=1n1x1-7xt2I2tCa

[Pages:14]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 for the constant of C

integration. Remember to use absolute values where appropriate.)

Z p

(a)

5 5+4 x

2 5+3 2

1

xex x

dx

x

} ?y 5 a

.

4e? " 2

x

+

-

2

.

' 4+3

3

x

tgx .

-

In l x I

+C

=Qx"2+4e?+?54tx3-1n1x#?

fly (b)

Z

2+7 xx

3

4

=

dx

+7?-2-4?-3 dx

x

=1n1x1-7x"t2I2tCa

f7 (c)

Z

7 e

x + 13

dx

x

+

?

Be

dx

e

=7xtl3e?#

(d)

Z

2

+

3 54

xx

1

7 dx x

4 '

} 'd 2g +

x

-

x

x

=2ln1x1-?x-3+{? Is To 6t I he = z

? 1 + 25 a

?

-

-

?

C

=fj (e)

Z

48 + 2 u

8

du

+ tsu du

u

? = 6 h I u I + 's

+C

a 6hrlult.lu#

2. For the following functions, evaluate the integral. (Use for the constant of integration. ReC

member to use absolute values where appropriate.)

Z

(a) ( 4 + 2)(10 + 5 + 1)3

x

xx

dx

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

(.bi)mZ 4#x5 yU=x5=fxTeuyxl=Heudu x e dx

?55534

d?=?: = tseutc.ly#c Z

(c)

(3 3

9)

(3 4 x

36x)

xe

dx

-

3?4 -

36x

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

9? Z

(d)

(ln )36 x

dx

3-

NX

x

=fIy6Xdu=fu36du

du=t?d?

= 't

.is?(+e)C=Z@3p07e+26c/=x 7ddxx=xd:6/xdu=6fx2dx=7dx=x2du=*#.xoIdu=5Seudu=5eUtC=5@+c x

?

2 Fall 2016, Maya Johnson

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 = 6. n

Xo Xc Xz Xz XY Xs X6

( ) 0 2 4 6 8 10 12 ts

( ) 0 6 7 9 2 14 1 17 5 19 4 20 2

v f t/s

.. . . . .

[ ] 0,12

n =6

,

DX =

12/6=2

( Lb = 2 .

o

+6.2

+9.2+14.1+17.5+19.4 )

( R6 22 . 6.7+9.2

+14.1+17.5+19.4+20.2

)

-=1h3@3156 4. Speedometer readings for a motorcycle at 12-second intervals are given in the table.

() ts ()

Xo Xc Xz Xs X4 X5

0 12 24 36 48 60 32 27 24 22 25 28

v f t/s

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

sum with = 5. n

[0/60] ,

n=5

Bx=

60/5=12

( L 5=12 . 32

+27+24+22+25

)=

(b) Estimate the distance traveled by the motorcycle during this time period using a right-hand sum with = 5. n

( Rs = 12

22

-151 +24+22+25+28

).

3 Fall 2016, Maya Johnson

5. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of to approximate the integral. Round the answers to two decimal places.

n

[1/13] n =3 ,

Z 13

(2 2 + 1)

=3

x dx, n

1

DX = (13-1)/3 = 4

)=# L }

=

4.su/Seg(2x2H , X , 1,43 - 4), 4)

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

6. Use a left-hand sum and a right-hand sum with rectangles of equal width for the given value of to approximate the integral. Round the answers to four decimal places.

n

) [ 1,10 , n =3

D?=( 10-11/3=3

Z 10

2 ln( )

=3

x x dx, n

1

( L }

( =3 . Sun Seq

X2h(

x ) ,? ,

I ,

(10-3) ,

33=352.59090

-1043.36 ( )) ( R }

=3

. Sum

Seq

x2h(

x)

,

X

,

(1+3) ,

10,3

-

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

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

[

0,12

] ,

n=4

DX = (12-0)/4=3

Z 12

(2 3 + )

=4

x x dx, n

0

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

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

4 Fall 2016, Maya Johnson

8. Evaluate the following definite integrals:

(a) Z 1 6 . Assume

1

dx

A<

x

A

= 6 lnlxlljs 641- 16 TLIAI

= -6

Z

B

(b) (3 2 7 3 + 7 2) . Assume

2

x x x dx

B>

2

7? ( is -

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

+7? = B3 . 72,1

( ) - ZB

-

to

=

Z

B

(c) B3(1.0leyx#76xB4 +I22B) d@x. =As@sumeexB->60sI+2x)lFl0ek6sBIt2B-(10e0 0

61? ) -

+2 ( o )

Z

=1OeB-6B?+2B

A

(d) 10 9 2 + 10 . Assume 1

x x dx

A>

1

IF =@? - 3?3+5?2 )

( NA - 343+5 AY - 400 - 3aP+5NY

=1oA-3A3t5A2

Z6

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

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

f

f

f x dx

f

4

=f( 30 = Sdf '( x ) dx

(4) 6) f-

f 30 =

C 6) - 18

=@ fl 6) = 30+18

5

Fall 2016, Maya Johnson

10. Suppose the marginal cost function for a certain commodity is given by 0( ) = 0 5 and (0) = C x .x C

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

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

)

36 = C ( 12

-

Zoo

C ( 127=36+260

=@

11. Suppose the marginal revenue function for a certain commodity is given by 0( ) = 10 6 and Rx x

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

"

441=d4RxCR9=kIx0{sdx?= -RC66) - )

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

+100

? =

12. Find the average value of the following functions on the given interval. (Round answers to two decimal places as needed.)

(a) ( ) = 6 + 9 2 [0 5] fx x x, ,

(b) ( ) = 12 3x [5 7] fx e , ,

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

(c) ( ) = 12 3 10 2 [2 6] fx x x, ,

= 786.6

) , 2,6

6 Fall 2016, Maya Johnson

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

d ( ) = 15 0 57 2 St t . t

dt (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.

?Ft -

(b)a5F7itn2dtt-hfneInatvfel5rxaig5e7sxa3lXe.sqplyegr2m28o.n9t6hthfoorutshae@sWecMon5dt-rs5ix7tm2tot-ntt6hepfneIrnitof5dxo-fs7thx2e.aXdvertising campaign. Give answer to three decimal places.

,

6 ,

)12

=87.1Zth?usa

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

0( ) = 320

0 08 .t

Pt

e

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

Pt

t

(a) How many tons of copper is projected to be extracted during the second four year period? Give answer to three decimal places.

fyzzoeittdt =fnIntfs2?eYx , 4,87-795.4260

(b) How many tons of copper is projected to be extracted during the third four year period? Give answer to three decimal places.

"

"

fglzzoe

tdt = fn Int (320209?48,12)=577.5980

7 Fall 2016, Maya Johnson

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

(Assume and are two real numbers such that

.)

ab

a ................
................

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