xfx 2lim

1lim 2

2x

x122 5

xfx 2lim 5

xfxfxx

22

limlim 5lim2

xfx

Problem #1

5

2 1, for 2 2

1, for 2

5, for 2

x - x

f x x

x

EMathboat.com

1,1.5 ; 4 ² 1y x

ab

afbf

Change of Rate Avg.

10

15.1

11415.14

15.1

15.1 22

ff

Inst. Rate of Change f a

24 1 8f x x x

1.5 8 1.5 12f

21210 Change of Rate Inst. Change of Rate Avg.

Problem #2

CMathboat.com

2

22 3, 1, 0 ,

2 8

xx x y y

x x

1

32ln44ln3Area

xx

Use cover-up

method

22

4 2 4 2

x A B

x x x x

5ln41ln31ln45ln3 7 ln 5

1

3

22

4 2

xdx

x x

Problem # 3

1 1

3 3

1 1Area 3 4

4 2dx dx

x x

2

22

2 8

xy

x x

3x 1x

0

1

23

220

2 8

xdx

x x

2 22

For , set 2 : 42 4 2

B x Bx

4 22

For , set 4: 34 4 2

A x Ax

CMathboat.com

42 xy

2 2 0x y y x

02 2 ydx

dyxy

22dy

xy ydx

2

2

dy y

dx xy

2

2 2

1 1

2 2

d y y xy - yx

dx x x

22

2

2

1

x

ydxdy

x

dx

yd

2

2

3

2

d y

dx

2

2

1,2

1 1 21plug in:

1

2

d y

dx

1 2

2 11

plug in:

Problem #4

1

2

y

x

Implicitly differentiate

to find the first derivative, dy

dx

differentiate

2

2Find second derivative,

d y

dx 2 4; 1,2 xy

BMathboat.com

dxxydy 235

Cxxy

22

3

2

5 22

C 462

5

2

1510

2

5 C

2

152

2

3

2

5 22

xxy

15435 22 xxy

5

1543 22

xx

y

5

1543 2

xxy

Problem #5

0 since if it is negative,

it cannot be 1 when 2

y

x

3 2 ; 1 when 2

5

dy xy x

dx y

AMathboat.com

Use initial conditions:

C 222

23

2

15 22

2 ; 0, 3xy e x x

dxxf b

a

21 Length

dxe x

3

0

221

dxe x 3

0

2221

dxe x 3

0

441

Using the arc length formula…

Problem #6

AMathboat.com

Point of inflection occurs when

0or DNE and changes

its sign around

x c

f c f x f x

x c

is increasing, therefore 0

NO POINTS OF INFLECTI

,

there are ON

f x f x

y f x

Problem #7

-1 1

1

AMathboat.com

-4,4x

x

24 232 x

x

By Pythagorean Theorem

3222 bx232 xb

2322 xxA

2

22

322

4322

x

xxA

0

02322 22 xx

644 2 x

4

A max.

maximize

Area:2322 xxA

243242 A

32A

0

22 322322 xxxxA

Problem #8

b

radius = 4 2

is increasing, then decreasing, so local max absolute maxA x

CMathboat.com

2

1lim II.

02 dx

x

b

b

b

xbdx

e04

1lim III.

2

3

1

3lim

2

b

b

x

2 2

3 33lim 1

2bb

b

b x 02

1lim

2

1

2

1lim

bb

2

1

dx

x

b

b 1 3

1

1lim I.

bx

be 0

4

4

1lim

04

4

1lim ee b

b

4

1

Diverges

Converges Converges

11 3

20

40

1I.

1II.

2

1III.

x

dx

x

dxx

dxe

Problem # 9

AMathboat.com

6

4

5 ?x dx

6

4

5 dxx

1A 2A

6

5

5

4

55 dxxdxx

OR

triangles twoof areas Add

21 AA 2

11

2

11

1

2

1

2

1 1

Problem #10

DMathboat.com

Integration by parts: vduuvudvxeu xdxdv cos

xv sin dxedu x xsin dxe xxsinxe

Integration by parts again:

xeu xdxdv sinxv cos dxedu x

xcos dxe xxcosxe

xdxe x cos xsinxe xcos dxe xxcosxe

2 cos sin cosx x xe xdx e x e x

Cxxexdxe xx cossin2

1cos

Plug in:

Problem #11

Divide both sides by 2 and add the constant of integration .C

move it to the left wit +h

cos-xe xdx

AMathboat.com

1 , 2 , 0 from smallest to largest = ?f f f

Problem #12

At 2 : function increases, is positive 2 0x f x fx f

At 1: local min = 0 1 0x f x f

0 1 2f f f

At 0: concaves down 0 0 x f x f

-1 1

1

CMathboat.com

1 sin

dx

x

1 sin

1 sin 1 sin

x dx

x x

dxx

x

2cos

sin1

xdxxxdx sectansec2

Cxx sectan

2

1 sin

1 sin

xdx

x

dxx

x

x

dx22 cos

sin

cos

Problem #13

EMathboat.com

from the graph,

1 2 1, 2

2g f

f xh x

g x

2 slope of at 2g g x x

3from the graph, 2 1, 2

2g f

f x

xg

1

1

2

2 2 2 22

2

f g f gh

g

2 slope of at 2f f x x

2

1 31 1

2 22

1h

2 1h

2

Problem #14

BMathboat.com

dxxfab

1

valueaverage

10

0

010

1dxxf

252

1

10

1

20

25

4

5

2

2

1

10

1r

Area of Semicircle 2

2

1r

0 5 10

Problem #15

AMathboat.com

2 1xxe

!!3!2

132

n

xxxxe

nx

!

2

!3

2

!2

221

322

n

xxxxe

nx

13 42 32 22 2 2 2

2! 3! !

nnx e e e e e

xx xxe x x

ne

2 1x exe

Problem #16

CMathboat.com

4

114

4

2x

x

dt dt

dx

Chain Rule

Split into two integrals:

4244211431116 xxx

dttdx

ddtt

dx

d xx

4

0

114

0

114 22

4

dttdx

ddtt

dx

d x

x

4

0

1140

4

114 22

Flip integral and make it negative

Use 1st Fundamental Theorem of Calculus

xxxx 424211441116

Problem #17

AMathboat.com

3

328

3

88

3

8

43

2

2

3

yy

dyy

2

2

2 40 Area

42 yx

0x

Problem #182 4 ; -axis x y - y

C

2

Find Limits of Integration

0 4

0 2 2

2

y

y y

y

Mathboat.com

10

0

2cos dtt

10

0

2cos tdt

10

0 2

2cos1dt

t

10

0 2

2cos

2

1dt

t

10

04

2sin

2

tt

5

2 cos ; 0 to 10v t t t

dtb

a

tv Distance

02

10

Problem #19

EMathboat.com

Always 0

yy exexy2tan

dx

dyxee

dx

dyy yy 22sec2

yy edx

dyxe

dx

dyy 2sec2 2

y

y

xey

e

dx

dy

2sec2 2

yy exeydx

dy2sec2 2

Problem #20

Implicitly differentiate.

tan 2 yy xe

BMathboat.com

3

4log 3 5f x x

9

4 3 5 ln10f x

x

53log

4

3x

10ln

53ln

4

3 x

353

1

10ln

1

4

3

x

Problem #21

34 5log (3 5) ,

3f x x x

DMathboat.com

1

Use Ratio test

lim

:

n

nn

u

u

11

lim1 1

n

nn

x n

n x

1 x

n

xxxx

nn 11

3

1

2

11

132

1lim

1n

x n

n

11

lim

xn

nn

02 x

x 11 converges, series If

111 x

1

1 1

1 1 12 :

Harmonic, Divergent

n n

n n

xn n

Check Endpoints

1 1

1 1

1 1 10 :

alternating harmonic,conditionally

convergent

n n n

n n

xn n

Problem #22

) 2 0a x AMathboat.com

2 2 2

n

1 1 2lim ...

n

n n n n

3 2

1 1 1

n n n

nx2x

It looks like a limit of Riemann sums

n

k

kkp

xwf1

)(lim0

0 1

dxx1

0

2

nx

1

0n

1

n

2

n

4

n

31

n

n

n

1

n

1

n

1

n

1

n

11x

Problem #23

2 2 23

1 lim 1 2 ... ?

nn

n

EMathboat.com

2 2 2

n

1 1 2lim ...

n

xn n n n

uaaa uu

ln use

52arctan4 x

52arctan 2arctan4ln45

xx

duu

u2

1-

1

1tan use

5arctan 2 525

14 ln 4 2

1 2

xx

x

5arctan 2 4

25

14 ln 4 10

1 2

xx

x

Problem #24

5arctan 2 4

xf x

EMathboat.com

0

0

ln 1

lim

x

x

tdt

t

x

0

0

x

dtt

tx

x

0

0

1ln

lim

use L'Hôpital's rule

0

ln 1

lim1x

x

x

11

11

lim0

xx

ln 1 0

0 0

Problem #25

0

ln 1limx

x

x

use L'Hôpital's rule again

0

ln 1limx

x

x

D

Mathboat.com

1

5lim

2 x

xx

xx

xx

x 1

5

lim2

22

2 1

5lim

xxxx

01

5

5

1

5lim

2 x

xx 1)100000(

)100000(52

100000

500000 5

5 and 5 are asymptotes Horizontal yy

2

5

1

xf x

x

Problem #26

DMathboat.com

2 2 5 3. 15II g x x x 215 0 or DNEx

0x

2

6 1 I.

2II h x

x x

2 3

6 1dh

dx x x

2

2 3 4

12 30 or DNE

d h

dx x x

12 3x 1

4

x

2sin 2I. 2f x x xxf 2cos4

02sin8 xxf

,2

3 ,,

2,0

x

DNE at 0, but not on

the domain not a PI

x

Pts. of inflection

Pt. of inflection

no sign change, so

0 is not a PIx

3 4

12 30

x x

y

0 1

4

Problem #27

32

6 1I. 2sin 2 2 II. 5 III. , 0

2f x x g x x h x x

x x

Or : by the shape of Sin graph, it concaves up and down

I and III onlyDMathboat.com

Shown here is the slope field for which differential equation?

x

yy

dx

dy 2 a.

2

x

x

dx

dy 43 b.

2

4

43 c.

2

y

dx

dy

x

y

dx

dy 3 d.

x

y

dx

dy

2

2 e.

No asymptote at y=3

Asymptote at y=2

Slope is not same for every vertical

Slope is not same for every horizontal

Asymptote at y=0 & y=2

Problem #28

AMathboat.com

3 9 3f b

3

lim 2.121173cos 34x

f x

9 3 2.12117b

is continuous at if

lim limx a x a

f x a

f x f x f a

3 11.12117b 3.7071b

BMathboat.com

2 , 3

If   is a continuous function3cos , 3

4

x bx x

f xx x

Problem #29

928.3

12

0

1

12 0avf f x dx

Problem #30

CMathboat.com

1 Average value of

b

a

f x f x dxb a

)5(

4

15453

2

1

12

1 2

x y

0

0.5

1

dx

dyxy

110

1

2

1 0

e

4

1

15.02

1

2

1 5.02 ee

4

12

e

4

9 e

Make a chart

1

2

211

New y

y Δy

Problem #31

21

point 0,1 ; , starting at 0; step size of 0.52 1

xef x x

x

EMathboat.com

is concave downf x

is decreasingf x

first derivative 0,

second derivative <0

f x

f x

Lower SumUpper SumTrapezoidal

Problem #32

B

Mathboat.com

tvta

3)7(7

3

vvdttv

377

3

vvdtta

7

3

2cossin3sin67 dtttttv 639.43

= 6

Problem #33

sin 3sin cos 2 ;

the velocity of the particle is 6 m / sec at time 3 sec

a t t t t t

t

A

Mathboat.com

By the Second Fundamental Theorem of Calculus,

4cos 32

tx t

tvGraph

txtv

4at greatest is ttv

tvSpeed

4 5 6 7 84 5 6 7 8

tv

2sin2

32

cos4

t

t

Problem #34

AMathboat.com

Problem #35

axbxdttxb

a

aybydttyb

a

244

2

xxdttx 244

2

yydtty

4

2

3 97236.3sin244 dtty

9723631333 is 4 at timeobject ofPosition .,.t

By Second Fundamental Theorem of Calculus:

4

2

2 133.3cos34 dttx

B

2 3 cos ; 2sin for 0 4 At 2 the object has position 3 4 .dx dy

t t t . t ,dt dt

Mathboat.com

V t S tWhen the sandbag hits

the ground, it is 0 feet

above the ground.

016160)( 2 tts

16016 2 t

102 t

1010 t

103210

V

Problem #36

32V t t

2 160 16S t t

AMathboat.com

kdtdyy

80

1

Ckty 80lnktCey 80

080100 Ce

Use initial conditions:

20 C

8020 ktey

2

112 ke

12

1

2

1

ke

802

120

12

t

y

802

12085

12

t

21

ln

41

ln12t

2

1ln

124

1ln

t

:Cooling of Law sNewton'802090 12 ke

24t

12

2

1205

t

Problem #37An outdoor thermometer registers a temperature of 100˚F. Twelve minutes after it is brought into a room where the temperature is 80˚F, the thermometer registers 90˚F. When will the thermometer register 85˚F?

80 ykdt

dy

85?

9012

1000

y

y

y

BMathboat.com

)35(15

111 xy

tt

t

dtdxdtdy

dx

dy

15

2

15

22

3

2

2 5 2 5 35

2 2 7 11

x

y

13015 xy

Find the slope:

2 12, so slope is

30 15t

Plug the coordinates of x and y and the slopeinto point-slope equation of the line:

2

3 2

7 2

5 5 15

dyy t t

dtdx

x t tdt

Problem #38

3 2 5 5 and 7; 2x t y t t

AMathboat.com

24 2 xy

2

13

xy

Use the solve button on your calculator

or graph to find the values of the points

of intersection.

x

76556.6or 73444.2 xx

dxrRVb

a

: Method sWasher'

22

R 26.76556 22

2.73444

3 14 2

2

xx dx

99.342

Rr

Problem #39

24 2 x y

2 3 1 0 y x

24 2 ; 2 3 1 0x y y x

r

CMathboat.com

d

2

2

1 Semicircle of Area πr

)(xA

21

10

xy

3y

x

y

3

7

3

7

dxxAVolume )(

31

10set , and Find

2

x

2

31

10

2

3 2

xy

2

2

2

31

10

2

1

xA

2

diameterr

3

7,

3

7

3

7 x

dxxV

3

7

3

7

2

2

2

31

10

2

1 009.21

Calculator Use

r

Problem #40

2

10 ; 3

1y y

x

AMathboat.com

minft5000

dt

dy

ft 12500y

?dt

dr

2 2 21000 12500r 629500r

222 1000 yr

2 0 2dr dy

r ydt dt

2 500 629 0 2 12500 5000dr

dt

4984.08dr

dt

ft 1000x

ft 1000

y r

Boy Girl

Problem #41

C

Implicitly differentiate

Mathboat.com

b

a

innerouter dxrRV 22

130.52132

2

222

dxxV

2a 2b

Solve for intersections a and b:

3outerR

22 123 xxrinner

022 x 2x

Substitute into formula

BMathboat.com

Problem #42

2

704 ; initial population 0 30. 33

For what value of is the population growing the fastest?

dP PP P

dt

P

33047

3PP

33704

2PP

dt

dPP

011

7042

P 88Pmax+ –

88P

dt

dPP

dt

dPP 2

11

1704

2

11704

P

dt

dP0

/ is the rate at which the population is growing.

Therefore, the population is growing the fastest

when / has a maximum.

dP dt

dP dt

Problem #43

To maximize , first find and set it 0P P

BMathboat.com

is always increasing,

is an odd function.

is increasing on -4,4 .

has a relative minimum at 0

has a relative maximum at 0.

0 on 4, 2 .

f x

A f x

B f x

C f x x

D f x x

E f x

-4 -2 0 2 4

-2 -1 0 1 2

to- from changes

and 0 when 0 True;

xf

xxf

False; 0 is decreasing on -4,0 .f f

false. is which 2be tohas2 andeven

bemust then odd, is if False;

ff

xfxf

False; has a minimum at 0.f x x

May not be true; don't know

the values of .f x

Problem #44

x

f x

CMathboat.com

dxxx 8

0

3

Area under the curve:

8

0

34

2

4

3

2

1

xx

441232

8 10 80 Area of 8 by 10 rectangle:

80 44 36 Area above the curve:

Probability:Area above curve

Area of 8 by 10 rectangle36

80

9

20

Problem #45

or use calculator

C

Mathboat.com