Test

Big 10 Unit 1 Review Instructions: In the box below are the numbers O - 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number exactly once!

,

V&:

I

/

I

4

>

I

7

Use the graph of J(x) below for all questions.

I

J(x)

4

_/.

_J

X

l _t

-1 -

3

I

5

4

71

I

t -18 I

6 j

-

1. Evaluate lim J(x).

QI 2. Evaluate lim

x-t4

x2

-

X -

16 4

(x ~)(x f ~ = C

)(+4

= '-1

+4

=@

-U)

x-3 - h a v e a vertical asymptote?

3 · For what value of x does g(x) =x-2-- -5x +6

)<, • Yi-::. '

',,f Y 'J :: 5 kx 2 - 6 x < 2

4. For what value of k is h(x) = { x 4

+ 10

~.:. 'J, 'f..1,

x 2: 2 continuous at x = 2?

t)(2..-l ='-lx.--tlO

fl.)( 'l-_4 -.,c =- IO -t t, x( k)(-~) -= 1,

5. For what value of x docs J(x) has a removable discontinuity?

0

=2

( 2 ,~ -"f) -;

't-

.2 k- 4 =..k -<\L::.J2.

[,~ ~0

7

)Y]

Big 10 Unit 1 Review

f(x )

4 3

2

I

•

1

---I--

6. Evaluate lim [2f(x) x---+2

2

1

-1

3

j

4 I

5

6

I I I 7

-;

X

8

I

+ 5].

?•:)._-,.S 4 -1f -=-Ii}

t5

. x 2 + ln(x) - cos(x) 7. What 1s the horizontal asymptote of the function m(x) = ?

2 ex+x-1

c)( ~o> 8. If n(x) is a continuous function, how many riots ar/guar~teed l>y the Intermediate Value Theorem on (0, 12)?

[I]

9. Let p and q be the functions defined by p(x) = cos(x - 2) + 6 and q(x) function that satisfies p(x) :=; r(x) :=; q(x) for all x, what is lim r(x)?

= (x -

X---+ 2

Co.({a) t6

1 10. Evaluate lim f(f(x)) x---+I

+ f(x) .

{;~ .(( ~)

t2.

(J;i-='2.

Jtlw

0

7

<7

2) 2 + 7 . If r is a

Cv

Big 10 Unit 2 Review Instructions: In the box below are the numbers O - 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number exactly once!

I% I

7

E7£ 7 K "JV ,X:Jt:L

8

Use the graph of J(x) and the table for g(x) below for all questions.

4

-

l

J(x) C

-

7

·1 .--

3

-

I

J

• 1

~_._-

f

-

-

X

1

-1 -

2

3

3

1. Evaluate Jim (

4

5

--j-

+ h)

2 -

9

-;-

a

I

6

I

7

'

.

X

g(x) g'(x)

0 2 -1

1 3 -3

2 5 -2

5 1 4

l.9-+ ~-)1.J),(~ -f")/ ,70) .h /, --~ ( f

,

:hf- ·52

(J

h-tO

I

I

i /

h.

--

2. What is the estimate of g'(6) using the average rate of change of g on [5 , 7]?

3. If h(x)

= 2../x + x 3 , find

h(>c::)-: J ·'IC .z.-t

h

the value of h'(l) .

X

1

l ~) = it + ? 'F-2

her) = I -+ 3 =- Oil

4. What is the slope of the line tangent to the graph of k(x) at x

k '{s-)

==

/t'{s).:

+ f /y) :J ( x)

7'{>o) · fr><)

4. J 4

/.

)i' (s) _: g--, , =-

I

[!)

5. What is the instantaneous rate of change of m(x) at x

' , ( -'l) -2- -\-

5 . (-1)

s .=-[j]

= 5 if k(x) = g(x) • J(x)?

= 2 if m(x) =

;~:~?

7 5 -1

Big 10 Unit 2 Review

4

J(t )

I

- 1----+------

1 3

•

2 -

I

I

-

I

·

I

'

-1

0>-------e

·- -- -

I

I

I

3

-1---- ; - - - ~ -

I

4

5

-----4

,-

I

-

7

6

X

I

8

X

0

1

g(x) g'(x)

2

3 -3

-1

2 5 -2

5

7

1

5

4

-1

6. Evaluate J'(7).

[Q] 7. What value of c does lim f(x) exist, but J(x) neither continuous or differentiable at x x-+c

= c?

\c-

8. If n(x)

= 4ln(x) + 3x, find n'(2).

V\ l (

f

l

9. If p(x)

s

).}

=- L/ • J_

T

J

](

( l)

L(,

J_

2,.

-i-S .~ l-t 5

= 3xcos(x) + 2x, find p'(O) .

r' C~)::

3 • ({ o 5

-y:

-

~S,\n

Y)

-f -z_

<-

- 3 't ., 5 ;h ~) -t} [ {OJ (y.)) i 10. Find the values of a and b such that h(x)

=

ax 2 + 7 { 2x + b

x

<l

x 2'.

1

-=-CD

differentiable at x

is the sum of a and b?

°' "-'2.-+1:;;: '1.

C\ ;-7::: '-,

b -+6

)(-t

~s

= l.

What

Big 10 Unit 3 Review Instructions: In the box below are the numbers O- 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number exactly once!

[ sY21/zf %~ V /2 f

/

7

7

7

fr

X

7

W/AJ

Use the graph of f'( x ) and the table for g(x ) below for all questions.

f'( x )

4 -t---~----'------+->------

3

2 --+---,/---+--~~ 1-X

1. If h(x)

= f(g(x)),

4 2 3

2 3 2

1 -1

X

g(x) g'(x)

-8

7 1

9 2

-3

3

find h'(9) .

.(''( ~(~· j'(,o

BJ

r'(2), 3 ; 2. If j(x)

d.

= e2x- 2 + 2ln(x 2 + x), find j'(l) .

e2

'I~

d l(

2-+ , l (

"

-x

-i.

3. Consider the curve xy - 2x

y::. 2'1<-<'-\ -

"

-t)(J

=

-__ze.,

21< - z

-+

,, ( 2.., -lo,) ""' ,.

L

d ¥ -tk' -9. Evaluate d~ at the point (1, -7).

£1. l~...:t.:.q'

_ G,_---t -t) (2 v-"1) J ~-r- =

d.y C

q U-J - )("2

4 - - · t:-1 / ::;,t]J 4. If k(x) = g(x) - f'(7x) , what is the slope of the line tangent to the graph of k(x) x = l?

X

k'( 1c,) _- t ::

f(

1/) · q f{ 1.,)-t

C- 2 ) (

lt

-l I(!'I.) • j (K)

1) t c-1) C1)(-z)

=-(Q

g(3x - 2) 5. If m(x) = , what is the instantaneou rate of change of m(x) at x = 2?

4:z; l-2-:.-'i

ln'[icJ: 3--J{3-x-Z)1Lfy -j(s'i<-Z)(y) (_Lf ~-i

q

,g _ ~·l-f btf

Big 10 Unit 3 Review

!'(±)

I

43

- t -- i - - -

21-

-,C-4--I

I

-1 -

6. If n ( x )

I

2

1-

4

3

I

5

i

I

6

X

g(x) g'(x)

1 -1

-8

2 3 2

7 1 -3

4

2 3

2

= 3x 3 + 2x 2 - 3x - 1, evaluate n 111 (6).

~'(>--')~ ~--,c 2-+4'l<-5

I\"(~)-:-'-' 'i(-+ h'•' (~") =- l/

4

h ,n(, ):::- {[jJ

7. Let p and s be functions that p(s(x))

= x for all x. If p(5) = 3 and p'(5) = ~- Find s'(3).

P( S ( '»)) ::

f ( s) =<!> 8. If q(x) = arcsin(x)

+ (2x -

y

(l)

1)3, find the instantaneous rate of change of q(x) at x = 0.

J

9. If r(x) = sin 2 (3x), find the slope of the line tangent to the graph of r(x) at x = ~-

r ' (\:):: l v- ,(

S11' ( :?x)-

D = ::I.. S/iJ • (

10. Consider the curve y 2 + 8x 2

ci

(a~c:1.,.). 5 )- (a J

=

d2y

[5 7 1

d~

ti)- =@]

1. Find dx 2 at the point (0, -1).

~2-::-S-')(¾I

-

6

9

2 3

Big 10 Unit 4 Review Instructions: In the box below are the numbers 0 - 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number exactly once!

l W: £

.c

I

Y';Z.,--- X ,.6 , . --% - Y I

'

,g.----

•

Use the graph of j(x) and the table for the differentiable function g(x) below for all questions.

4-

J(x)

I

1 I

-I '

3-

I

I

t-+--

2 j

1

- ---1

I

I

I

2

l. Evaluate Jim

x-+3

I

3

I

4

5

X

6

&--

X

g(x) g'(x)

0 2 -1

g(2x + 1) - 5 X2 - 9

7

5 3

5

4

9

le !~

--I..'~ : '

i '\ ')

2. Approximate the value of g(6) using the tangent line to g(x) at x

= 5.

3. If the position of a particle moving along the x-axis is given as x(t) t 0 is the particle at rest?

vet-)-=

3 0 -1

1 3 -3

= t 2 - 4t + 5, at what time

2+--Y

1 4. If the side length of a cube is i~creasing at a rate of inch per second when the cube has a 3 side length of 2 inches, at what rate is the volume of the cube increasing with respect to time when the side length is 2 inches?

d

V

en-

5. Suppose a particle is moving along the x-axis with velocity v(t) = 2t - 6t + 5 where t What is the only integer time, t > 0, where the particle is slowing down?

2

0.

-9-t

Big 10 Unit 4 Review

f(x) 4-3 , -----,--2

1 -- -

-1

i

X

I

2

3 4 1- --

1 3 -3

0

X

2

g(x) g'(x)

-1

7

3 0

5 3

5

-2

4

9

6. A water tank is filling up with water at a rate of w(t) = t 2 - t . What is the rate of change of the rate at which the water is filling up in the tank at t = 5 seconds?

t---1 /0-1 7. Given that f'(l) = 4, approximate the value of f(3) using the tangent line to f(x) at x How much of an overapproximation is your estimate of the true value o f f ~

lj - B ::: L/ { K -

1)

= ~Y. -Lt-ti

4)(-

8. If the velocity of a particle is given as v(t) =

t

= 1T.

s

ecos(x),

/l

-?

=

1.

=W

q - 5 -=-~

find the acceleration of the particle at

9. The area of an irregular shape, A, with given side length S is given the the equation A= S 2 - 2S. When the side length S is 3 feet, it is known that the side length is increasing at a rate of 2 feet per second. What is the rate of change of the area of the shape with respect to time at the instant S = 3 feet?

d:s. =- :z d-+-

A =-s 2:_z5 2.s - 2 10. Evaluate lim

x-+O

x2

+ lOx

. ( ) . sin 2x

.A-

&-

~~-+ lb

::(9;

Big 10 Unit 5 Review Instructions: In the box below are the numbers O - 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number

-,L/. __--_ ~£__--:--:-------..£...-- - - --

r-ex_a_c_tl~y,L.L:...... n_ce_!_ _

lff V '.i_L :f/ 7 Y

VA7 % Fl

- ----,,.- ----::...-----,;i~- -~/

t

l

Use the graph of J'(x), the derivative of J(x), for all questions.

2 - J'_ (x) 1

1 -1 -

7

2

I

5

3

9

6

7-

'

I

-2

-·-

1. For what value of x on O < x

< 9 does f(x) have a relative maximum?

2. What i ~ of xis included in the interval of where the graph of f(x) is both increasing and concave up?

CD

3. Let h(x) = x 2 - 4. On the interval (1 , 5) , what is the value of c guaranteed by the Mean Value Theorem when applied to h(x)?

t~-l

4 4. For what value of x does g(x) = x

B'( \-)::: S){2: q \l( )c).:::.

J

5. For what value of x

3

-

6x

2

+ 8 hav

-=--3

of inflection?

)L)(

'b't-l2.

~ -0 ft:, V-=- ~L ',- -=:.) does the curve x + xy = y ave a

v1 -t .J

horizontal tangent?

v ~j = 4 d1 d->

J:,,-

Y-f:) -=-0

'} -= -2.x

'1...

'x ..::..

)(. X

= ._ ~, y y

2_]y. 2.... = ~ .f-k'

)(~@)

Big 10 Unit 5 Review 6. Let m(x)

= cos ( ~;

-

4;)

relative maximum?

ft'\ l ( ") ,.

=- - )lh

on the interval [0, 10]. For what value of x does m(x) have a

(::!.! - 4~) r ' To -11

t(J

J1 '-l

c;;:;;i

fi -

-0

- 3

to

7. What is the absolute minimum value of j(x) = ex - x + 8 on the interval [-2, 4]?

) ( )ot :::.- e.., '(_ )( + f

e

j ( XJ = ex_,

C>

.

-o--1P---

-=-N

1 -fl

k' ::.a

8. How many relative extrema does k(x)

= ln(x 2 + x + l)

a')( + I

have?

J )(-ti 'f..

=-6 !

I

=- - -

[ :1.J

ot...

YL(,J,'I(_

e;of.<-A.-.

9. If two numbers whose difference is 8 and whose product is a minimum, what is the larger number?

b { b -f$)

0\-b ~g

~. f"2._

,-g

o.. -:::-. b -t g

-t- f

h =- o

;2 ~-t O

:::;.Q

~ o ~"\~=)~) 10. For what x-value does the curve y2 + x

= 6 have a vertical tangent?

'[(-=-

I

;..O

-+

-1-, .

CIJ -

Big 10 Unit 6 Review Instructions: In the box below are the numbers O - 9. Complete the following and cross off the number for each answer. If you complete all problems correctly, you will cross off each number exactly once!

[ 0

2

1 I

I

t

4

3

-~- ~+

'-4-- -g(x) -~

-J- -

2

9I

+-

,-X I

4

-6 - 5 -4 -3 7 2_

_ [

8

_._,

B

----r--

7

6

5

f--

5

I

I I

I

6

-t-

'2

Let f be a twice differentiable function defined as J(x) = twice-differentiable function defined as k(x)

=

lax g(t)dt . Let k be a

3x

1 -6

g(t)dt. The graph of g(x) consists of three line

segments and a semi-circle as shown above on the closed interval [-6, 5].

1. What is the value of f'(-4)?

2. If m'(x)

h-'\(x--)::3. If n'(x)

I

= 3x 2 and m(l) = -1 what is m(2)?

= 2sin(2x)

~>...((_

and n(O)

= 0 what is the absolute maximum of n(x ) on [O, 21r]?

..57 4. What is the value of k"(-1)?

I

J

t\(vJ =-Cct5C2.20)tc

m

·~''(>-) = ,, '(?~) 5. What is the value of f"(-4)?

Big 10 Unit 6 Review

6. What is the approximation of

1 6

h(x) dx when a left rectangle Riemann sum is used with 3

subintervals as indicated by the table?

!

6

7. Evaluate

h'(x) dx.

8. Evaluate fo\6x 2

1 5

9. If

p(x)dx

-

16x + 2) dx.

= 3 and

1 5

p(x)dx

= 1, then what is the value of

1 8

(p(x)

+ l)dx?

r12 10. Evaluate Jo [9sin(3x)]dx .

.f\/2

j .,'(

./ 4

..

<17 ,-{]_d5. (3'$',~);

21Cod1£'))- ':27cuJ(6) - \'s-5

-27-:: - ]O.> r;> ,

[Do"~

Cfl\(w-0

1J

Big 10 Unit 7 Review Instructions: In the box below are the numbers O - 9. Complete the following and cross off t he number for each answer. If you complete all problems correctly, you will cr~ss off each number e."l:actly once!

<

/

8

}

1. Consider the differential equation ddy = X

I

Jtl

2 x . Let y = J(x ) be the particular solution to the y

differential equation with initial condition f (3) = 5. Use separation of variables to find an eA'J)ression for f( x ) of the form f( x ) = Jax 2 + b. What is a+ b?

'u2 J ::: ')(.'J.... tc__

i

t (~)

,2.) = q--i~ C

?,S

rn

~

'f."2+7

2. Consider the differential equation dy = 3(x - 2)(y - 3) where y < 0. Let y = J(x) be the dx particular solution to this differential equation. In what quadrant does f(x) have a horizontal tangent?

5 (K-2) ('j-3) ::6 y= 4 @) -~ ?

2 . Let y = J(x) be the particular solution to the x x- l differential equation with initial condition f (3) = 3. What is J(l)?

3. Consider the differential equation ddy = y -

lfl lj -2)

-· I ·J_t

I

CJ!)

I

c,I\

If _s-qf./1~

7 ~ + . Let y' = f (x) be the particular solution to t:e (:..., +4 differential equation with initial condition J(2) = 1. What is the slope of the line tangent to the graph off at the point (2, 1)?

=

4. Consider the differential equation ddy X

4

y

L!_! =.-Ji -

151

5 -LV

I t4

5. The rate at which a lemur's weight in pounds, L, changes after t months since birth, can be modeled by the differential equation dL

dt

= 2( 40 -

weighs 5 pounds, 6 pounds, or 7 pounds?

L). Is the lemur gaining weight faster when it

L{ 0 -5 L(o~

'-(d-7

r- F' ..-.~,,-'d/ I

:J

Big 10 Unit 7 Review 6. Consider the differential equation dy

dx

.

y. '

= x 2 y2 .

J2../

dy -d = l are in what quadrant?

X

8. Consider the differential equation dy what is the approximation off (2)?

dx

2

•2 J

y

O. All points in the xycplane such that

V w = 2x 2 + y.

'2.

I

at the point (1, -1)?

;

dx

l:f,; wW>

dx

2

\ . -2 · l · I

~- I 7. Consider the differential equation

What is d

::- 3 'y._

Using the line tangent to the curve at (1, 2)

:>

2-{2.:=:-y

-I

t,3= J

d 9. Consider the differential equation dy

X

2

= YX3 _ Let y = f(x)

'j - 2

-- 3

~lzL_

J

be the particular solution to the

differential equation with initial condition f (2) = 4. What is lim

x-+oo

;

f (x) dx?

Y (~-1) 4('>i2.

'~ ::JI

I

[ } 0 /l ly .!0J/f1QI'"\ I~-J:/ 10. Consider the differential equation :~ = cos(x). Let y = f(x) be the particular solution to the 3n differential equation with initial condition f (O) = 8. What is f ( )?

2

j =1 Si/'\ (>c) -u. s,-r'I ( o) -+ (s[_ <;?-:: ut<-

c::.

Lj :;.<;,·t1

(>c)-tf

_f ( ;_ ) = -/

-I.I'

{ti