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[solved] hutchinson (lrh2266) - Problem Set 9 - pavlovic - (53505) This print-out should have 25 questions. Multiple-choice questions may continue on the next...


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problem 1-25


hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

This print-out should have 25 questions.

 

Multiple-choice questions may continue on

 

the next column or page ? find all choices

 

before answering.

 

001 1 5. c = 6 , 7

 

6. c = 12 4.0 points 003 4.0 points The function f is defined on [1, 3] by

 

Determine if Rolle?s Theorem can be applied to

 

f (x) = x2 + 3x ? 18

 

x+3 on the interval [?6, 3], and if it can, find all

 

numbers c satisfying the conclusion of that

 

theorem. 2 At which point c in (1, 3), if any, does

 

f (3) ? f (1)

 

= f ? (c)

 

2

 

hold?

 

1. c = 1. c = ?1

 

2. c = ?3, ? 13

 

5 2. no such c exists 3

 

2 3. c = 2 3. c = ?3, ?15

 

4. Rolle?s Theorem not applicable

 

3

 

5. c = ?

 

2 4. c = 7

 

3 5. c = 5

 

2

 

004 6. c = ?3

 

002 1 f (x) = 2x + (x ? 1) 3 (3 ? x) 3 . 4.0 points Determine if the function

 

?

 

f (x) = x 18 ? x

 

satisfies the hypotheses of Rolle?s Theorem

 

on the interval [0, 18], and if it does, find all

 

numbers c satisfying the conclusion of that

 

theorem. Find the value of f (2) when f, g are differentiable functions such that

 

 ?

 

f

 

f (x)g ?(x)

 

= ?

 

,

 

g(x) > 0 ,

 

g

 

(g(x))2

 

for all x, while f (0) = 6 .

 

1. None of these

 

2. f (2) = 12 1. c = 7

 

2. hypotheses not satisfied 4.0 points 3. f (2) = 6 3. c = 12 , 13 4. f (2) = 4 4. c = 13 5. f (2) = 2 hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

005 4.0 points Which of the following functions

 

A. f (x) = 1

 

on [0, 2],

 

x?1 B. f (x) = x1/3 on [0, 1],

 

C. f (x) = |x| on [0, 1],

 

satisfy the hypotheses of the MVT? The derivative of a function f is given for

 

all x by

 



 



 

f ? (x) = (2x2 + 6x ? 8) 1 + g(x)2

 

where g is some unspecified function. At

 

which point(s) will f have a local maximum?

 

1. local maximum at x = 4

 

2. local maximum at x = ?4

 

3. local maximum at x = ?4, 1

 

4. local maximum at x = ?1 1. A only 5. local maximum at x = 1 2. B and C only 008 3. C only 4.0 points Let f be the function defined by

 

4. all of them f (x) = 1 + x2/3 . 5. none of them Consider the following properties: 6. A and C only A. concave up on (??, 0) ? (0, ?) ; 7. B only B. has local minimum at x = 0 ; 8. A and B only C. derivative exists for all x 6= 0 . 006 Which does f have?

 

4.0 points 1. A and C only How many real roots does the equation

 

x5 + 3x + 5 = 0 2. B and C only

 

3. C only have?

 

1. exactly four real roots

 

2. no real roots 4. A and B only

 

5. A only 3. exactly three real roots

 

4. exactly one real root

 

5. exactly two real roots

 

007 4.0 points 2 6. B only

 

7. None of them

 

8. All of them hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

009 3

 

1 4.0 points 1. The derivative, f ? , of f has graph

 

f ? (x) a b c 2. f ? (x)

 

graph of f ?

 

1

 

Use it to locate the critical point(s) x0 at

 

which f has a local maximum?

 

1. 3. x0 = c f ? (x) 2. x0 = a

 

3. x0 = b , c 4. x0 = c , a

 

1 5. x0 = b

 

6. x0 = a , b , c 7. x0 = a , b 8. 1

 

4.

 

f ? (x) none of a , b , c

 

010 4.0 points If f is decreasing and its graph is concave

 

up on (0, 1), which of the following could be

 

the graph of the derivative, f ? , of f ? 011 4.0 points The graph of a twice-differentiable function

 

f is shown in hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

C. f is concave up on (0, 2).

 

1. A only

 

2. A and C only b 1 3. none of them

 

4. A and B only

 

5. C only Which one of the following sets of inequalities

 

is satisfied by f and its derivatives at x = 1?

 

1. f ? (1) < f ?? (1) < f (1)

 

?? 6. all of them

 

7. B only ? 2. f (1) < f (1) < f (1)

 

8. B and C only 3. f ? (1) < f (1) < f ?? (1)

 

4. f ?? (1) < f (1) < f ? (1) 013 5. f (1) < f ? (1) < f ?? (1) On which interval(s) is 6. f (1) < f ?? (1) < f ? (1)

 

012 4.0 points f (x) = x4 ? 2x2 ? 8 4.0 points decreasing? When Sue uses first and second derivatives

 

to analyze a particular continuous function

 

y = f (x) she obtains the chart

 

? ?? y

 

y

 

y

 

x < ?3

 

+

 

?

 

x = ?3

 

4

 

0

 

?3 < x < 0

 

?

 

?

 

x=0

 

1

 

?1

 

0<x<2

 

?

 

+

 

x=2

 

?1 DNE

 

x>2

 

+

 

+ Which of the following can she conclude from

 

her chart?

 

A. f is concave down on (??, 0).

 

B. f has a point of inflection at x = 0. 1. (?? , 0 )

 

2. ( 0 , ? )

 

3. ( ?? , ?1 ) ,

 

4. (?1 , 0 ) , (1, ?) (1, ?) 5. (?1, 1 )

 

6. ( ?? , ?1 ) ,

 

014 (0, 1)

 

4.0 points When the graph of f is 4 hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

8

 

7

 

6

 

5

 

4

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8 6

 

4

 

2

 

?6 ?4 ?2

 

?2 2 4 6 8 10 ?4

 

?6 which of the following is the graph of f ?? ?

 

8

 

7

 

6

 

1. 54

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8

 

8

 

7

 

6

 

2. 54

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8

 

7

 

6

 

3. 54

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8 4 8

 

7

 

6

 

4. 54

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8

 

8

 

7

 

6

 

5. 54

 

3

 

2

 

1

 

0

 

-1

 

-2

 

-3

 

-4

 

-5

 

-6

 

-7

 

-8 4

 

?4 4 8 ?4 4 8 4 8 ?4 4

 

?4

 

?4 015

 

?4 5 4.0 points A polynomial function f has the properties

 

(i) f (?7) = 0 = f (?4),

 

(ii) f ?? (x) > 0 on (?7, ?4). Which of the following statements is a consequence of these properties?

 

1. f (x) ? 0 on [?7, ?4] 4

 

?4 4 8 2. f ? (x) ? 0 on (?7, ?4)

 

3. f (x) = 0 on [?7, ?4] ?4 4. f ? (x) ? 0 on (?7, ?4)

 

5. f (x) ? 0 on [?7, ?4]

 

4

 

016

 

?4 4

 

?4 8 4.0 points If the graph of

 

f (x) = ax3 + bx2 + cx + d

 

has a local maximum at (0, 1) and a local

 

minimum at (2, ?3), compute the value of

 

f (1). hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

1. f (1) = 1 019 (part 1 of 2) 4.0 points 2. f (1) = 2 Lete f be the function defined by 3. f (1) = 3 f (x) = 4. f (1) = 0 017 4.0 points Find all values of x at which the graph of

 

1

 

f (x) = x3 ? 2x2 + 7

 

3

 

has a point of inflection.

 

4

 

3 4

 

3 2. 2x

 

?1 x2 3. ? (x2 (x2 2x

 

? 1)2 2x

 

? 1)2 (x2 (x + 2)

 

? 1)(x ? 1) 020 (part 2 of 2) 4.0 points 5. x = 2 (ii) Find the interval(s) on which f is increasing. 6. x = 0, ?4

 

018 1. (??, 0)

 

4.0 points Find the interval(s) where

 

4 4

 

x ? 2x3 ? 2x2 ? x + 8

 

3

 

is concave down.

 



 

1

 

1. ?1, ?

 

4

 



 

  1

 



 

2. ??, ?1 , ? , ?

 

4

 

 1 

 

3. ? , 1

 

4

 



 



 

1 

 

, 1, ?

 

4. ??,

 

4

 



 



 

1 

 

5. ??, ? , 1, ?

 

4

 

f (x) = (x2 5. ? 3. x = ?2 x

 

? 1)2 1. ? 4. 2. x = 0, 4 4. x = x2

 

.

 

x2 ? 1 (i) Which of the following functions is the

 

derivative of f ? 5. f (1) = ?1 1. x = 0, ? 6 2. (??, ?1), (?1, 0)

 

3. (?1, 1), (1, ?)

 

4. (?1, ?)

 

5. (??, ?1), (?1, 1)

 

021 4.0 points Let f be the function defined by

 

f (x) = x ? 2 sin(x), 0 ? x ? ?. On which interval(s) is f increasing? hutchinson (lrh2266) ? Problem Set 9 ? pavlovic ? (53505)

 

1.  1

 

0, ?

 

3  4. x = ? 3. 1

 

0, ?

 

3 1

 

21/2 5. x = ?1 2. (0, ?)

 

 7  ?  2

 

?, ?

 

3  024 (part 3 of 4) 4.0 points

 

Locate all the local maxima of f . 

 

1 5

 

?, ?

 

4.

 

6 6

 



 



 

1

 

5.

 

?, ?

 

3

 

 1. x = ? 1

 

21/4 2. x = ?1, ? 022 (part 1 of 4) 4.0 points 1

 

21/4 3. none of these Let f be the function defined by

 

f (x) = x2 + 1

 

x2 on (??, 0).

 

Find the derivative of f

 

2x4 + 1

 

1. f ? (x) =

 

x3

 

2(x4 + 1)

 

2. f (x) =

 

x3 4. x = ? 1

 

21/2 5. x = ?1

 

025 (part 4 of 4) 4.0 points

 

Determine the absolute minimum value

 

minf of f . ? 3. f ? (x) = 2(x4 ? 1)

 

x3 1. minf = 4

 

2. none of these 2x4 ? 1

 

4. f (x) =

 

x3 3. minf = 1 4x3 + 1

 

x4 4. minf = 3 ? 5. f ? (x) = 023 (part 2 of 4) 4.0 points

 

Find all the critical points of f .

 

1. x = ?1, ? 1

 

21/4 2. x = none of these 3. x = ? 1

 

21/4 5. minf = 2

 


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