1 !x 2 sin x ! x 2 ! x " 1

Use Newtons method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. earctan x ! sx 4e 3 " 1

(a) Apply Newtons method to the equation to 1#x ! a ! 0 derive the following square-root algorithm xn"1 ! 2xn ! axn 2 used by the ancient Babylonians to compute : (b) Use part (a) to compute correct to six decimal places.

(a) Apply Newtons method to the equation to derive the following reciprocal algorithm: (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute correct to six decimal places.

Explain why Newtons method doesnt work for finding the root of the equation x 3 ! 3x " 6 ! 0 if the initial approximation is chosen to be x1 ! 1

(a) Use Newtons method with to find the root of the equation correct to six decimal places. (b) Solve the equation in part (a) using as the initial approximation. (c) Solve the equation in part (a) using . (You defi- nitely need a programmable calculator for this part.) ; (d) Graph and its tangent lines at , 0.6, and 0.57 to explain why Newtons method is so sensitive to the value of the initial approximation.

Explain why Newtons method fails when applied to the equation with any initial approximation . Illustrate your explanation with a sketch.

If f!x" ! + sx !s!x if x * 0 if x + 0 hen the root of the equation is . Explain why Newtons method fails to find the root no matter which initial approximation is used. Illustrate your explanation with a sketch.

(a) Use Newtons method to find the critical numbers of the function correct to six decimal places. (b) Find the absolute minimum value of correct to four decimal places.

Use Newtons method to find the absolute maximum value of the function , correct to six decimal places.

Use Newtons method to find the coordinates of the inflection point of the curve , , correct to six decimal places.

Of the infinitely many lines that are tangent to the curve and pass through the origin, there is one that has the largest slope. Use Newtons method to find the slope of that line correct to six decimal places.

Use Newtons method to find the coordinates, correct to six decimal places, of the point on the parabola that is closest to the origin.

In the figure, the length of the chord is 4 cm and the length of the arc is 5 cm. Find the central angle , in radians, correct to four decimal places. Then give the answer to the nearest degree.

A car dealer sells a new car for . He also offers to sell the same car for payments of per month for five years. What monthly interest rate is this dealer charging? To solve this problem you will need to use the formula for the present value of an annuity consisting of equal payments of size with interest rate per time period: Replacing by , show that Use Newtons method to solve this equation

The figure shows the sun located at the origin and the earth at the point . (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU km.) There are five locations , , , , and in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravi tational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If is the mass of the sun, is the mass of the earth, and , it turns out that the -coordinate of is the unique root of the fifth-degree equation and the -coordinate of is the root of the equation p!x" " 2rx 2 ! 0

Find the most general antiderivative of the function. f!x" ! x " 3

Find the most general antiderivative of the function. f!x" ! x " 3 2 " 2x ! 6

Find the most general antiderivative of the function. f!x" ! 1 2 ! 3 4 x 2 " 4 5 x 3

Find the most general antiderivative of the function. f!x" ! 8x 9 " 3x 6 ! 12x 3

Find the most general antiderivative of the function. f!x" ! !x ! 1"!2x " 1"

Find the most general antiderivative of the function. f!x" ! x!2 " x" 2

Find the most general antiderivative of the function. f !x" ! 5x 1$4 " 7x 3$4

Find the most general antiderivative of the function.f !x" ! 2x ! 3x 1.7

Find the most general antiderivative of the function. f!x" ! 6sx " s 6 x

Find the most general antiderivative of the function. f!x" ! s 4 x3 ! s 3 x 4

Find the most general antiderivative of the function. f !x" ! 10 x 9

Find the most general antiderivative of the function. t!x" ! 5 " 4x 3 ! 2x 6 x 6

Find the most general antiderivative of the function. f!u" ! x u4 ! 3su u2

Find the most general antiderivative of the function.f!x" ! 3ex ! 7 sec2 f!u" ! x

Find the most general antiderivative of the function. t!-" ! cos - " 5 sin -

Find the most general antiderivative of the function. f!t" ! sin t ! 2 sinh t

Find the most general antiderivative of the function. f!x" ! 5e f!x" ! 2sx ! 6 cos x x " 3 cosh x

Find the most general antiderivative of the function. f!x" ! 5e f!x" ! 2sx ! 6 cos x x " 3 cosh x

Find the most general antiderivative of the function.f!x" ! x 5 " x 3 ! 2x x 4

Find the most general antiderivative of the function. f!x" ! 2 ! x 2 1 ! x 2

Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of f and F. f !x" ! 5x 4 " 2x 5 21. , F!0" ! 4

Find the antiderivative of that satisfies the given condition. Check your answer by comparing the graphs of f and F. f !x" ! 4 " 3!1 ! x 2 " "1 , F!1" ! 0

Find f. f )!x" ! 6x ! 12x 2

Find f. f )!x" ! 2 ! x 3 ! x 6

Find f. f )!x" ! f )!x" ! 6x ! sin x 2 3 x 2$3

Find f. f )!x" ! f )!x" ! 6x ! sin x 2 3 x 2$3

Find f. f +!t" ! e f +!t" ! t " st t

Find f. f +!t" ! e f +!t" ! t " st t

Find f. f#!x" ! 1 " 6x, f!0" ! 8

Find f. f#!x" ! 8x 3 ! 12x ! 3, f!1" ! 6

Find f. f#!x" ! sx !6 ! 5x" f!1" ! 10

Find f. f#!x" ! 2x " 3$x x & 0 f !1" ! 3

Find f. f#!t" ! 2 cos t ! sec ",$2 ' t ' ,$2 f !,$3" ! 4

Find f. f#!x" ! !x 2 " 1"$x f!1" ! f!"1" ! 0

Find f. f#!x" ! x f!1" ! 1 f!"1" ! "1

Find f. f#!x" ! 4$s1 " x 2 , f( 1 2 ) ! 1

Find f. f )!x" ! 24x f!1" ! 5 f #!1" ! "3 2 ! 2x ! 10

Find f. f )!x" ! 4 " 6x " 40x f !0" ! 2 f#!0" ! 1

Find f. f )!-" ! sin - ! cos - f!0" ! 3 f#!0" ! 4

Find f. f )!t" ! 3$st f !4" ! 20 f #!4" ! 7

Find f. f )!x" ! 2 " 12x f !0" ! 9 f !2" ! 15

Find f. f )!x" ! 20x , f!0" ! 8, f !1" ! 5 3 ! 12x 2 ! 4

Find f. f )!x" ! 2 ! cos x f !0" ! "1 f!,$2" ! 0

Find f. f )!t" ! 2e f!0" ! 0 f!," ! 0 t ! 3 sin t

Find f. f )!x" ! x x & 0 f !1" ! 0 f !2" ! 0

Find f. f +!x" ! cos x f!0" ! 1 f#!0" ! 2 f )!0" ! 3

Given that the graph of passes through the point and that the slope of its tangent line at is , find

Find a function such that and the line is tangent to the graph of f

Find a function such that and the line is tangent to the graph of

Find a function such that and the line is tangent to the graph of

The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F , given that F!0" ! 1

The graph of the velocity function of a particle is shown in the figure. Sketch the graph of the position function.

The graph of is shown in the figure. Sketch the graph of if is continuous and f!0" ! "1

(a) Use a graphing device to graph . E I (b) Starting with the graph in part (a), sketch a rough graph of the antiderivative that satisfies . (c) Use the rules of this section to find an expression for . (d) Graph using the expression in part (c). Compare with your sketch in part (b).

Draw a graph of and use it to make a rough sketch of the antiderivative that passes through the origin. f!x" ! "2, . x . 2, sin x 1 ! x 2

Draw a graph of and use it to make a rough sketch of the antiderivative that passes through the origin. f!x" ! sx "1.5 . x . 1.5 4 " 2x 2 ! 2 " 1

A particle is moving with the given data. Find the position of the particle. v!t" ! sin t " cos t, s!0" ! 0

A particle is moving with the given data. Find the position of the particle. v!t" ! 1.5st , s!4" ! 10

A particle is moving with the given data. Find the position of the particle. a!t" ! t " 2, s!0" ! 1, v!0" ! 3

A particle is moving with the given data. Find the position of the particle. a!t" ! cos t ! sin t s!0" ! 0 v!0" ! 5

A particle is moving with the given data. Find the position of the particle. a!t" ! 10 sin t ! 3 cos t s!0" ! 0 s!2," ! 12

A particle is moving with the given data. Find the position of the particle. a!t" ! t s!0" ! 0 s!1" ! 20 2 " 4t ! 6

A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, m above the ground. (a) Find the distance of the stone above ground level at time . (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 m$s, how long does it take to reach the ground?

Show that for motion in a straight line with constant acceleration , initial velocity , and initial displacement , the displacement after time is s ! 1 2 at 2 ! v0 t ! s0

An object is projected upward with initial velocity meters per second from a point meters above the ground. Show that )v!t"*2 ! v0 2 " 19.6)s!t" " s0 *

Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of ft$s and the other is thrown a second later with a speed of ft$s. Do the balls ever pass each other?

A stone was dropped off a cliff and hit the ground with a speed of 120 ft$s. What is the height of the cliff?

If a diver of mass stands at the end of a diving board with length and linear density , then the board takes on the shape of a curve , where EIy ) ! mt!L " x" ! 1 2 /t!L " x" 2 are positive constants that depend on the material of the board and is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use to estimate the distance below the horizontal at the end of the board.

A company estimates that the marginal cost (in dollars per item) of producing items is . If the cost of producing one item is , find the cost of producing items

The linear density of a rod of length m is given by , in grams per centimeter, where is measured in centimeters from one end of the rod. Find the mass of the rod.

Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m$s and its downward acceleration is a ! ' 9 " 0.9t 0 if 0 . t . 10 if t & 10 If the raindrop is initially m above the ground, how long does it take to fall?

A car is traveling at 50 mi$h when the brakes are fully applied, producing a constant deceleration of 22 ft$s . What is the distance traveled before the car comes to a stop?

What constant acceleration is required to increase the speed of a car from 30 mi$h to 50 mi$h in 5 s?

A car braked with a constant deceleration of 16 ft$s , producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

A car is traveling at when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is , at which time the fuel is exhausted and it becomes a freely falling body. Fourteen seconds later, the rockets parachute opens, and the (downward) velocity slows linearly to ft$s in 5 s. The rocket then floats to the ground at that rate. (a) Determine the position function and the velocity function v (for all times t). Sketch the graphs of s and v. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land?

A high-speed bullet train accelerates and decelerates at the rate of . Its maximum cruising speed is 90 mi$h. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?