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Transcript of Problem #1 E Mathboat.com. Problem #2 C Mathboat.com.
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