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Description

TEST BANK To Accompany

SALAS / HILLE / ETGEN

CALCULUS One and Several Variables Eighth Edition

PREPARED BY

Deborah Betthauser Britt

John Wiley & Sons New York Ÿ Chichester Ÿ Weinheim Ÿ Brisbane Ÿ Singapore Ÿ Toronto

CONTENTS Chapter

Introduction

Chapter

Limits and Continuity

Chapter

Differentiation

Chapter

The Mean-Value Theorem and Applications

Chapter

Integration

Chapter

Some Applications of the Integral

Chapter

The Transcendental Functions

Chapter

Techniques of Integration

Chapter

Conic Sections

Polar Coordinates

Parametric Equations

Chapter 10

Sequences

Indeterminate Forms

Improper Integrals

Chapter 11

Infinite Series

Chapter 12

Vectors

Chapter 13

Vector Calculus

Chapter 14

Functions of Several Variables

Chapter 15

Gradients

Extreme Values

Differentials

Chapter 16

Double and Triple Integrals

Chapter 17

Line Integrals and Surface Integrals

Chapter 18

Elementary Differential Equations

TO THE INSTRUCTOR The Test Bank has been prepared for the eighth edition of Salas,

and Etgen’s Calculus: One and Several Variables

Each chapter is divided into sections that correspond to the sections in the textbook

In all,

the Test Bank consists of 2,223 problems,

together with their solutions that appear immediately after the end of each chapter

Numerous charts and illustrations have also been drawn where appropriate to strengthen the presentation

Far more items are offered within each chapter than any user of the Test Bank would ordinarily need

Due to differences in rounding,

some of the answers requiring calculations by students may differ slightly from the ones given in the Test Bank

The entire Test Bank has been incorporated into MICROTEST,

a computerized test preparation system,

This software has been designed to retrieve the questions from the Test Bank and print your tests and answer keys using a PC running on Windows 3

The questions on your tests printed by the software will be exact duplicates of those shown here

The software includes a host of powerful features and flexibility for selecting questions and producing multiple versions of tests without your having to retype the items or draw the pictures

To obtain MICROTEST software for this textbook,

please contact your John Wiley sales representative

I wish to thank Edwin and Shirley Hackleman of Delta Software for their composition and editorial assistance in preparing the hard copy manuscript and Donald Newell of Delta Software for developing and checking the computerized version of the Test Bank

We have endeavored as a team to produce test items that we trust instructors will find a useful supplement to Calculus: One and Several Variables

DEBORAH BETTHAUSER BRITT

CHAPTER 1 Introduction 1

Notions and Formulas from Elementary Mathematics

Is the number 132 − 12 2 rational or irrational

Is the number 5

Write 6

27272727

upper and lower bounds for the set S = {x : x3 > 1}

upper and lower bounds for the set S =   n 

Rewrite 27 – 8x3 in factored form

Rewrite x4 – 18x2 + 81 in factored form

Evaluate

Evaluate

What is the ratio of the surface area of a cube of side x to the surface area of a sphere of diameter x

Inequalities

Solve x + 3 < 2x – 8

Solve 8 – x2 > 7x

Solve x2 – 5x + 5 ≥ 1

Solve 3x 2 − 1 ≥

2x + 3 < 1

4x − 1

Calculus: One and Several Variables 19

Solve x(2x – 1)(3x + 2) < 0

Solve x2(x – 1)(x + 2)2 > 0

Solve |x| > 2

Solve x + 1 ≥

Solve x − 1 ≤

Solve 0 < x −

Solve |2x – 1| < 5

Solve |3x – 5| ≥

Find the inequality of the form x − c'< δ whose solution is the open interval (–1,

Find the inequality of the form x − c'< δ whose solution is the open interval (–2,

Determine all values of A > 0 for which the following statement is true

If |2x – 5| < 1,

Determine all values of A > 0 for which the following statement is true

If |2x – 3| < A then |6x – 9| < 4

x2 − 9

2 1 < 1

Coordinate Plane

Analytic Geometry

Find the distance between the points P0(2,

Find the midpoint of the line segment from P0(a,

Find the slope of the line through P0(–4,

Find the slope of the line through P0(–2,

Find the slope and the y-intercept for the line 2x + y – 10 = 0

Find the slope and the y-intercept for the line 8x + 3y = 6

Introduction 41

Write an equation for the line with the slope –3 and y-intercept –4

Write an equation for the horizontal line 4 units below the x-axis

Write an equation for the vertical line 2 units to the left of the y-axis

Find an equation for the line that passes through the point P(2,

–1) and is parallel to the line 3y + 5x – 6 = 0

Find an equation for the line that passes through the point P(1,

–1) and is perpendicular to the line 2x – 3y – 8 = 0

Find an equation for the line that passes through the point P(2,

Find an equation for the line that passes through the point P(3,

Determine the point(s) where the line y = 2x intersects the circle x2 + y2 = 4

Find the point where the lines l1 and l2 intersect

Find the area of the triangle with vertices (–1,

Find the area of the triangle with vertices (–1,

Find an equation for the line tangent to the circle x2 + y2 – 4x – 2y = 0 at the point P(4,

Functions

If f ( x ) =

If f ( x ) =

f ( x ) = 1 − cos x Find the number(s),

Find the exact value(s) of x in the interval [0,

2π) which satisfy cos 2 x = −

Find the domain and range for f ( x ) = 2 − x − x 2

Find the domain and range for f ( x ) =

Find the domain and range for h( x ) = − 4 − x 2

Find the number(s),

Find the domain and range for f ( x ) =

Calculus: One and Several Variables 1

Find the domain and range for f ( x ) =

Find the domain and range for f ( x ) = cos x −

Sketch the graph of f ( x ) = 3 − 4 x

Sketch the graph of f ( x ) = − 6x − x 2

Sketch the graph of f ( x ) = x −

Sketch the graph of g ( x ) = 2 + sin x

2  Sketch the graph of g ( x) = 3 2 x − 1 

if x < 1 if x = 1 and give its domain and range

x 2 Sketch the graph of f ( x) =  2 x

if x < 1 and give its domain and range

Is an ellipse the graph of a function

Determine whether f ( x ) = x 4 − x 2 + 1 is odd,

Determine whether f ( x ) = x 5 + x 3 − 3x is odd,

Determine whether f ( x ) =

Determine whether f ( x ) = cos( x + π / 6) is odd,

Determine whether f ( x ) = 3x − 2 sin x is odd,

Determine whether f ( x ) = cos x + sec x is odd,

A given rectangle is twice as long as it is wide

Express the area of the rectangle as a function of the (a) width,

x 3 − 2 x 2 + 5x + 1 is odd,

The Elementary Functions

Find all real numbers x for which R ( x ) =

Find all real numbers x for which R ( x ) =

Find the inclination of the line x − 3 y + 2 3 = 0

Introduction 82

Write an equation for the line with inclination 45° and y-intercept –2

Find the distance between the line 4x + 3y + 4 = 0 and (a) the origin (b) the point P(1,

Find the distance between the line 2x – 5y – 10 = 0 and (a) the origin (b) the point P(–2,

In the triangle with vertices (0,

which vertex is farthest from the centroid

Combinations of Functions x2 − x − 6 and g ( x ) = x − 3 ,

Given that f ( x ) =

Sketch the graphs of the following functions with ƒ and g as shown in the figure

(a) Form the composition of f o g

(b) Form the composition of g o f

x (a) Form the composition of f o g

(b) Form the composition of g o f

x (a) Form the composition of f o g

(b) Form the composition of g o f

x (a) Form the composition of f o g

(b) Form the composition of g o f

(a) Form the composition of f o g

(b) Form the composition of g o f

Form the composition of f o g o h if f ( x ) =

Calculus: One and Several Variables 94

Form the composition of f o g o h if f ( x) = x 2 ,

Form the composition of f o g o h if f ( x) =

Find ƒ such that f o g = F given that g(x) = 2x2 and F(x) = x + 2x2 + 3

Find ƒ such that f o g = F given that g( x ) =

Find g such that f o g = F given that f ( x ) = x 2 − 1 for all real x and F ( x ) = 3x − 1 for x ≥ 0

Find g such that f o g = F given that f ( x ) = x 2 and F ( x ) = (2 x + 5) 2

Form f o g and g o f given that f ( x ) = 4 x + 1 and g ( x ) = 4 x 2

1  Form f o g and g o f given that f ( x ) =  x 2 x − 1

Decide whether f ( x ) = 4 x + 3 and g ( x ) =

Decide whether f ( x ) = ( x − 1) 5 + 1 and g( x ) = ( x − 1) 1 / 5 + 1 are inverses of each other

A Note on Mathematical Proof

Mathematical Induction

Show that 3n ≤ 3n for all positive integers n

Show that n(n + 1)(n + 2)(n + 3) is divisible by 8 for all positive integers n

Show that 1 + 5 + 9 +

+ (4n – 3) = 2n2 – n for all positive integers n

2 x and that g ( x) =  2 x if x ≥ 0 if x < 0

Introduction

Answers to Chapter 1 Questions 1

1] ∪ [4,

1] ∪ [1,

 2 5 4 5  − 2 5 − 4 5  ,

The point of intersection is P(–1,

0) ∪ (0,

0) ∪ (0,

Calculus: One and Several Variables

(2n + 1)

5π/12,

7π/12,

17π/12,

19π/12

1) ∪ (1,

0) ∪ (0,

Introduction 69

θ = π/6

(4x2 + 1)2

3x2 + 2

ƒ(x) = x + 3

ƒ(x) = x2 – 1

2 2 d'5

2 x +1 2

1  (b) 3 + 1 + 2 x 

2 4 + x2

Calculus: One and Several Variables

( f o g )( x ) = 16x 2 + 1 ( g o f )( x ) = 4(4 x + 1) 2 1 if x < 0  2x  101

( f o g )( x ) = 4 x − 1 if 0 ≤ x < 1  2 2 x − 1 if x ≥ 1  2 / x if x < 0  ( g o f )( x ) = 2(2 x − 1) if 0 ≤ x < 1  2 if x ≥ 1 (2 x − 1) 102

True for n = 1: 3 ≤ 3

Assume true for n

Then 3(n + 1) = 3n + 3 ≤ 3n + 3 ≤ 3n + 3n = 2(3n) < 3(3n) = 3n + 1,

so the inequality is true for n + 1

Therefore,

True for n = 1: 1 • 2 • 3 • 4 = 3 • 8

Assume true for n

Then (n + 1)(n + 2)(n + 3)(n + 4) = n(n + 1)(n + 2)(n + 3) + 4(n + 1)(n + 2)(n + 3)

The first term is divisible by 8 by the induction hypothesis,

and the second term is divisible by 8 since at least one of (n + 1),

Hence the result is true for n + 1

Therefore,

True for n = 1: 1 = 2(1)2 – 1

Assume true for n

Then 1 + 5 + 9 +

+ [4(n + 1) – 3] = 1 + 5 + 9 +

+ (4n – 3) + [4(n + 1) – 3] = 2n2 – n + (4n + 1) = 2(n + 1)2 – (n + 1),

so the result is true for n + 1

Therefore,

CHAPTER 2 Limits and Continuity 2

The Idea of Limit For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

Calculus: One and Several Variables 4

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function g graphed below,

Use the graph of g to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

Limits and Continuity 7

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

For the function ƒ graphed below,

Use the graph of ƒ to find (a) lim ƒ(x) (b) lim ƒ(x) (c) lim ƒ(x)

Calculus: One and Several Variables Consider the function ƒ graphed below

State the values of c'for which lim ƒ(x) does not exist

Consider the function ƒ graphed below

State the values of c'for which lim ƒ(x) does not exist

Consider the function ƒ graphed below

State the values of c'for which lim ƒ(x) does not exist

Limits and Continuity 13

Consider the function ƒ graphed below

State the values of c'for which lim ƒ(x) does not exist

Evaluate lim (2x – 5),

Evaluate lim π2,

Evaluate lim x3,

Evaluate lim

Evaluate lim (x3 + 6x2 – 16),

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

x2 − 1

3x − 6 ,

x3 − 8 ,

x2 − 3 ,

Calculus: One and Several Variables x2 + 2x ,

Evaluate lim

Evaluate lim f ( x ) ,

Evaluate lim f ( x ) ,

Evaluate lim f ( x ) ,

Evaluate lim x→1

Definition of Limit 2x ,

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

1 − x2 ,

Limits and Continuity

x < 1  Evaluate the right hand limit at x = 1,

x < 0 Evaluate the right hand limit at x = 0,

for f ( x) =  3  x − 1,

x < 1  Evaluate lim f ( x ) ,

x = 1 x →1  2 2 x − 1,

Evaluate the largest δ that “works” for a given arbitrary ε

Evaluate the largest δ that “works” for a given arbitrary ε

3 x = 2 5

Give an ε,

σ proof for lim (3x − 2) = 4

Give an ε,

σ proof for lim (5x − 2) = 3

Give an ε,

σ proof for lim x − 3 = 2

Give the four equivalent limit statements displayed in (2

Give an ε,

σ proof for lim x 2 = 9

if f ( x) =  x→3 ( x − 1)3 ,

Some Limit Theorems Given that lim f (x) = 0,

evaluate the limits that exist

(a) lim f ( x ) − g( x ) x→c

g( x ) h( x ) f ( x) (d) lim x → c'g ( x) (c) lim

4 ,c = 2

Evaluate lim 5,

(g) lim [ 3 f ( x ) − 2 g ( x ) − h( x )] x→c

Calculus: One and Several Variables Evaluate lim (2 − 3x)2,

Evaluate lim (2x3 − 3x2 + 2),

Evaluate lim 22x – 1,

Evaluate lim

Evaluate lim 2 x −

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

x3 − 8 ,

x−2 2 x − 3x 2 (2 x − 1)( x 2 − 4)

x+5 x → −5 x 2 + 2 x − 15

( x + 5) 2 ( x + 5) 2 x + 2 x − 15 2

Limits and Continuity 70

x3 + a3 ,

Evaluate lim 2 x→ 2 1− x Evaluate lim

Evaluate lim 3 x→0 x− x

Evaluate lim  + x − 3 x→3  x − 3

−6   2x Evaluate lim  +  ,

Evaluate the following limits that exist

 3 1 (a) lim  −  2 x→2  x  3 1   1   (b) lim  −    2   x − 2   x → 2  x

Given that ƒ(x) = x2 – 3x,

evaluate the limits that exist

 3 1   (c) lim  −  ( x − 2) 2 x → 2  x  2  3 1   1   (d) lim  −    2   x − 6   x → 2  x 

Continuity

Determine whether or not ƒ(x) = 2x2 – 3x – 5 is continuous at x = 1

If not,

determine whether the discontinuity is a removable discontinuity,

Determine whether or not f ( x ) = ( x − 2) 2 + 2 is continuous at x = 2

If not,

determine whether the discontinuity is a removable discontinuity,

Determine whether or not ƒ(x) = |5 – 2x2| is continuous at x = 3

If not,

determine whether the discontinuity is removable discontinuity,

Determine whether or not f ( x ) =

If not,

determine whether the 3x − 2 discontinuity is a removable discontinuity,

Calculus: One and Several Variables − 3,

x < −1  Determine whether or not f ( x) =  1,

x = −1 is continuous at x = –1

If not,

x > −1  discontinuity is a removable discontinuity,

x < 1  Determine whether or not f ( x) =  3,

If not,

determine whether the 2 x + 1,

x > 1  discontinuity is a removable discontinuity,

x < 2  Determine whether or not f ( x ) =  1,

If not,

determine whether the  2 x,

x > 2   discontinuity is a removable discontinuity,

x ≠1  Determine whether or not f ( x) =  x − 4 is continuous at x = 4

If not,

determine whether the  1,

x = 4 discontinuity is a removable discontinuity,

If not,

determine whether the x ( x + 1) discontinuity is a removable discontinuity,

Determine whether or not f ( x ) =

If not,

determine whether the ( x − 1) 3 discontinuity is a removable discontinuity,

Determine whether or not f ( x ) =

Sketch the graph of f ( x ) =

Sketch the graph of f ( x ) =

Sketch the graph of ƒ(x) = | x – 3 | and classify the discontinuities,

x ≤ −2  Sketch the graph of f ( x) =  x + 1,

− 2 < x < 1 and classify the discontinuities,

x < 1  Sketch the graph of f ( x) =  1,

x = 1 and classify the discontinuities,

and classify the discontinuities,

Limits and Continuity

x < 1  Sketch the graph of f ( x) =  0,

x = 1 and classify the discontinuities,

Define f ( x ) =

Define f ( x ) =

Define f ( x ) =

x ≥ −1  Let f ( x) =  x + 1

Find A given that ƒ is continuous at –1

Prove that if ƒ(x) has a removable discontinuity at c,

The Pinching Theorem

Trigonometric Limits sin 7 x ,

Evaluate lim

x → 0 1 − cos 3 x Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim θ cot 4θ ,

Evaluate lim

θ →0 θ sin 2θ ,

θ → 0 tan θ

α →0

θ →0

θ →0

Calculus: One and Several Variables

Evaluate lim

Evaluate lim

θ csc θ

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

For ƒ(x) = sin x and a = π/3,

θ →0

θ →0

θ →0

θ2 sin 3θ 2

α → 0 cos α

θ → 0 cos 2θ sin 2 θ ,

θ → 0 tan θ

f ( a + h) − f (a) and give an equation for the tangent line to the h

ƒ (a))

Use the pinching theorem to find lim

Some Basic Properties of Continuous Functions

Sketch the graph of a function ƒ that is defined on [0,

Sketch the graph of a function ƒ that is defined on [0,

Sketch the graph of a function ƒ that is defined on [0,

takes on the values ½ and 1 but does not take on the value 0

Sketch the graph of a function ƒ that is defined on [0,

Sketch the graph of a function ƒ that is defined on [0,

Show the equation x3 – cos2 x = 0 has a root in [0,

Given that ƒ (x) = x4 – x2 + 5x + 2,

show that there exist at least two real numbers x such that ƒ (x) = 3

Limits and Continuity

Answers to Chapter 2 Questions 1

(a) 4 (b) does not exist (−∞)

(a) does not exist (+∞) (b) does not exist (−∞)

(c) does not exist (d) does not exist

(a) does not exist (+∞) (b) does not exist (+∞)

(c) does not exist (+∞) (d) does not exist

Since |(3x – 2) – 4| = |3x – 6| = 3|x – 2|,

we can 1 1 take δ = ε : if 0 < |x – 2| < ε ,

then 3 3 |(3x – 2) – 4| = 3|x – 2| < ε

Calculus: One and Several Variables Since |(5x – 2) – 3| = |5x – 5| = 5|x – 1|,

we can 1 1 take σ = ε : if 0 < |x – 1| < ε ,

then 5 5 |(5x – 2) – 3| = 5|x – 1| < ε

Since |(x – 3) – 2| = |x – 5|,

we can take δ = ε : if 0 < |x – 5| < ε,

then |(x – 3) – 2| = |x – 5| < ε

If |x – 3| < 1,

2 < x < 4,

Take δ = minimum of 1 and ε/7

If 0 < |x – 3| < σ,

then2 < x < 4 and |x – 3| < ε/7

Therefore,

|x2 – 9| = |x + 3||x – 3| < 7|x – 3| < 7(ε/7) = ε

(e) does not exist (f) 1/7 (g) −2

(a) 3 (b) 7 (c) does not exist

removable discontinuity at x = 2/3

removable discontinuity at x = −3

Limits and Continuity 88

nonjump discontinuity at x = 2

ƒ(−1) = 3

ƒ(−3) = −6

ƒ(2) = 5

Since ƒ has a removable discontinuity at c,

for lim f ( x ) = L'some real number L

Then x →c

jump discontinuity at x = −2 and x = 1

8/9 101

1/3 108

3/2 110

Calculus: One and Several Variables

If ƒ (x) = x3 – cos2 x,

ƒ (2) = 8 – cos2 (2) ≥ 7 > 0,

so by the intermediate-value theorem ƒ (c) = 0 for some c'in [0,

ƒ is continuous,

ƒ (–2) = 4,

ƒ (1) = 7,

so by the intermediate-value theorem ƒ (x) = 3 for some x in [–2,

CHAPTER 3 Differentiation 3

The Derivative f ( x + h) − f ( x ) and taking the limit as h tends to 0

h f ( x + h) − f ( x ) Differentiate ƒ(x) = 3 – 4x by forming a difference quotient and taking the limit as h tends h to 0

Differentiate ƒ(x) = 7 by forming a difference quotient

Differentiate ƒ(x) = 2x2 + x by forming a difference quotient

f ( x + h) − f ( x ) and taking the limit as h tends h

Differentiate ƒ(x) = x3 by forming a difference quotient

Differentiate ƒ(x) =

f ( x + h) − f ( x ) and taking the limit as h tends to 0

x + 2 by forming a difference quotient

f ( x + h) − f ( x ) and taking the limit as h h

Differentiate ƒ(x) =

Differentiate ƒ(x) =

by forming a difference quotient

f ( x + h) − f ( x ) and taking the limit as h tends h

Find ƒ ′ (2) for ƒ(x) = (2x + 3)2 by forming a difference quotient h → 0

Find ƒ ′ (2) for ƒ(x) = x3 – 2x by forming a difference quotient h → 0

f (2 + h) − f (2) and taking the limit as h

f (2 + h) − f (2) and taking the limit as h

Find ƒ ′ (2) for ƒ(x) = 2 x + x + 2 by forming a difference quotient

f (2 + h) − f (2) and taking the limit as h

Find ƒ ′ (2) for ƒ(x) = h → 0

Find equations for the tangent and normal to the graph of ƒ(x) = 2x3 + 1 at the point (1,

Find equations for the tangent and normal to the graph of ƒ(x) = x3 – 3x at the point (2,

Calculus: One and Several Variables

Find equations for the tangent and normal to the graph of ƒ(x) =

Find equations for the tangent and normal to the graph of ƒ(x) =

ƒ(3/2))

Draw a graph of ƒ(x) = |2x – 1| and indicate where it is not differentiable

x ≤ 1 Draw a graph of f ( x) =  and indicate where it is not differentiable

Find ƒ ′ (c) if it exists

x ≤ −1 Find ƒ ′ (c) if it exists

Sketch the graph of the derivative of the function with the graph shown below

Sketch the graph of the derivative of the function with the graph shown below

Determine ƒ and c

sin h represents the derivative of a function ƒ at a point c

Determine ƒ and c

Differentiation

Some Differentiation Formulas

Differentiate F(x) = 1 – 3x

Differentiate F(x) = 4x5 – 8x2 + 9x

Differentiate F(x) =

Differentiate F(x) = (2x2 – 1)(3x + 1)

Differentiate F(x) =

Differentiate F(x) =

 Differentiate F(x) = 1 + 

Find ƒ ' (0) and ƒ ' (1) for ƒ (x) = x3(2x + 3)

Find ƒ ' (0) and ƒ ' (1) for ƒ (x) =

Given that h(0) = 4 and h ' (0) = 3,

find ƒ ' (0) for f(x) = 2x2h(x) – 3x

Find an equation for the tangent to the graph of ƒ(x) = 2 x 2 −

Find the points where the tangent to the curve is horizontal for ƒ(x) = (x + 1)(x2 – 3x – 8)

Find the points where the tangent to the curve for ƒ(x) = –x3 + 2x is parallel to the line y = 2x + 5

Find the points where the tangent to the curve for ƒ(x) = 3x + x2 is perpendicular to the line 3x + 2y + 1 = 0

Find the area of the triangle formed by the x-axis and the lines tangent and normal to the curve ƒ(x) = 2x + 3x2 at the point (–1,

3x 4 + 5

x −1 2 2  1 + 2 

2 x −1

The d/dx Notation

Derivatives of Higher Order

d [−2( x 2 − 5x )(3 + x 7 )]

ƒ(–1))

Calculus: One and Several Variables d'dx

Evaluate

Find the second derivative for f ( x ) =

Find the second derivative for ƒ(x) = (x2 – 2)(x3 + 5x)

Determine the values of x for which (a) ƒ ′ ′ (x) = 0,

and (c) ƒ ′ ′ (x) < 0 for ƒ(x) = 2x4 + 2x3 – x

d4y dx 4

dy at x = 2 for y = (x2 + 1)(x3 – x)

2 dx dx

The Derivative as a Rate of Change

Find the rate of change of the area of a circle with respect to the radius r when r = 3

Find the rate of change of the volume of a cube with respect to the length s'of a side when s'= 2

Find the rate of change of the area of a square with respect to the length z of a diagonal when z = 5

Find the rate of change of the volume of a ball with respect to the radius r when r = 4

Find the rate of change of y = 6 – x − x2 with respect to x at x = −1

Find the rate of change of the volume V of a cube with respect to the length w of a diagonal on one of the faces when w = 2

The volume of a cylinder is given by the formula V = π r2 h where r is the base radius and h is the height

(a) Find the rate of change of V with respect to h if r remains constant

(b) Find the rate of change of V with respect to r if h remains constant

(c) Find the rate of change of h with respect to r if V remains constant

An object moves along a coordinate line,

its position at each time t ≥ 0 given by x(t) = 3t2 – 7t + 4

Find the position,

An object moves along a coordinate line,

its position at each time t ≥ 0 given by x(t) = t3 – 6t2 – 15t

Determine when,

An object moves along the x-axis,

its position at each time t ≥ 0 given by x(t) = t4 – 12t3 + 28t2

Determine the time interval(s),

during which the object moves left

An object moves along the x-axis,

its position at each time t ≥ 0 given by x(t) = 5t4 – t5

Determine the time interval(s),

during which the object is speeding up to the right

Differentiation

An object is dropped and hits the ground 5 seconds later

From what height was it dropped

A stone is thrown upward from ground level

The initial speed is 24 feet per second

(a) In how many seconds will the stone hit the ground

? (c) With what minimum speed should the stone be thrown to reach a height of 40 feet

An object is projected vertically upward from ground level with a velocity of 32 feet per second

What is the height attained by the object

If C( x ) = 700 + 5x +

is the cost function for a certain commodity,

find the marginal cost at a production x level of 400 units,

and find the actual cost of producing the 401st unit

If C(x) = 25,000 + 30x + (0

find: (a) the profit function (b) the marginal profit (c) the production level(s) at which the marginal profit is zero

The Chain Rule

Differentiate ƒ(x) = (x3 + 1)3 : (a) by expanding before differentiation,

Then reconcile the results

Differentiate ƒ(x) = (x – x3)3

 x + 1 Differentiate f ( x ) =  

 x2 + 7  Differentiate f ( x ) =  2 

Differentiate ƒ(x) = (x + 4)4(3x + 2)3

dy 1 at x = 0 for y = and u = (3x + 1)3

dx 1+ u dy 1− x at t = 1 for y = u3 – u2,

Find dt 1+ x Find

dy 1+ 5 at x = 1 for y = ,and s'= dx 1− 5

Given that ƒ(1) = 2,

ƒ ' (1) = 3,

Given that ƒ(x) = (1 + 2x2)–2,

determine the values of x for which (a) ƒ ' (x) = 0,

(b) ƒ ' (x) > 0 and (c) ƒ ' (x) < 0

Calculus: One and Several Variables

An object moves along a coordinate line,

its position at each time t ≥ 0 given by x(t) = (t2 – 3)3(t2 + 1)2

Determine when the object changes direction

Differentiate ƒ(x) = [(x3 – x–3)2 – x2]3

Find ƒ ′ ′ (x) for ƒ(x) = (x2 + 2x)17

The edge of a cube is decreasing at the rate of 3 centimeters per second

How is the volume of the cube changing when the edge is 5 centimeters long

The diameter of a sphere is increasing at the rate of 3 centimeters per second

How is the volume of the sphere changing when the diameter is 6 centimeters

Differentiating the Trigonometric Functions

Differentiate y = x tan x

Differentiate y = sin x tan x

Differentiate y =

Differentiate y =

Differentiate y = sec x tan x

Find the second derivative for y = x sin x

Find the second derivative for y = 5 cos x + 7 sin x +

Find an equation for the tangent to the curve y – sin x at x = π/6

Determine the numbers x between 0 and 2π on y = sin x,

where the tangent to the curve is parallel to the line y = 0

An object moves along the y-axis,

its position at each time t given by x(t) = sin 2t

Determine those times from t = 0 to t = π when the object is moving to the right with increasing speed

A rocket is launched 2 miles away from one observer on the ground

How fast is the rocket going when the angle of elevation of the observer’s line of sight to the rocket is 50° (from the horizontal) and is increasing at 5 °/sec

An airplane at a height of 2000 meters is flying horizontally,

directly toward an observer on the ground,

with a speed of 300 meters per second

How fast is the angle of elevation of the plane changing when this angle is 45°

d3 dx 3

dy 1  for y =  (1 + u) ,

Differentiation

Implicit Differentiation: Rational Powers

Use implicit differentiation to obtain

Use implicit differentiation to obtain

dy in terms of x and y for x2 – 4xy + 2y2 = 5

dy in terms of x and y for x2y + y2 = 6

dx dy Use implicit differentiation to obtain in terms of x and y for xy 2 + xy = 2

dx dy in terms of x and y for y = sin (x + y) + cos x

Use implicit differentiation to obtain

Express

Express

Express

Express

2 dx dx

Find the equations for the tangent and normal at the point P(–1,

–1) for 2x2 – 3xy + 3y2 = 2

dy for y = (x2 + 1)1/4(x2 + 2)1/2

Compute

Compute

Find the second derivative for y =

Find the second derivative for y = 4 4 − x 3

Compute

d2y dx 2

in terms of x and y for x2 + 3y2 = 10

in terms of x and y for x2 + 2xy – y2 + 8 = 0

3x 3 + 2

2x − 5

Calculus: One and Several Variables

In economics,

the elasticity of demand is given by the formula ε =

P dQ where P is price and Q quantity

 inelastic where ε < 1  The demand is said to be  unitary where ε = 1

Describe the elasticity of Q = (400 – P)3/5

Rates of Change Per Unit Time

A shark,

is swimming parallel to a straight beach and is 90 feet offshore

The shark is swimming at a constant speed of 30 feet per second

At time t = 0,

the shark is directly opposite a lifeguard station

How fast is the shark moving away from the lifeguard station when the distance between them is 150 feet

A boat sails parallel to a straight beach at a constant speed of 12 miles per hour,

How fast is it approaching a lighthouse on the shoreline at the instant it is exactly 5 miles from the lighthouse

A ladder 13 feet long is leaning against a wall

If the base of the ladder is moving away from the wall at the rate of ½ foot per second,

at what rate will the top of the ladder be moving when the base of the ladder is 5 feet from the wall

A spherical balloon is inflated so that its volume is increasing at the rate of 3 cubic feet per minute

How fast 4 is the radius of the balloon increasing at the instant the radius is ½ foot

Sand is falling into a conical pile so that the radius of the base of the pile is always equal to one-half of its altitude

If the sand is falling at a rate of 10 cubic feet per minute,

how fast is the altitude of the pile 1 increasing when the pile is 5 feet deep

A spherical balloon is inflated so that its volume is increasing at the rate of 20 cubic feet per minute

How 4 fast is the surface area of the balloon increasing at the instant the radius is 4 feet

S = 4πr 2 3

Two ships leave port at noon

One ship sails north at 6 miles per hour,

and the other sails east at 8 miles per hour

At what rate are the two ships separating 2 hours later

A conical funnel is 14 inches in diameter and 12 inches deep

A liquid is flowing out at the rate of 40 cubic 1 inches per second

How fast is the depth of the liquid falling when the level is 6 inches deep

A baseball diamond is a square 90 feet on each side

A player is running from home to first base at the rate of 25 feet per second

At what rate is his distance from second base changing when he has run half way to first base

A ship,

proceeding southward on a straight course at a rate of 12 miles/hr

which is sailing west at 15 miles/hr

(a) How fast are the ships approaching each other 1 hour later

? (b) Are the ships approaching each other or are they receding from each other at 2 o’clock and at what rate

An angler has a fish at the end of his line,

which is being reeled in at the rate of 2 feet per second from a bridge 30 feet above water

At what speed is the fish moving through the water towards the bridge when the amount of line out is 50 feet

? (Assume the fish is at the surface of the water and that there is no sag in the line

Differentiation

A kite is 150 feet high and is moving horizontally away from a boy at the rate of 20 feet per second

How fast is the string being payed out when the kite is 250 feet from him

An ice cube is melting so that its edge length x is decreasing at the rate of 0

How fast is the volume decreasing when x = 2 meters

Consider a rectangle where the sides are changing but the area is always 100 square inches

One side changes at the rate of 3 inches per second

When that side is 20 inches long,

how fast is the other side changing

The sides of an equilateral triangle are increasing at the rate of 5 centimeters per hour

At what rate is the area increasing when the side is 10 centimeters