BCPreCalc Final Review Problems

8/10/2019 BCPreCalc Final Review Problems

1/11

,,' X

7 5 '

'2.

1m ' 2 - 25 =

, x- iS X

(A) -1

B)

0

1

(C) 10

D)

1

E) does

not exist

3 liII}, x

=

x ~

(A) 0

B)

1

(C) , ;

(D) ;

(E)

does not

exist

1

.

sin

2x

4 1m x cos x =

x .... o

(A) -1

B)

0

(C) 1

(D) 2

(E)

does

not exist

5 [ x) = {X + 1,x 0)

(B) (x

I)

(E) (all

real

numbers)

11

The

inverse ofy = lnx - 1 is

(A) y = e

x

-

1

B) y = e

-

1

(C)

y = e +

1

(D) y =

+1

- 1

E) y = e

x

+

1

12 Find the slope of a line parallel to

the

line

with equationy- 2x = 7.

(A)

2

(B) 0

(C) 1

D) 2

E) 7

13

For

what value(s)

of

x is [ x) =

:2 =

discon

tinuous?

(A) x

=

0

B) x =

1

(C) x = -1

(D)

x

= 1

and x

= -1

(E) all real numbers

14

[(x) =

is

continuous for all

real numbers

x

EXCEPT

(A) x = 0

(B) x = 1

only

(C) x = 1 and x = -1

(D) x = -1

only

E)

x = 2

15

lim .

1 -

~ s x

=

x-+3

(A) ~

B) ~

(

C) ~

211

(

D) 3(1 -

V3)

211

E)

0

Chapter

Assessment

55


2/11

~ ~

.........

........

........

........

16 1

sin 3x

.

Im =

x

o 7x

(A)

(B)

i

(C) 3

(D) 7

(E)

does

not

exist

. 2 x

17 hm

~ 4 =

x

2 x -

1

(A) -"2

1

(B) -4

(C)

(D)

(E)

does not exist

18

The slope of 3x + 4y

=

7 is

4

(A)

-3

3

(B) -4

(C) t

(D) i

(E)

4

9

1

X - 4x + 4 =

.1 m 4 2 - 1

x ..... oo

x

(A)

(B)

1

(C) 4

(D) 8

(E) does

not

exist

20

Find the equation

of the vertical

asymptote

of {(x) = 4xx+ 8

(A)

y

=

(B) y = 2

(C) x = 2

(D) Y

= 2

E)

x =

2

-:' ,' '-

.

t f i i : i ] 5 t ~ F g : ; ; ; t i r f t i t S ~ n d C o n t i h t . i i t y

21 Find the equation of the horizontal asymp-

x

tote

of

y

= X -

2

(A)

y =

0

(B)

x =

(C)

y

=

(D)

x

=

1

E ) x = l

{

X + 4 x < 0

22 Find {(2)

for

{(x) = 3 _ x, x

;: ::

O

(A) 1

(B) 3

(C)

4

(D) 8

(E)

does not exist

23 Which of the

following graphs shows a

nmc

tion that

is continuous for all real numbers?

(A)

y

= ~ ~ ~ ~ X

(B)

y

- - - - - - - - - - ~ - - - - - - - x

(C)

y

I I 1 _X


3/11

D)

y

~ ~ 4 ~ __ X

E)

y

~ ~ ~ X

24.

lim

Vx=2 =

, x-+oo

X -

2

A) 2

B) 0

(C)

1

D) 2

E) does not exist

1 1

. :X- 3

25. hm

3

=

x-+3 X

(A) 00

1

B)

- 3

1

(C) - 9

D) 0

E)

1

Free-Response Questions

No calculator is allowed for Questions 1-3.

{

X

+ a x l

limf x)

x

-->

2

f

3

)

-3

-1 34

6. Sketch a

function

f(x) with all of the follow

ing properties:

f(x)

is odd.

f(1) = 2

The

x-axis is a

horizontal

asymptote.

Chapter Assessment

57


4/11

If f(x) = X4

-

4x,

evaluate lim f(x) - f( -1)

x

..... l

x+1

(A) 8

(B) 0

(C) 1

(D) 2

(E) 4

8 f(x) = (x -

1)3.

Find f (0).

(A) 6

B)

3

3

(C) - 2

1

(D) - 2

E)

9 f(x) =

In sin

x).

Find

f ( ).

(A)

- I n

2

B) V2

2

(C) 0

(D) 1

(E)

undefined

10 y sin 2x - x. Find

y,

.

(A)

3

(B) -1

(C) 0

(D) 1

(E)

undefined

11 y =

3 ta n

2

(

t

. Find y (11').

(A) 1

(B) 8 V3

3

(C) 9

(D) 8V3

(E) 27

12

y =

In(e

x2

-

1

). Find y (l).

(A)

0

(B)

(C) 1

(D) 2

(E) undefined

13 y = e

x3

Find y (1).

(A)

3e

2

B) 3e

(C)

e

D)

i

E) e

3

14

For what

values of x

continuous?

(A) x =f 0

(B)

x

> 0

(C) x > 1

(D)

x

;:::

1

E) all real x

is

f(x); ;' s i ~ x

15

For

what

values

ofx

is

f(x) =

Y lnx)2

differ

entiable?

(A) x > 0

B)

x ;::: 1

(C) x =f 0

(D)

x >

0

and x =f 1

E)

all

real

x

16 Find

2

or xy

+

x

-

y = 2

when

x = 0:

(A)

2

(B) -1

(C) 0

(D) 1

(E) 2

17 A

particle

moves along .the y-axis with a

motion defined by the

equation s(t)

= t for

t

>

O Find

the

velocity

at t

=

2.

(A)

(B) 0.0767

(C) 0.693

(D) 1

(E) 2

18 Find the derivative of f(x)

=

x

2

- I n

x.

(A)

2x

- 1

(B) 2x - I n x

1

(C) 2x - : x

(D)

x - k

E) 2x x- 1

19

Find the second derivative of Y x

2

+ 1).

3

(A) x

2

+

l)-Z

1

(B) x

2

+ l)-Z

1

(C) x(x

2

+ 1fz

3

(D) 2x x

2

+

l)-Z

3

E)

2(x

2

+ l)-Z

Chapter Assessment

9


5/11

20.

Find

y

fory

= x n x - 3x.

(A)

k -

3

(B)

l l nx

(C) I n x - 2

(D) k

(E) k - 2

21. Find the slope of the

tangent to

the graph of

y = Vex + 2) at

x =

-1.

V2

(A) 2

1

(B)

-2

(C) 1

(D)

Y

(E)

undefined

22. Write the

equation

of the

line

tangent to

the

graph of

x

= y2 +

4 at

the point

(5,

1 .

(A) 2y = x - 2

(B)

y =

x

-

3

(C) 2y = x

+

9

(D)

y = x 7

.

(E) x

=

5

23. Write the

equation of

the line tangent

to

the

1

graph

off(x) =

x

2

+ 2) at

x

= 0,

(A)

y =

1

B) x =1

16 1

(C) y

= -SIx +

2

(D)

y = 1

E) y =

--i-

x

+

1

24. Find the equation of

the

line

tangent

to

y = arctan x atx = 1.

(A)

x -

y= 1 -

'IT

(B) 2x -

4y

= 2 -

IT

(C) 2x 4y =

'IT

- 2

(D)

4x - 4y =

4 -

'IT

E)

x -

2y =

2 -

IT

96

Chapter

4

The

Qerivative

25.

f(x) = x\' x > o. I f g(x) is the inverse of f X),

find

g (2).

(A) -4V2

1

B) - V2

(C) :'V2

V2

(D)

--s-

E)

-s

For questions

26 and 27,

use

x(t)

=

2t2 and

yet) = 4t - 1.

26. Find the slope of the line tangent to the curve

at t

=

1.

(A)

4

(B)

-

(C) 1

(D) 4

(E) undefined

27.

Find the

equation of the line

tangent

to the

curve

at

t =

1.

(A)

y =

x

+

1

(B) y = 4x - 5

(C) Y = i x +

(D)

y = -4x + 11

E) x= 2

Free Response Questions

A graphing calculator is required for

some

ques-

tions.

1.

(a)

Sketch the

graph of

y = In x on the

inter

val

[i

e ].

(b) Write the range of the function 0

interval.

(c) Find the average rate of

,

y = In x on the interval

[

e

1

I OG

(d)

Estimate

the

slope of

the graph

L .

Y

= In

x

at

x

= i

x

=

1, and x

=

e.

(e)

Plot

the slopes at

the

corresponding

x-

values on a separate graph.


6/11

; 2 ~ Use

the standard

definition of the derivative

to find f x) for:

2

a) { x)

x

1

b)

rex) = x

2

-

2x

. d sinx)

3 Use the fact that dx = cos x and

d cosx) . fi

h . . f

dx

=

- smx

to

n t e envatIVe

0 :

a)

tanx

b) cotx

c) sec x

d) csc

x

{

-x, for x

?:

1

4 GIVen

r x)

= k

< 1:

x

, lorx

a)

Find

the value ofk

so that rex)

willbe

continuous

at

x =

1

b) Using the value ofk

found

n

part a),)s

r x)

also differentIable at x = I? .

5. What is the minimum initial veloc:ity nee


7/11

. o f e q t l ~ l size are

cut

off

the

COrners

an 8

X

10 piece of c a r d b o a ~ d The sides

. .

are

then turned

up to form an open. box.

.What is

the

largest possible volume of the

box?

A) 1.472

B) 1.5

(C) 23.986

(D) 52.50

E)

52.514

F""i -Response Questions

Xgraphirig calculator

is

required for

some

ques-

1. A

rectangle

is

inscribed above the

x-axis

in

the parabola

y = a

2

- x

2

Find

the

area of the

largest

possible

rectangle.

.Chapter ssessment

Multiple-Choice Questions

A

graphing calculator

is

required [or some ques

tions.

1.

Find the value of c

guaranteed

by the Mean

Value Theorem for

[ x)

= x .: 1 on the inter

val

[3,5].

(A) 1

+

2V2

B) 2V2

(C) 1 + V 2

(D) 2

E)

1 -

2V2

2.

Which of

the following is

a hypotheses of

Rolle s

Theorem

not satisfied by

[ex

= x .: 1

on the interval

[3, 5]?

(A)

[is

continuous on [3,5].

(B)

[is

differentiable

on

3,5).

(C) . [(3) = [(5)

(D) f (c)

=

0

E)

f (x)

is defined on

[3,5].

3.

Write

the

equation of

the

line tangent to

[(x) = x .: 1 at x = 3.

y

2. A rectangul ar plot is to be

fenced

n using the

side of an existing barn

that

is

50

feet

long

as

one

side

of the plot. Two hundred feet of

fencing are available

for the

.other three sid,es

of the plot. Find the

largest possible area

that

can be enclosed.

3. Twenty feet of

wire

are to be used to create a

wire sculpture that consists

of

a square and a

circle.

Find

the

largest

number of

square feet

of

area that

can be enclosed

by the

square

and the circle

(A) x + 2y = 5

B)

x

- 2y= 1

(C) 2x

- y = 5

(D) 2x +

y

= 7

E) y = 1

4.

Which of the

following

are

true about the

function

[ x)

=

1

?

I [ x) is

an

odd

function.

II

[(x) has a

horizontal asymptote.

III [(x) has a relative maximum at x =

1.

(A) I only

B)

II only

(C) I and II

only

(D) II and III

only

eE) I, II, and III

5.

Find

the value of c guaranteed

by

the Mean

Value Theorem for [(x)

= x 2 ~

1 on

the

inter

val [0, 1].

(A) 0.475

(B) 0.486

eC) 0.488

(D) 0.577

(E) 1.000

Chapter Assessment

12 1


8/11

6.

Find

the

equation of

the

line perpendicular

to the line tangentto y = 2x - x

2

at

x =

l-

(A) 4y - 3 = 0

(B) 4x

+ 4y

- 5 = 0

(C)

2x - 2y -

1

=

0

(D) 4x -

4y +

1 = 0

(E)

2x + 2y -

1

=

0

7. Find

the.

value of c guaranteed by

Rolle s

Theorem for [ x) = x

-

x

3

on [0, 1].

A)

0.638

(B) 0.6

(C)

0.577

D) 0.5

(E) Rolle s Theorem does not apply.

8. Find the absolute

maximum

of [ x) =X x

3

on the

interval [ -1,

1].

(A) 0.375

(B) 0.384

(C)

0.385

(D)

0.577

. E) 0.6

Find the absolute maximum of

f x)

=

x

3

on

the interval [-1, 3].

(A) 1

(B) 0

(C) 3

(D) 27

(E) none

10. How many critical values are there for

[ x)

=

x

3

-

eX

(A) 0

B) 1

(C) 2

(D) 3

(E) 4

11. How many

points of

inflection

are there

for

[ x) = x

3

- eX

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

12.

How

many critical values are there for

[ x)

=

X} -:-

3x

5

?

(A) 0

}3) 1

(C) 2

(D) 3

E) 4

122

Chapter 5

4

Applications o f the Derivative

13. How

many points

of

inflection

are there

for

[ x) = x

3

- 3x

5

?

(A) 0

.

B)

1

(C) 2

(D) 3

(E) 4

14. I f [ x) = x -

3

+ x

2

,

which

of the follqwing

is

true about

[ x)? .

(A) The function

is

not continuous and not

differentiable at x =

3.

(B) The function

is

not continuous

but

dIf

ferentiable

at x

= 3.

(C) The function

is

continuous but not dif

ferentiable

at

x = 3.

(D) The function is continuous and differen

tiable at x = 3,

(E) f '(3)

=

6

15.

Which

of

the

following is true

about the

function

[ x)

=

9x

3

-

In

x?

A) Its

domain is all

real numbers.

(B)

It

is always concave up.

(C) It has no

relative minima.

(D)

It

has one relative maximum.

(E) I t

has one point

of inflection.

16. Find the

interval(s)

where

[ x)

=

4

+ 3x

2

-

2

is greater

than zero.

(A) (- 00,

-1.225)

and (0, 1.225)

(B) ( -00,

-

~ and

0, ~

(C)(Yz, -1) and

(1,

Yz)

(D) (-0.707,0.707)

(E) (-1.414,

-1)

and

(1, 1.414)

17. I f [(x) =

4

+ 3x

2

- 2, find the interval(s)

where { (x) is greater than zero.

A)

(-00,

-1.225)

and (0, 1.225)

(B) (-00,

- ~

and

0, ~

(C) - Yz, 1) and (1, Yz)

(D)

(-0.707,0.707)

(E) (-1.414, -1 ) and (1, 1.414)

18.

I f [ x)

=

4

+ 3x

2

-

2, find the interval(s)

where [ (x) is greater than zero.

A)

(-00,

-1.225)

and (0, 1.225)

(B) - O O , - ~ ) and ( 0,

~

(C) - Yz, -1)

and

(1, Yz)

(D) (-0.707,0.707)

(E)

(-1.414,

-1)

and

(1, 1.414)


9/11

i I f f x)= x sin x

2

, which of he

following is

true?

A) Thereis a relative

maximum

at x = o.

B) There

isa

relative

minimum

at

x =

O.

C) There is no point of inflection at x = o.

D) fC21T) = 0

E)

reO

= 0

Find the coordinates of the point oJ 1 the curve

y

=

t th t

is

closest

to

the point withcQordi

nates 0, 3).

A) 1, 1)

B) 0.329,3.036)

C) 0.25,4)

D)

t,

3

E) There is

no

closest point.

21. Which of

the

following are

true

for

the

func

tion [(x) =

3x

3

- x?

I

[(x)

is an

odd

function.

II [(x)

has

one

relative maximum

and

one

relative

minimum.

III Its

point of

inflection is at x = o

A) I

only

B)

II

only

C) I and II

D) I and III

E) I, II, and III

22. Which

of the

following are

true for the

func-

3x

2

tion[ x) = x - 1?

I

Its

domain is

all

real

num.bers;

II Its range is

y

; 0).

III

It

has

two relative

extrema.

A) I only

B)

II

only

C)

III only

D) I

and

III

E) I, II, and

III

23. What is

the

relative mInImUm value of

[(x) = x

- In sin x))

on the

interval 0,

1T)?

A) 1.132

B) 1.136

C) 1.768

D) 4.798

E)

7.415

24. [(x)

=

x

2

- 3x

3

has

a point

of inflection at

A) x

= 0

B) x =

C) x

=

D) x =

1

3

E) There is no

point

of inflection.

25. The

graph

of

[ x)

= lnlxl

has

A)

domain

= x >

0)

B)

range =

y

> 0)

C) range = all real numbers)

D) symmetry with the respect to the origin

E) a

vertical and

a horizontal asymptote

Free-Response Questions

A graphing calculator is required [or

some

ques

tions.

1. a)

Find the

value

of

c

guaranteed by th

Mean

Value

Theorem for

y

=

cos x

o

the interval [ 0, ~ ].

b)

Draw

a sketch of the graph of y = cos x o

the interval [ 0, ~ ].

Draw

the line

joinin

the

endpoints

of

the

graph

and a

tangen

line at the

value

of

x found

in part

a).

2. Given

the graph of [ (x) on [-2;

3] as show

y

( (x)

-4/ )

X

-2

a)

Find the critical

values

of [(x) and iden

fy

each critical value

as

a relative max

mum, a relative minimum, or neither.

b) Find

the

intervals

where

[(x) is increa

ing.

c) Find the points of inflection of [(x).

d)

f [ 0) =

1, sketch a graph of

[ x)

[-1,3].

Chapter Assessment

2


10/11

f . .

plei-GhQllce uestions

< L U I ' L L ' ' ' '' talciilator is required {or

some

ques-

1.'

The lower sum of { x)

=Vx

on

.the interval

[0, 1] with four equal

subintervals

is

(A) 025

B) 0.518

(C) 0.667

(D) 0.768

E) 3.073

2. The

limit of

the right-hand sum lim i

n -->00

( n

n )

3] represents the area of

which

function

on

which

interval?

(A) { x) = x

3

0n [0, 1]

(B) { x) = - \- on [0,

1]

x

(C) { x) = x

3

on [1,2]

(D) { x) = x

3

on [0, 2]

E)

{ x) = x

3

on [0, n]

3. The

lowel'

sumoff(x) = -;- x-

1)2

+

l o n

the

interval

[0, 2] with

four equal subintervals

is

(A)

B) i

(C)

(D)

(E) 2

4.

Which of

the

following is

true

for

{ x) = 4 - x

2

on the

interval

[0, 2]

with

n

equal

subintervals?

(A)

the left-hand

sum =

the

right-hand sum

(B)

the left-hand sum> the right-hand

sum

(C) the

left-hand

sum < the right-hand sum

(D)

the area

under

the curve>

the

left-hand

sum

(E)

the area

under the

curve

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BCPreCalc Final Review Problems