Suppose that functions f and g and their first derivatives

Chapter 3: Problem 1 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 2 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 3 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 4 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 5 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 6 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 7 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 8 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 9 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 10 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 11 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 12 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 13 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 14 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 15 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 16 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 17 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 18 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 19 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 20 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 21 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 22 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 23 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 24 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 25 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 26 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 27 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 28 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 29 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 30 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 130, find the derivative of the function

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Chapter 3: Problem 31 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3134, find all values of x for which the function is differentiable.

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Chapter 3: Problem 32 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3134, find all values of x for which the function is differentiable.

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Chapter 3: Problem 33 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3134, find all values of x for which the function is differentiable.

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Chapter 3: Problem 34 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3134, find all values of x for which the function is differentiable.

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Chapter 3: Problem 35 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3538, find dy/dx.

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Chapter 3: Problem 36 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3538, find dy/dx.

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Chapter 3: Problem 37 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3538, find dy/dx.

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Chapter 3: Problem 38 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3538, find dy/dx.

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Chapter 3: Problem 39 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, find dy/dx by implicit differentiation.

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Chapter 3: Problem 40 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, find dy/dx by implicit differentiation.

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Chapter 3: Problem 41 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, find dy/dx by implicit differentiation.

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Chapter 3: Problem 42 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, find dy/dx by implicit differentiation.

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Chapter 3: Problem 43 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 43 and 44, find all derivatives of the function.

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Chapter 3: Problem 44 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 43 and 44, find all derivatives of the function.

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Chapter 3: Problem 45 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4548, find an equation for the (a) tangent and (b) normal to the curve at the indicated point

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Chapter 3: Problem 46 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4548, find an equation for the (a) tangent and (b) normal to the curve at the indicated point

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Chapter 3: Problem 47 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4548, find an equation for the (a) tangent and (b) normal to the curve at the indicated point

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Chapter 3: Problem 48 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4548, find an equation for the (a) tangent and (b) normal to the curve at the indicated point

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Chapter 3: Problem 49 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4952, find an equation for the line tangent to the curve at the point defined by the given value of t.

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Chapter 3: Problem 50 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4952, find an equation for the line tangent to the curve at the point defined by the given value of t.

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Chapter 3: Problem 51 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4952, find an equation for the line tangent to the curve at the point defined by the given value of t.

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Chapter 3: Problem 52 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 4952, find an equation for the line tangent to the curve at the point defined by the given value of t.

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Chapter 3: Problem 53 Calculus: Graphical, Numerical, Algebraic 3

(a) Graph the function x, 0 # x # 1 f!x" ! { 2 " x, 1 ' x # 2. (b) Is f continuous at x ! 1? Explain. (c) Is f differentiable at x ! 1? Explain.

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Chapter 3: Problem 54 Calculus: Graphical, Numerical, Algebraic 3

For what values of the constant m is sin 2x, x # 0 f!x" ! { mx, x ) 0 (a) continuous at x ! 0? Explain. (b) differentiable at x ! 0? Explain.

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Chapter 3: Problem 55 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 5558, determine where the function is (a) differentiable, (b) continuous but not differentiable, and (c) neither continuous nor differentiable

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Chapter 3: Problem 56 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 5558, determine where the function is (a) differentiable, (b) continuous but not differentiable, and (c) neither continuous nor differentiable

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Chapter 3: Problem 57 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 5558, determine where the function is (a) differentiable, (b) continuous but not differentiable, and (c) neither continuous nor differentiable

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Chapter 3: Problem 58 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 5558, determine where the function is (a) differentiable, (b) continuous but not differentiable, and (c) neither continuous nor differentiable

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Chapter 3: Problem 59 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 59 and 60, use the graph of f to sketch the graph of f *

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Chapter 3: Problem 60 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 59 and 60, use the graph of f to sketch the graph of f *

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Chapter 3: Problem 61 Calculus: Graphical, Numerical, Algebraic 3

The following graphs show the distance traveled, velocity, and acceleration for each second of a 2-minute automobile trip. Which graph shows (a) distance iii? (b) velocity? i (c) acceleration? ii

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Chapter 3: Problem 62 Calculus: Graphical, Numerical, Algebraic 3

Sketch the graph of a continuous function f with f !0" ! 5 and "2, x ' 2 f*!x" ! { "0.5, x ) 2

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Chapter 3: Problem 63 Calculus: Graphical, Numerical, Algebraic 3

Sketch the graph of a continuous function f with f !"1" ! 2 and "2, x ' 1 f*!x" ! { 1, 1 ' x ' 4 "1, 4 ' x ' 6.

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Chapter 3: Problem 64 Calculus: Graphical, Numerical, Algebraic 3

Which of the following statements could be true if f !!x" " x1#3? Answer is D: i and iii only could be true i. f !x" " $2 9 8 $ x7#3 % 9 ii. f&!x" " $2 9 8 $ x7#3 ' 2 iii. f&!x" " $ 3 4 $ x 4#3 % 6 iv. f !x" " $ 3 4 $ x 4#3 ' 4 A. i only B. iii only C. ii and iv only D. i and iii only

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Chapter 3: Problem 65 Calculus: Graphical, Numerical, Algebraic 3

The following data give the coordinates of a moving body for various values of t. (a) Make a scatter plot of the !t, s" data and sketch a smooth curve through the points. (b) Compute the average velocity between consecutive points of the table. (c) Make a scatter plot of the data in part (b) using the midpoints of the t values to represent the data. Then sketch a smooth curve through the points. (d) Writing to Learn Why does the curve in part (c) approximate the graph of ds#dt?

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Chapter 3: Problem 66 Calculus: Graphical, Numerical, Algebraic 3

Suppose that a function f and its first derivative have the following values at x " 0 and x " 1. Find the first derivative of the following combinations at the given value of x. (a) #x$ f !x", x " 1 '13/10 (b) #f$!x$", x " 0 '1/3 (c) f !#x$", x " 1 1/10 (d) f !1 ' 5 tan x", x " 0 '1 (e) $2 % f ! c x o " $s x , x " 0 '2/3 (f ) 10 sin ($ p 2 x $) f 2!x", x " 1 '12

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Chapter 3: Problem 67 Calculus: Graphical, Numerical, Algebraic 3

Suppose that functions f and g and their first derivatives have the following values at x " '1 and x " 0. Find the first derivative of the following combinations at the given value of x. (a) 3f !x" ' g!x", x " '1 5 (b) f 2!x"g3!x", x " 0 0 (c) g! f !x"", x " '1 8 (d) f !g!x"", x " '1 2 (e) $g!x f " !x % " $2 , x " 0 6 (f) g!x % f !x"", x " 0

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Chapter 3: Problem 68 Calculus: Graphical, Numerical, Algebraic 3

Find the value of dw#ds at s " 0 if w " sin !#r$ ' 2" and r " 8 sin !s % p#6". #3

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Chapter 3: Problem 69 Calculus: Graphical, Numerical, Algebraic 3

Find the value of dr#dt at t " 0 if r " !u2 % 7"1#3 and u2t % u " 1. '1/6

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Chapter 3: Problem 70 Calculus: Graphical, Numerical, Algebraic 3

The position at time t ) 0 of a particle moving along the s-axis is s!t" " 10 cos !t % p#4". (a) Give parametric equations that can be used to simulate the motion of the particle. (b) What is the particles initial position !t " 0"? (c) What points reached by the particle are farthest to the left and right of the origin? (d) When does the particle first reach the origin? What are its velocity, speed, and acceleration then?

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Chapter 3: Problem 71 Calculus: Graphical, Numerical, Algebraic 3

On Earth, if you shoot a paper clip 64 ft straight up into the air with a rubber band, the paper clip will be s!t" " 64t ' 16t 2 feet above your hand at t sec after firing. (a) Find ds#dt and d2s#dt 2. (b) How long does it take the paper clip to reach its maximum height? 2 sec (c) With what velocity does it leave your hand? 64 ft/sec (d) On the moon, the same force will send the paper clip to a height of s!t" " 64t ' 2.6t 2 ft in t sec. About how long will it take the paper clip to reach its maximum height, and how high will it go?

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Chapter 3: Problem 72 Calculus: Graphical, Numerical, Algebraic 3

Suppose two balls are falling from rest at a certain height in centimeters above the ground. Use the equation s " 490t 2 to answer the following questions. (a) How long does it take the balls to fall the first 160 cm? What is their average velocity for the period? $ 4 7 $ sec; 280 cm/sec (b) How fast are the balls falling when they reach the 160-cm mark? What is their acceleration then?

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Chapter 3: Problem 73 Calculus: Graphical, Numerical, Algebraic 3

If a hemispherical bowl of radius 10 in. is filled with water to a depth of x in., the volume of water is given by V " p%10 ' !x#3"&x2. Find the rate of increase of the volume per inch increase of depth

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Chapter 3: Problem 74 Calculus: Graphical, Numerical, Algebraic 3

A bus will hold 60 people. The fare charged ( p dollars) is related to the number x of people who use the bus by the formula p " %3 ' !x#40"& 2. (a) Write a formula for the total revenue per trip received by the bus company. (b) What number of people per trip will make the marginal revenue equal to zero? What is the corresponding fare? (c) Writing to Learn Do you think the bus companys fare policy is good for its business?

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Chapter 3: Problem 75 Calculus: Graphical, Numerical, Algebraic 3

The figure shows a boat 1 km offshore sweeping the shore with a searchlight. The light turns at a constant rate, du!dt " #0.6 rad!sec. (a) How fast is the light moving along the shore when it reaches point A? #0.6 km/sec (b) How many revolutions per minute is 0.6 rad!sec?

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Chapter 3: Problem 76 Calculus: Graphical, Numerical, Algebraic 3

The graph of y " sin !x # sin x" appears to have horizontal tangents at the x-axis. Does it?

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Chapter 3: Problem 77 Calculus: Graphical, Numerical, Algebraic 3

We measure the frequencies at which wires vibrate in cycles (trips back and forth) per sec. The unit of measure is a hertz: 1 cycle per sec. Middle A on a piano has a frequency 440 hertz. For any given wire, the fundamental frequency y is a function of four variables: r: the radius of the wire; l: the length; d: the density of the wire; T: the tension (force) holding the wire taut. With r and l in centimeters, d in grams per cubic centimeter, and T in dynes (it takes about 100,000 dynes to lift an apple), the fundamental frequency of the wire is y " $2 1 rl$#$p T d $$. If we keep all the variables fixed except one, then y can be alternatively thought of as four different functions of one variable, y!r", y!l", y!d", and y!T ". How would changing each variable affect the strings fundamental frequency? To find out, calculate y%!r", y%!l", y%!d", and y%!T ".

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Chapter 3: Problem 78 Calculus: Graphical, Numerical, Algebraic 3

The spread of measles in a certain school is given by P!t" " $1 & 20 e 0 $5#t , where t is the number of days since the measles first appeared, and P!t" is the total number of students who have caught the measles to date. (a) Estimate the initial number of students infected withmeasles. P(0) % 1.339, so initially, one student was infected (b) About how many students in all will get the measles? 200 (c) When will the rate of spread of measles be greatest? What is this rate?

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Chapter 3: Problem 79 Calculus: Graphical, Numerical, Algebraic 3

Graph the function f !x" " tan#1 !tan 2x" in the window &#p, p' by , 4'. Then answer the following questions. (a) What is the domain of f ? x ' k$ ! 4 $, where k is an odd integer (b) What is the range of f ? (#!!2,!!2) (c) At which points is f not differentiable? (d) Describe the graph of f%.

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Chapter 3: Problem 80 Calculus: Graphical, Numerical, Algebraic 3

If x2 # y2 " 1, find d2 y!dx2 at the point !2, (3)".

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Chapter 3: Problem 81 Calculus: Graphical, Numerical, Algebraic 3

A particle moves along the x-axis so that at any time t ( 0 its position is given by x(t) " t 3 # 12t & 5. (a) Find the velocity of the particle at any time t. (b) Find the acceleration of the particle at any time t. (c) Find all values of t for which the particle is at rest. (d) Find the speed of the particle when its acceleration is zero. (e) Is the particle moving toward the origin or away from the origin when t " 3? Justify your answer.

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Chapter 3: Problem 82 Calculus: Graphical, Numerical, Algebraic 3

Let y " . (a) Find $ d d y x $. (b) Find . (c) Find an equation of the line tangent to the curve at x " 1. (d) Find an equation of the line normal to the curve at x " 1. (e) Find any points where the tangent line is horizontal.

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Chapter 3: Problem 83 Calculus: Graphical, Numerical, Algebraic 3

Let f (x) " ln (1 # x2). (a) State the domain of f . (b) Find f%(x). (c) State the domain of f%. (d) Prove that f)(x) * 0 for all x in the domain of f .

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