PDF- -Jerrold Marsden - CaltechAUTHORS - Calculus Salas

- s Salas

TEST BANK To Accompany

CALCULUS One and Several Variables Eighth Edition

PREPARED BY

Introduction

Chapter

Chapter

Differentiation

Integration

Chapter

Some Applications of the Integral

Chapter

Techniques of Integration

Polar Coordinates

Parametric Equations

Chapter 10

Improper Integrals

Chapter 11

Gradients

Differentials

Chapter 17

Chapter 18

Elementary Differential Equations

and Etgen’s Calculus: One and Several Variables

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

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

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- by Delta Software

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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

CHAPTER 1 Introduction 1

Notions and Formulas from Elementary Mathematics

- 121122111222
- rational or irrational

Write 6

27272727

- in rational form p / q
- if any,

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

- 2n − 1 : n = 1,
- if any,

upper and lower bounds for the set S = n

Rewrite x4 – 18x2 + 81 in factored form

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

Solve 8 – x2 > 7x

Solve x2 – 5x + 5 ≥ 1

2x + 3 < 1

4x − 1

- < x x +1
- 1 2 (3x + 2) < 2 − (5 − 3x )
- −2 1 3 x ≥ − x
- 1 (3 + x 2 )

- < x+2 x−1
- 2x − 3 ≥ 1

Solve x(2x – 1)(3x + 2) < 0

Solve 0 < x −

Solve |2x – 1| < 5

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

If |2x – 5| < 1,

- then |4x – 10| < A

x2 − 9

- x+2 > 0
- x ( x + 3) 2

2 1 < 1

- –4) and P1(1,

- 2b) to P1(3a,

- 2) and P1(3,

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

- –4) and P1(3,

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

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

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

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

- 2) and is parallel to the line 2x + 3y = 18

- 5) and is perpendicular to the line 6x – 7y + 17 = 0

- l1: x + y – 2 = 0
- l2: 3x + y = 0

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

- calculate (a) f ( 0) ,
- (b) f (1) ,
- (c) f ( −2) ,
- (d) f ( −5 / 2)
- x2 + 4 x3 x +2 2
- calculate (a) f ( − x ) ,
- (b) f (1 / x ) ,
- (c) f ( a + b)
- f ( x) = 1 − 2 x

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

- if any,
- where ƒ takes on the value 1

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

Find the domain and range for f ( x ) =

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

- 1 Find the domain and range for f ( x ) =

- if any,
- where ƒ takes on the value 1

- ( x + 3) 2

Calculus: One and Several Variables 1

Sketch the graph of f ( x ) = x −

2 Sketch the graph of g ( x) = 3 2 x − 1

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

- if x > 1

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

if x < 1 and give its domain and range

- if x ≥ 1

Is an ellipse the graph of a function

- or neither

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

- or neither

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

- or neither

- or neither

- or neither

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

- (b) length,
- (c) diagonal

x 3 − 2 x 2 + 5x + 1 is odd,

- or neither

- 3x 2 + 2 x + 5 is undefined
- x − x2

- x 3 − 4 x 2 + 3x is zero
- x2 + x − 2

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

- find (a) f + g x

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

- 1 (a) f (b) –g (c) g − f 2
- f ( x) = x + 5 ,
- g ( x) = x

(a) Form the composition of f o g

- f ( x) = x 2 + 1,
- g ( x) =
- (b) f − 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

- g ( x) = 3x 2 + 2

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

- f ( x) = x ,
- g ( x) = x 3 + 1

(a) Form the composition of f o g

(b) Form the composition of g o f

- f ( x) =
- f ( x) = 4 + x 2 ,
- g ( x) =

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

- g ( x ) = 2 x − 1 ,
- and h( x ) = 3x 2

Calculus: One and Several Variables 94

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

- g ( x) = 2 x + 1 ,
- and h( x ) = 2 x 2

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

- g ( x) = ,
- and h( x ) = x 2
- x 3x + 2
- x + 1 and F ( x ) = x + 2 x

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

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

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

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

- if x < 1 if x ≥ 1
- 1 x − 3 are inverses of each other

Introduction

- rational
- irrational
- (–∞,
- 13/9) ∪ (17/9,
- |x – 2| < 3
- lower bound 1,
- no upper bound
- |x – 1/2| < 5/2
- lower bound 1,
- upper bound 2
- (3 – 2x)(9 + 6x + 4x2)
- 0 ≤ A ≤ 4/3
- (x – 3) 2(x + 3)2
- 7 2a ,
- b 2
- m = 3/7
- m = 9/5
- m = –2
- y-intercept: 10
- m = –8/3
- y-intercept: 2
- (–∞,

1] ∪ [4,

- y = –3x – 4
- (–∞,

1] ∪ [1,

- y = –4
- (–∞,
- 1/4] ∪ (2,
- x = –2
- −5 7 x+ 3 3
- ∞) 45
- −3 1 x+ 2 2
- −2 10 x+ 3 3
- −7 17 x+ 6 2

2 5 4 5 − 2 5 − 4 5 ,

- ,
- 5 5 5 5

The point of intersection is P(–1,

- y = –2x + 10
- (–∞,
- –2) ∪ (1,
- (–∞,
- –2) ∪ [5,
- (–∞,
- –2/3] ∪ (0,

0) ∪ (0,

- (–∞,
- –3) ∪ (–2,

0) ∪ (0,

- (–∞,
- –2) ∪ (2,
- (–∞,
- –4/3] ∪ [–2/3,
- (–3/4,
- 1/4) ∪ (1/4,

Calculus: One and Several Variables

- (a) 1/2
- (b) 3/5

(2n + 1)

17π/12,

19π/12

- domain: (ƒ) = (–∞,
- ∞) range: (ƒ) = (–∞,
- domain: (ƒ) = [–4/3,
- ∞) range: (ƒ) = [0,
- domain: (ƒ) = [–2,
- 2] range: (ƒ) = [–2,
- − x3 x +2 2
- (c) 0 1
- (d) 2/41 (c)
- y = − 6x − x 2
- (a + b) 3 (a + b) 2 + 2
- n = integer 2
- domain: (ƒ) = [0,

1) ∪ (1,

- ∞) range: (ƒ) = [–1,

0) ∪ (0,

- domain: (ƒ) = (–∞,
- –3) ∪ (–3,
- ∞) range: (ƒ) = (0,
- domain: (ƒ) = (–∞,
- ∞) range: (ƒ) = (0,
- domain: (ƒ) = (–∞,
- ∞) range: (ƒ) = [0,
- y = 3 – 4x
- y = 2 + sin x

- domain: (ƒ) = (–∞,
- +∞) range: (ƒ) = [2,

- x–y–2=0
- (a) 4/5
- (b) 17/5
- 2 x 2 − 4x − 6 x 2x − 6 (b) x
- x + 26 ,x ≠ 3 x
- domain: (ƒ) = (–∞,
- +∞) range: (ƒ) = [0,
- neither
- x > 0 x
- (a) |x3 + 1|
- 3x 2 + 2 4 x
- neither
- 1 (6x 2 − 1 ) 4

(4x2 + 1)2

- (a) A = 2w2

3x2 + 2

ƒ(x) = x2 – 1

- (b) A =
- (c) A =

2 2 d'5

2 x +1 2

1 (b) 3 + 1 + 2 x

2 4 + x2

- (b) |x|3 + 1

- g ( x ) =
- g(x) = 2x + 5 100

( 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

- not inverses 103
- inverses 104

True for n = 1: 3 ≤ 3

Assume true for n

so the inequality is true for n + 1

- by induction,
- it is true for n ≥ 1

True for n = 1: 1 • 2 • 3 • 4 = 3 • 8

Assume true for n

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

- (n + 2),
- (n + 3) is even

- by induction,
- it is true for all n ≥ 1

+ [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

- by induction,
- it is true for all n ≥ 1

- (d) ƒ(c)

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

- (d) ƒ(c)

- c = −3

- (d) ƒ(c)

Calculus: One and Several Variables 4

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

- (d) ƒ(c)

- c = −2

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

- (d) ƒ(c)

- c = −1

- (d) ƒ(c)

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)

- (d) ƒ(c)

For the function ƒ graphed below,

- (d) ƒ(c)

- c = −1

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

- (d) ƒ(c)

Consider the function ƒ graphed below

Consider the function ƒ graphed below

Consider the function ƒ graphed below

- if it exists

- if it exists

- if it exists
- x→–3
- if it exists

- if it exists
- x → −2
- x→–2

Evaluate lim

- x2 − a2 ,
- if it exists

Evaluate lim

- x 2 − 16 ,
- if it exists
- x 2 + x − 20

x2 − 1

- if it exists

3x − 6 ,

- if it exists
- 2 x→2 x − 4 7 x − 5x 2 ,
- if it exists
- x x→0

x3 − 8 ,

- if it exists

x2 − 3 ,

- if it exists
- 2 x → −1 x + 1

- if it exists
- x2 − 2 x2

- if it exists
- x ≠ 2 f ( x) = 2,

- if it exists
- 2 x − 1,
- x < −1 f ( x) = 3x,
- x ≥ −1

- if it exists
- 1 ,
- x → −1
- x2 + 4 − 5 ,
- if it exists

Definition of Limit 2x ,

- if it exists

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

Evaluate lim

- x−3 ,x 3
- x 3 (4 x + 1) 5x 2
- if it exists
- if it exists

1 − x2 ,

- if it exists
- 2 x → 1 x + 5x − 6
- x2 + x − 2 x2 − 4x + 3
- if it exists
- x 3 − 3x 2 + 2 x ,
- if it exists
- x−2 x →1
- h−2 −2 ,
- if it exists
- x→2−
- x → 1+
- x−2 x−2
- if it exists
- x −1 ,
- if it exists

- 1 + x,

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

- if it exists,
- for f ( x) = 6,
- x = 1 1 − x,
- x > 0
- x + 1,

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

- if it exists,

for f ( x) = 3 x − 1,

- x ≥ 0
- 3x 2 ,

x < 1 Evaluate lim f ( x ) ,

- if it exists,
- if f ( x) = 5,

x = 1 x →1 2 2 x − 1,

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

- lim 5x = 5

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

3 x = 2 5

Give an ε,

Give an ε,

Give the four equivalent limit statements displayed in (2

- taking f ( x ) =

- x 2 − 1,
- x < 3 Evaluate lim f ( x ) ,
- if it exists,

if f ( x) = x→3 ( x − 1)3 ,

- lim g(x) = 3,
- lim h(x) = −4,

evaluate the limits that exist

(a) lim f ( x ) − g( x ) x→c

- [ x→c
- (b) lim h( x )

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

4 ,c = 2

- if it exists
- (e) lim
- g ( x) f ( x)
- (f) lim
- 1 g ( x ) − h( x )

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

- if it exists

- if it exists

- if it exists
- if it exists
- x → 1 3x − 1

Evaluate lim

- 3 Evaluate lim x 2 x + ,
- if it exists
- x x→0

Evaluate lim

Evaluate lim

- 1 1 − Evaluate lim x a ,
- if it exists
- x→a x − a
- 1 1 − Evaluate lim 2 + h 2 ,
- if it exists
- h→0 h
- if it exists
- h 3 − 4h 3 2 ,
- if it exists
- h → 2 h − 2h
- x−2 x3 − 8
- if it exists

x3 − 8 ,

- if it exists

x−2 2 x − 3x 2 (2 x − 1)( x 2 − 4)

- if it exists
- ( x 2 + 2 x − 15) 2 ,
- if it exists

x+5 x → −5 x 2 + 2 x − 15

- x → −5
- x → −3
- ( x + 5) 2
- if it exists
- x 2 + 2 x − 15 ,
- if it exists

( x + 5) 2 ( x + 5) 2 x + 2 x − 15 2

- if it exists

x3 + a3 ,

- if it exists
- x→ − a x + a 4 1− 2 x ,
- if it exists

Evaluate lim 2 x→ 2 1− x Evaluate lim

- if it exists

- 5 2x ,
- if it exists

−6 2x Evaluate lim + ,

- if it exists
- x − 3 x − 3 x→3

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

- (a) lim
- f ( x ) − f (3) x−3
- (d) lim
- (b) lim
- f ( x ) − f (5) x−5
- (e) lim
- (c) lim
- f ( x ) − f (5) x+3
- (f) lim
- x → −3

3 1 (c) lim − ( x − 2) 2 x → 2 x 2 3 1 1 (d) lim − 2 x − 6 x → 2 x

- f ( x ) − f (1) x−1
- f ( x ) − f (1) x−2
- x → −2
- f ( x ) − f ( −2) x+2

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,

- a jump discontinuity,
- or neither

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,

- a jump discontinuity,
- or neither

determine whether the discontinuity is removable discontinuity,

- a jump discontinuity,
- or neither

- 9x 2 − 4 is continuous at x = 2/3

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

- a jump discontinuity,
- or neither

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

x = −1 is continuous at x = –1

- determine whether the 2,

x > −1 discontinuity is a removable discontinuity,

- a jump discontinuity,
- or neither
- x2 ,

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

- x = 1 is continuous at x = 1

determine whether the 2 x + 1,

x > 1 discontinuity is a removable discontinuity,

- a jump discontinuity,
- or neither
- 1 3 2 x ,

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

- x = 2 is continuous at x = 2

determine whether the 2 x,

x > 2 discontinuity is a removable discontinuity,

- a jump discontinuity,
- or neither
- 1 ,

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

determine whether the 1,

x = 4 discontinuity is a removable discontinuity,

- a jump discontinuity,
- or neither
- x −1 is continuous at x = 0

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

- a jump discontinuity,
- or neither

- 1 is continuous at x = 1

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

- a jump discontinuity,
- or neither

Sketch the graph of f ( x ) =

- x + 1 ,

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

− 2 < x < 1 and classify the discontinuities,

- x ≥ 1
- 2 x + 1,

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

x = 1 and classify the discontinuities,

- 2 x − 1,
- x > 1
- x2 − 9 and classify the discontinuities,
- x+3 x+2 x2 − 4

and classify the discontinuities,

Limits and Continuity

- x2 ,

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

x = 1 and classify the discontinuities,

- 2 x ,
- x > 1

Define f ( x ) =

- x2 − 9 at x = –3 so that it becomes continuous at x = –3

Define f ( x ) =

- x2 + x − 6 at x = 2 so that it becomes continuous at 2
- x2 − x − 2 ,

x ≥ −1 Let f ( x) = x + 1

Find A given that ƒ is continuous at –1

- 1 A x < −

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

- then lim (x – c) ƒ(x) = 0
- x3 + 1 at x = –1 so that it becomes continuous at x = –1

The Pinching Theorem

Trigonometric Limits sin 7 x ,

- if it exists
- x x→0

Evaluate lim

- if it exists
- x sin 2 4x 2 Evaluate lim ,
- if it exists

x → 0 1 − cos 3 x Evaluate lim

- 1 − cos 3x 2
- if it exists

Evaluate lim

Evaluate lim

Evaluate lim

- if it exists

- tan θ ,
- if it exists

- if it exists

α →0

- sin α − tan α sin 3 α
- if it exists

θ →0

- sin 2θ θ
- if it exists

Calculus: One and Several Variables

Evaluate lim

- if it exists

- sin 3θ ,
- if it exists
- sin 2θ

Evaluate lim

Evaluate lim

- find lim

θ →0

- if it exists
- if it exists

α → 0 cos α

- t2 1 − cos 2 t
- if it exists
- if it exists

- if it exists

θ → 0 tan θ

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

- graph of ƒ at (a,

Use the pinching theorem to find lim

Some Basic Properties of Continuous Functions

- 1] and meets the following conditions (if possible): ƒ is continuous on [0,
- minimum value ½,
- maximum value 1

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

- 1] and meets the following conditions (if possible): ƒ is continuous on (0,
- no minimum value,
- maximum value ½

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

- 1] and meets the following conditions (if possible): ƒ is continuous on (0,

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

- 1] and meets the following conditions (if possible): ƒ is continuous on [0,
- does not take on the value 0,
- minimum value –1,
- maximum value ½

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

- 1] and meets the following conditions (if possible): ƒ is discontinuous at x = ¾ but takes on both a minimum value and a maximum value

Given that ƒ (x) = x4 – x2 + 5x + 2,

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

(a) 4 (b) does not exist (−∞)

- (c) does not exist (d) 0
- does not exist
- (a) 3 (b) 3
- (c) 3 (d) 5
- (a) −2 (b) −4
- (c) does not exist (d) −2
- (a) 1 (b) 1
- (c) 1 (d) 2
- (a) −1 (b) 1
- (c) does not exist (d) −1

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

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

- (a) does not exist (b) 0
- (c) does not exist (d) 0

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

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

- (a) 1 (b) 2
- (c) does not exist (d) 0
- c = −3 and c'= 0
- c = −2 and c'= 1
- 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

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

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

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

then |(x – 3) – 2| = |x – 5| < ε

- 4 4 (i) lim = 5 x →2 2x + 1 4 4 (ii) lim = 5 h → 0 2 (2 + h) + 1 4 4 (iii) lim − = 0 5 x →2 2x + 1 (iv)
- 4 4 − = 0 2x + 1 5

- then –1 < x – 3 < 1,

2 < x < 4,

- 5 < x + 3 < 7,
- and |x + 3| < 7

then2 < x < 4 and |x – 3| < ε/7

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

- (a) (b) (c) (d)
- −3 16 −3/4 0

(e) does not exist (f) 1/7 (g) −2

- does not exist
- does not exist
- −1 a2
- does not exist
- (a) 1 (b) does not exist
- (c) 0 (d) −1/4

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

- (d) −1/4 (e) 1 (f) −7
- continuous
- continuous
- continuous

removable discontinuity at x = 2/3

- jump discontinuity
- jump discontinuity
- removable discontinuity
- discontinuity of neither type
- discontinuity of neither type
- discontinuity of neither type
- does not exist

removable discontinuity at x = −3

Limits and Continuity 88

- nonremovable,

nonjump discontinuity at x = 2

- no discontinuities
- jump discontinuity at x = 1

ƒ(−1) = 3

ƒ(2) = 5

Since ƒ has a removable discontinuity at c,

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

- lim ( x − c') f ( x) = L'
- lim ( x − c) = L'• 0 = 0

jump discontinuity at x = −2 and x = 1

8/9 101

- −½ 105
- jump discontinuity at x = 1

1/3 108

3/2 110

- limit: ½
- tangent line: y =
- 1 π 3 x − + 2 3 2

If ƒ (x) = x3 – cos2 x,

- then ƒ is continuous,
- and ƒ (0) = –1 < 0,

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

- andƒ (0) = 2,

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

- 0] and for some x in [0,
- impossible 118
- impossible

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

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

- tends to 0

- 1 f ( x + h) − f ( x ) by forming a difference quotient and taking the limit as h tends x+1 h

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) = 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

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

- h → 0

Find ƒ ′ (2) for ƒ(x) = h → 0

- 3 f (2 + h) − f (2) by forming a difference quotient and taking the limit as 2x + 1 h

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,

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

- 2x at the point (2,

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

- 2 x + 1 at the point (3/2,

- x 2 + 1,

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

- x > 1
- 2 x 2 + 2,
- x ≤ 1 c'= 1

- f ( x) = 4 x + 3,
- 3x + 1,

x ≤ −1 Find ƒ ′ (c) if it exists

- f ( x) = c'= –2
- 2 2( x + 1) ,
- x > −1

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

- (8 + h ) 2 / 3 − 4 represents the derivative of a function ƒ at a point c

- h h→ 0

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

- h→ 0 h

Differentiate F(x) = 4x5 – 8x2 + 9x

Differentiate F(x) = 1 +

Find ƒ ' (0) and ƒ ' (1) for ƒ (x) =

find ƒ ' (0) for f(x) = 2x2h(x) – 3x

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,

- ( 2 x 2 − 5) x4

3x 4 + 5

x −1 2 2 1 + 2

- x x

2 x −1

The d/dx Notation

- 3x 3 + 5 y 2 + 2 dy for y =
- 1 dy for y = 5x 3 +
- x 2 + 3x dy
- for y = dx 7 − 2x

d [−2( x 2 − 5x )(3 + x 7 )]

- 5 at the point (–1,

ƒ(–1))

Calculus: One and Several Variables d'dx

- x2 − 5 2
- 3x − 1

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

- (b) ƒ ′ ′ (x) > 0,

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

- d 3y dx

d4y dx 4

dy at x = 2 for y = (x2 + 1)(x3 – x)

- for y = x 3 −
- for y =
- −1 − 5x − 2
- d 2 d2 [x (3x 2 − x 5 )]

2 dx dx

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

- velocity,
- acceleration,
- and speed at time t0 = 4

An object moves along a coordinate line,

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

- if ever,
- the object changes direction

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

- if any,

during which the object moves left

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

- if any,

during which the object is speeding up to the right

- ? Neglect air resistance

A stone is thrown upward from ground level

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

- ? (b) How high will it go

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

What is the height attained by the object

- ? (Take g as 32 ft/sec2)

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

- 003)x2 is the cost function for a certain commodity and R(x) = 60x – (0
- 002)x2 is the revenue function,

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

- (b) by using the chain rule

x + 1 Differentiate f ( x ) =

- x − 1
- 1 1 Differentiate f ( x ) = + 2
- x x

x2 + 7 Differentiate f ( x ) = 2

- x −7

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,

- u = and x = 2t – 5

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

- t + 1 ,
- and t =

- g(1) = 1,

ƒ ' (1) = 3,

- g ' (1) = 2,
- and ƒ ' (2) = 0,
- evaluate (f • g) ' (1)

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

- d [ f ( x 3 − 1)]

Calculus: One and Several Variables

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

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

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

- 1 − cos x

Differentiate y = sec x tan 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

directly toward an observer on the ground,

with a speed of 300 meters per second

d3 dx 3

- (sin x )

dy 1 for y = (1 + u) ,

- u = sin x,
- and x = 2π t
- dt 2

Implicit Differentiation: Rational Powers

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

- dy at the point P(–1,
- –1) for 3x2 + xy = y2 + 3

Express

- d2y dy and at the point P(2,
- –1) for x2 – xy + y2 = 7

2 dx dx

–1) for 2x2 – 3xy + 3y2 = 2

- dy for y = (x4 + x3)3/2
- dy for y = dx

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

Compute

- d 4 1 x + 4
- dx x

Compute

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

- d f ( x − 1)

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

- elastic where ε > 1

Rates of Change Per Unit Time

- looking for dinner,

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,

- staying 4 miles offshore

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

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

- ? V = πr 3 3

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

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

- ? V = πr 2 h 3

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

- ? V = πr 3 ,

Two ships leave port at noon

and the other sails east at 8 miles per hour

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

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

- ? V = πr 2 h 3

A ship,

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

- at noon,
- 40 miles due north of a second ship,

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

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

- 1 meters per second

One side changes at the rate of 3 inches per second

how fast is the other side changing

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