=1n1x1-7xt2i2tcamayaj/review142_exam3_answers.pdf · t(s) 0 2 4 6 8 10 12 v(ft/s) 0 6.7 9.2 14.1...

Exam 3 Review (Sections Covered: 6.1-6.6, 6.7topic and 8.1-8.2)

1. Find the most general antiderivative of the following functions. (Use C for the constant of

integration. Remember to use absolute values where appropriate.)

(a)

Z ✓5px

5 + 4ex � 2x�5 + 3x2 � 1

x

◆dx

(b)

Z ✓x

2 + 7x� 4

x

3

◆dx

(c)

Z ✓7e�x + 13

e

�x

◆dx

(d)

Z ✓2

x

+3

5x4� 1

x

7

◆dx

(e)

Z ✓48 + u

2

8u

◆du

a 5 . } x" 2

+ 4e× - 2 . ÷y x' 4+3

. tgx3

-In l x I + C

=Qx"2+4e×+£54tx3-1n1x#÷= fly +7×-2-4×-3 dx

=1n1x1-7x"t2I2tCaf7 + Be

× dx

=7xtl3e×#2g + } x

' 4- x 'd x

= z he I × 1 + 25 a Is ×→

- To ×- 6t C =2ln1x1-§x-3+{×€

=fj + tsu du

= 6 h I u I + 's ¥ + C

a 6hrlult.lu#

2. For the following functions, evaluate the integral. (Use C for the constant of integration. Re-

member to use absolute values where appropriate.)

(a)

Z(x4 + 2)(10x+ x

5 + 1)3 dx

(b)

Zx

4e

x

5

dx

(c)

Z(3x3 � 9)e(3x

4�36x)dx

(d)

Z(ln x)36

x

dx

(e)

Z30e�6/x

x

2dx

2 Fall 2016, Maya Johnson

Ua1Oxtx5t1du-4ot5x4Jdx-Slx4t2Tu3K_g.lfu3duFx4Tt5x4atsetyu4tC.z@xs4ct7dx.s

.im#yU=x5=fxTeuyxl=Heudu¥55534

= tseutc.ly#c⇒d×=¥:- 3×4 - 36x

=fl3x#H.de#=tyfeududu=d2x3.36)dx=tyeutC=lge3@x+c=7dx=du*

3-9÷NX

=fIy6Xdu=fu36du du=t×d×

= 't

.is?+C=@p7+c=7dx=xd:6/xdu=6fx2dx=7dx=x2du=*#.xoIdu=5Seudu=5eUtC=5@+c

•

3. The speed of a runner increased steadily during the first twelve seconds of a race. Her speed at

two-second intervals is given in the table. Find lower and upper estimates for the distance that

she traveled during these twelve seconds using a left-hand sum and a right-hand sum with n = 6.

t(s) 0 2 4 6 8 10 12

v(ft/s) 0 6.7 9.2 14.1 17.5 19.4 20.2

4. Speedometer readings for a motorcycle at 12-second intervals are given in the table.

t(s) 0 12 24 36 48 60

v(ft/s) 32 27 24 22 25 28

(a) Estimate the distance traveled by the motorcycle during this time period using a left-hand

sum with n = 5.

(b) Estimate the distance traveled by the motorcycle during this time period using a right-hand

sum with n = 5.


Xo Xc Xz Xz XY Xs X6

[ 0,12 ],

n = 6 ⇒ DX = 12/6=2Lb = 2 . ( o +6.2 +9.2+14.1+17.5+19.4 )

-133€R6 22 . ( 6.7+9.2 +14.1+17.5+19.4+20.2 )

=h@€Xo Xc Xz Xs X4 X5

[0/60] ,n=5 ⇒ Bx= 60/5=12

L 5=12 . ( 32 +27+24+22+25 ) =

156€

Rs = 12 ( 22 +24+22+25+28 ) .-151€

5. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of

n to approximate the integral. Round the answers to two decimal places.

Z 13

1

(2x2 + 1) dx, n = 3


n to approximate the integral. Round the answers to four decimal places.

Z 10

1

x

2 ln(x) dx, n = 3


n to approximate the integral. Round the answers to two decimal places.

Z 12

0

(2x3 + x) dx, n = 4


[1/13] ,n =3 ⇒ DX = (13-1)/3 = 4

L} = 4.su/Seg(2x2H , X , 1,43 - 4) , 4) )=#

R }= 4 . Sun (Seq (2×2+1,14+4) , 1314 ))a2@

[ 1,10 ), n =3

⇒ D×=( 10-11/3=3

L } =3 . Sun ( Seq ( X2h( x ) ,× ,I

, (10-3) , 33=352.59090

R } =3 . Sum ( Seq ( x2h( x) , X, (1+3) , 10,3 )) --1043.36€

[ 0,12 ] ,n=4

⇒ DX = (12-0)/4=3

Lot =3 . Sun (Seq ( 2×3 tx, × , 0,42 - 3) , 3 ))=5@

Ry =3 . Sun ( Segkistx , x , (0+3312,3)=162900

8. Evaluate the following definite integrals:

(a)

Z 1

A

6

x

dx. Assume A < 1

(b)

ZB

2

(3x2 � 7x3 + 7x� 2) dx. Assume B > 2

(c)

ZB

0

(10ex � 6x4 + 2) dx. Assume B > 0

(d)

ZA

1

10� 9x2 + 10x dx. Assume A > 1

9. If f(4) = 18, f 0 is continuous, and

Z 6

4

f

0(x) dx = 30, what is the value of f(6)?


= 6 lnlxlljs 6411T- 6 LIAI

= -6€

( is - 7¥ +7¥ . 2x)K=B3 - ZBTHBI . ZB - (23-2,129+22125-44)

= B3 . 72,1 +7¥ - ZB - ( to ) =

B3.ly#7BI2B@=@ex-6sI+2x)lFl0ek6sBIt2B-(10e0- 61¥ +2 ( o ))

=1OeB-6B÷+2B€

=@× - 3×3+5×2) IF ( NA - 343+5 AY - 400 - 3aP+5NY

=1oA-3A3t5A2€

30 = Sdf'( x ) dx =f( 6) - f (4) ⇒ 30 = f C 6) - 18

⇒ fl 6) = 30+18 =@

10. Suppose the marginal cost function for a certain commodity is given by C

0(x) = 0.5x and C(0) =

200, find the cost to make 12 units of this commodity.

11. Suppose the marginal revenue function for a certain commodity is given by R

0(x) = 10x� 6 and

R(1) = 100, find the revenue when 10 units of this commodity are sold.

12. Find the average value of the following functions on the given interval. (Round answers to two

decimal places as needed.)

(a) f(x) = 6x+ 9x2, [0, 5]

(b) f(x) = 12e3x, [5, 7]

(c) f(x) = 12x3 � 10x2, [2, 6]


36 a ftp.5xdx#Ckx)dx=C( 12 ) - Cco )

⇒ 36 = C ( 12 ) - Zoo ⇒ C ( 127=36+260

=@

441=490×-6dx={"

Rkxsdx = RC 6) -RCI )

441 = R ( 10 ) - 100 ⇒ R ( 10 ) = 441 +100

= µ

atfj6xt9x2dx-ts.fnIntL6xt9x3x.o.s7-o@Baasjl2exdx-tz.fnIntf12eYx.5.7t5689.JWY2x3tox2dx-ytcfnInt42x3tox2.x

, 2,6 )= 786.6€

13. The rate of sales of a certain brand of bicycle by a retailer in thousands of dollars per month is

given by

d

dt

S(t) = 15t� 0.57t2

(a) Find the amount of sales, in thousands of dollars, for the first six months after the start of

the advertising campaign. Give answer to three decimal places.

(b) Find the average sales per month for the second six month period of the advertising campaign.

Give answer to three decimal places.

14. Suppose that copper is being projected to be extracted from a certain mine at a rate given by

P

0(t) = 320e�0.08t

where P (t) is measured in tons of copper and t is measured in years.

(a) How many tons of copper is projected to be extracted during the second four year period?


(b) How many tons of copper is projected to be extracted during the third four year period?



§Ft -

a57t2dt-fnIntfl5xi57x3X.qlyg228.96thousa@WM5t-r57t2tt-t6efnIntf5x-s7x2.X

,6

,12) =87.1Zth°usa€

fyzzoeittdt =fnIntfs2°e→Yx, 4,87-795.4260

fglzzoe" "

tdt = fn Int (320209×48,12)=577.5980

15. Use properties of the definite integral and information listed below to solve the following problems:

(Assume a and b are two real numbers such that a < b.)

Zb

a

f(x) dx = 20

Zb

a

g(x) dx = �12

Z 0

�3

u(x) dx = �16

Z 3

0

u(x) dx = 50

(a) Evaluate

Z 2a

2a

f(x) dx.

(b) Evaluate

Za

b

g(x) dx.

(c) Evaluate

Zb

a

[f(x)� 3

2g(x)] dx.

(d) Evaluate

Z 3

�3

u(x) dx.

16. Determine the area that is bounded by the graphs of the following equations.

y = 64x, y = x

3


÷ a - ( - 12 ) @

= 20 - El - 14=20+18 -@sjulxldx ={{

lxidxtfuhdx= - 16+50 =µ

X 3=64 × ⇒ × 3- 64× =o ⇒ × ( x +8 Xx - 8) -0 ⇒ × a 0,8 ,-8

↳ x

3-64×6=1024⇒ Area = µ 24+1024 =2@

/×3 -

64×4=-1024

17. Determine the area that is bounded by the graphs of the following equations. (Round answer to

three decimal places.)

y = 3x, y = 9x� x

2

18. Determine the area that is bounded by the graphs of the following equations on the interval below.

(Round answer to three decimal places.)

y = x

2 + 7x, y = 8x+ 56

19. The graph of f is shown. Use the graph to evaluate each integral.

(a)

Z 28

20

f(x) dx

(b)

Z 36

0

f(x) dx

(c)

Z 12

0

f(x) dx


9 x - XZ =3 × ⇒ 0=3×-9×+6 ⇒ X 2- 6× =0

⇒ :X ( x - 6) =D ⇒ X = 0,

6

Area =§!x2+6×dx=fnIut(yZ6x ,X

,Q 6)#

X 2+7 × = 8×+56 ⇒ XZ - x -56=0

⇒ ( X -84×+3=0 ⇒ X= -7,8

Area a €×2+× +56 dx =

562.5¥

a= ÷ ( 8) fl 2) =@

= (81/4)+248 ) (8) + (4×12) + } (12718) - tz ( 12) (8) - El4)(8) - (8×8)=320

= ( 8 )( 4) ttz 18 )( 8) +141112 ) = @

20. Calculate the producers’ surplus at the indicated price level for the supply equation below. (Round

answer to the nearest cent.)

p = S(x) = 130 + 0.2x2, p

o

= $194.80

21. Calculate the consumers’ surplus for the demand equation at the given number of units demanded.

(Round answer to the nearest cent.)

p = D(x) = 27� 2x1/3, x

o

= 343


194.80 = 130+-2×2 ⇒ .

2×2=64¥ - ⇒ XE 324

⇒ X±H . =§Fq 4.80 - (130+-2×2) dx =

fn Int ( 64.80=2×3 ×,

0,

1 8) =$7@

Po =D ( 343 ) = 27 - 2 ( 34343=13

{3If -2×113-13 dx=fnIat( 27-2×4313 , × , 0,34 3)

=$12O@

22. Determine the consumers’ surplus for the demand function below at the indicated price level.

p = D(x) = 500� 0.06x, p

o

= $110

23. Determine the indicated values of the following functions.

f(x, y) = 4x2 � xy + y � 9

g(x, y) = x� 4

5� 2y2

(a) f(�5,�2)

(b) g(1,�9)

(c) f(1, 3)� 3g(7,�1)

24. Determine the indicated value of the function. (Round answer to one decimal place.)

W (0.9, 6, 1,�5) for W (a, b, c, d) =a(1 + b)� d

2

2c

25. Determine the indicated values of the function.

f(x, y, z) =3xy + 6z

3xz � 6y

(a) f(1, 0,�1)

(b) f(0,�1, 1)

(c) f(�1, 1, 0)


110=500 - .°6× ⇒ . 06 × = 500 - 110 ⇒ . OGX =39¥ 05-7×0=6500§5Foo - -06×-110 dx=fnInt( 500=06×-119×10,6500 )

=$1,26€

= 41-55-(-511-2) H - 2) -9=790

=i-day - ' - Ist =¥Dfl 1,31=4115-(4/3) t( 3) - 9 =

917 , -11=1 . ←4#§= . }→⇒f"

' 31-3917 '' D= - 5- 3ft )= - 5+1 #

A = .9

,b= 6

, C = 11,

D= - 5

Wt .9, 6,1 ,

- 5) = -917k¥ =

-9.35¥XO)+6¥states =D

= 3/0 )tDt6( 1)

TEE'

. at =D

=3 ( - IXDtask ,¥=D

26. Macrosoft produces two versions of its popular gaming console: the Elite and the Casual. The

weekly demand and cost functions for the consoles are

p = 300� 4x+ 2y

q = 225� x+ 9y

C(x, y) = 300 + 90x+ 120y

where x represents the weekly demand for the Elite version; y represents the weekly demand

for the Casual version; p and q represent the price (in dollars) of an Elite console and a Casual

console, respectively; and C(x, y) is the cost function.

(a) Determine R(x, y), the weekly revenue function.

(b) Determine P (x, y), the weekly profit function.

(c) Find P (4, 2).

27. HeadsRock produces two versions of its popular headphones: the RockUrWorld and the Silen-

tRocker. The weekly demand and cost functions for the headphones are

p = 300� 7x+ 2y

q = 225� x+ 8y

C(x, y) = 400 + 90x+ 120y

where x represents the weekly demand for the RockUrWorld version; y represents the weekly

demand for the SilentRocker version; p and q represent the price (in dollars) for a pair of Rock-

UrWorld headphones and a pair of SilentRocker headphones, respectively; and C(x, y) is the cost

function.

(a) Determine R(x, y), the weekly revenue function.


Rkiy) =

pxtqy.3oox.ch/2+xy+225y+9TPlxiy)=RLx

, y ) - C ( x , y ) =2lOx-4×2txy+l05yt9yI3=

p ( 4,4 =$#

R l ×, y) = pxtqy =3oox-7x2txy+225y+8Ty

(b) Determine P (x, y), the weekly profit function.

(c) Find P (7, 2).

28. Find the first partial derivatives of the function.

w = 9z + 10exyz

(a)@w

@x

(b)@w

@y

(c)@w

@z

29. Find the first partial derivatives of the function.

f(x, y) = x

6y

5 + 7x4y

(a) f

x

(x, y)

(b) f

y

(x, y)


Plxsykrlky ) - Clay) =Zwx-7x4xy+lo5y+8y440£

P 17,4

=$983±

yze×Y⇒O×ze×@

=9+1Oxye@

=

6×555+28×7

5x6y4+€

30. Find all the second partial derivatives.

f(x, y) = x

9y

5 + 3x9y + x

4 + 30y2

(a) f

xx

(b) f

yy

(c) f

yx

= f

xy

31. Find all the second partial derivatives.

f(x, y) = 200 + 3x5y

3 + 2x10y � 2x6 + 12y3

(a) f

xx

(b) f

yy

(c) f

yx

= f

xy


fx = 9×35+27×9 +4×3

fxx -

-72×75>+216×3+12×7= 5×9y 4+3×9 + 6 Oy

fyy =2ox9y3#

fyx = fag

=45x8y4t2¥Ga

fx = 15 x 4g3 +20 x9y - 12×5

fxx - -60×33+180×9.6=4

fy = 9×592+2×10 t 3 6y2

fyy218×59+77

-

yx = fxy =45x4y2t€

=1n1x1-7xt2i2tcamayaj/review142_exam3_answers.pdf · t(s) 0 2 4 6 8 10 12 v(ft/s) 0 6.7 9.2 14.1...

Documents