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1
When y=
ou
see…
Fin=
d the
zeros
You
think…
![](3D"slide0001_image002.gif")
![](3D"slide0001_image004.gif")
2
To find the zeros...
![](3D"slide0064_im=)
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3
When y=
ou
see…
Find
equation of the line tange=
nt to f(x) at (a, )
![](3D"slide0002_image007.gif")
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You
think…
4
Equation of the tangent line
5
You
think…
![](3D"slide0003_image010.gif")
When y=
ou
see…
Find
equation of the line norma=
l to f(x) at (a, )
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6
Equa=
tion
of the normal line
7
You
think…
![](3D"slide0004_image013.gif")
When y=
ou
see…
Sho=
w that f(x) is even
8
Even
function
9
You
think…
![](3D"slide0005_image016.gif")
When y=
ou
see…
Sho=
w that f(x) is odd
10
Odd
function
11
You
think…
![](3D"slide0006_image019.gif")
When y=
ou
see…
Fin=
d the
interval where fx) is increasing =
12
f(x)
increasing
13
You
think…
![](3D"slide0007_image022.gif")
When y=
ou
see…
Fin=
d the
interval where the slope=
of f ( x) is increasing
14
Slope of f (x) is increasing
15
You
think…
![](3D"slide0008_image025.gif")
When y=
ou
see…
Fin=
d the
minimum value of a
function
16
Local Minimum value of a function
17
You
think…
![](3D"slide0010_image028.gif")
When y=
ou
see…
Find
critical numbers
18
Find critical numbers
19
You
think…
![](3D"slide0011_image031.gif")
When y=
ou
see…
Find
inflection points
20
Find inflection points
21
You
think…
![](3D"slide0012_image034.gif")
When y=
ou
see…
Show that =
;
exists
22
Show=
=
;
exists
<=
span
style=3D'visibility:hidden'>•
=
; &n=
bsp;
Show that
<=
span
style=3D'font-size:138%;visibility:hidden'>•
<=
span
style=3D'visibility:hidden'>•
23
You
think…
![](3D"slide0013_image040.gif")
When y=
ou
see…
Sho=
w that f(x) is continuous
24
.f(x) is continuous
25
You
think…
![](3D"slide0134_image043.gif")
When y=
ou
see…
Sho=
w that f(x) is differentiable
at x =3D a
26
f(x)=
is
differentiable
![](3D"slide0135_image045.gif")
27
You
think…
![](3D"slide0014_image046.gif")
When y=
ou
see…
Find
vertical
asymptotes of fx)
28
Find vertical asymptotes of f(x)
=
;
=
;
Factor/cancel f(x)
=
;
Set denominator =3D 0
29
You
think…
![](3D"slide0015_image047.gif")
When y=
ou
see…
Find
horizontal asymptotes of fx)
30
Find horizontal asymptotes of f(x)
31
You
think…
![](3D"slide0016_image050.gif")
When y=
ou
see…
Fin=
d the
average rate of chang=
e of f(x) at [a, ]
------=_NextPart_01C8AF5B.8E6927C0
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32
Average rate of change of f(x)
=
; &n=
bsp;
=
; &n=
bsp;
Find
=
;
f (b) -
f ( a)
=
; &n=
bsp;
b - a
33
You
think…
![](3D"slide0017_image051.gif")
When y=
ou
see…
Fin=
d the
instantaneous rate =
of
change of f(x)
at x =3D a
34
Instantaneous rate of change of f(x)
=
;
Find f ‘<=
span
style=3D'font-size:138%'> ( a)
35
You
think…
![](3D"slide0018_image052.gif")
When y=
ou
see…
&=
#13;
36
Average value of the function
37
You
think…
![](3D"slide0019_image057.gif")
When y=
ou
see…
Fin=
d the
absolute maximum of f(x) on [a, ]
38
Find the absolute maximum of f(x)
39
You
think…
![](3D"slide0020_image060.gif")
When y=
ou
see…
Show that a piecewise function is differentiable at the point a where the function
rule splits
40
Show a piecewise function is =
differentiable at x=3Da
41
You
think…
![](3D"slide0021_image063.gif")
When y=
ou
see…
Given st) (position function),
fin=
d
v(t)
42
Given position s(t), find v(t)
43
You
think…
![](3D"slide0022_image066.gif")
When y=
ou
see…
Giv=
en v(t), find how far a particle travels on [a, <=
/span>b]
44
Given v(t), find how far a particle travels on [a,b]
![](3D"slide0091_image068.gif")
45
You
think…
![](3D"slide0023_image069.gif")
When y=
ou
see…
Fin=
d the
average velocity of a pa=
rticle
------=_NextPart_01C8AF5B.8E6927C0
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on [=
a, b]
46
Find the average rate of change on [a,b]
![](3D"slide0088_image071.gif")
47
You
think…
![](3D"slide0024_image072.gif")
When y=
ou
see…
Giv=
en v(t), determine if a particle is speeding up at
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t =3D k
48
Given v(t), determine if the particle is <=
span
style=3D'position:absolute;top:12.75%;left:5.99%;width:88.2%;height:7.75%'=
>speeding up at t=3Dk
![](3D"slide0092_image074.gif")
49
You
think…
![](3D"slide0025_image075.gif")
When y=
ou
see…
Gi=
ven v(t) and sst)
50
Given v(t) and s(0), find s(t)
![](3D"slide0093_image077.gif")
51
You
think…
![](3D"slide0027_image078.gif")
When y=
ou
see…
Sho=
w that
the Mean Value Theorem ho=
lds
on [a, ]
52
Show that the MVT holds on [a,b]
![](3D"slide0095_image080.gif")
53
You
think…
![](3D"slide0026_image081.gif")
When y=
ou
see…
Sho=
w that
Rolle’s Theor=
em
holds on [a, b]
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54
Show that Rolle’s Theorem holds on <=
span
style=3D'position:absolute;top:12.75%;left:5.99%;width:88.2%;height:7.75%'=
>[a,b]
![](3D"slide0094_image083.gif")
55
![](3D"slide0028_image084.gif")
You
think…
When y=
ou
see…
Fin=
d the
domain
of fx)
56
Find the domain of f(x)
![](3D"slide0096_image086.gif")
57
You
think…
![](3D"slide0029_image087.gif")
When y=
ou
see…
Fin=
d the
range
of fx) on [ab
When you see…
![](3D"slide0097_image089.gif")
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When you see…
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58
Find the range of f(x) on [a,b]
59
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think…
When y=
ou
see…
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range
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60
Find the range of f(x) on
61
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think…
When y=
ou
see…
Find =
f ’(x) by definition
62
Find f R=
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63
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think…
When y=
ou
see…
Fin=
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derivative of the i=
nverse
of =
f(x) at xa
64
Derivative of the inverse of f(x) at x=3Da
65
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think…
When y=
ou
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y is increasi=
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66
.y is increasing proportionally to=
y
67
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When y=
ou
see…
Fi=
nd the
line <=
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/span>equal areas
68
Find the x=3Dc so the area under f(x) is <=
span
style=3D'position:absolute;top:12.0%;left:5.99%;width:88.2%;height:9.25%'>=
divided equally
![](3D"slide0103_image108.gif")
![](3D"slide0103_image110.gif")
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69
You
think…
When y=
ou
see…
70
Fundamental Theorem
71
You
think…
When y=
ou
see…
72
Fundamental Theorem, again
Given=
:
73
You
think…
When y=
ou
see…
The=
rate of
change of population is &#=
8230;
74
Rate of change of a population
75
You
think…
When y=
ou
see…
The=
line y =3D mx + b=
is tangent
to =
f(x) at (a, )
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76
.y =3D mx+b is tangent to f(x) at (a,b)
77
You
think…
When y=
ou
see…
Int=
egrate
78
1=
. Estimation:
LRAM
RRAM =
;
(Riemann Sums)
MRAM
Trapezoid
2=
. Geometry
3.An=
tiderivative
Straight Forward
Substitution
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Rewrite (Simplify)
Meth=
ods
for Integration
79
You
think…
When y=
ou
see…
Fin=
d area
using Left Riemann sums
80
Area using Left Riemann sums
81
You
think…
When y=
ou
see…
Fin=
d area
using Right Riem=
ann
sums
82
Area using Right Riemann sums
83
You
think…
When y=
ou
see…
Fin=
d area
using Midpoint rectang=
les
84
Area using midpoint rectangles
85
You
think…
When y=
ou
see…
Fin=
d area
using trapezoids=
86
Area using trapezoids
87
You
think…
When y=
ou
see…
Sol=
ve the
differential equa=
tion
…
88
Solve the differential equation...
89
You
think…
When y=
ou
see…
Me=
aning of
90
Meaning of the integral of f(t) from a to x
91
You
think…
When y=
ou
see…
Given
a base, cross secti=
ons
perpendicular to the =
x-ax=
is that
are squares <=
/span>
92
Semi-circular cross sections perpendicular to the x-axis
93
You
think…
When y=
ou
see…
Fin=
d where
the tangent line to f (x) is horizontal
94
Hori=
zontal
tangent line
95
You
think…
When y=
ou
see…
Fin=
d where
the tangent line to f (x) is vertical
96
Vertical tangent line to f(x)
97
You
think…
When y=
ou
see…
Find the minimum acceleration given v(t
When you see…
![](3D"slide0117_image164.gif")
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98
Given v(t), find minimum acceleration
99
You
think…
When y=
ou
see…
App=
roximate
the value ff at x
When you see…
![](3D"slide0118_image167.gif")
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100
Approximate f(0.1) using tangent line
t= o f(x) at x =3D 0
t= o f(x) at x =3D 0
101
You
think…
When y=
ou
see…
Giv=
en the
value of F(a) and
the fact that the
anti-derivative
of =
f is F, find Fb)
102
Given <= span style=3D'font-family:"Times New Roman";font-size:64%;text-shadow:auto'>= F(= a) and the that the
anti-derivativ= e of = f is F, find F(= b)
103
You
think…
When y=
ou
see…
Fin=
d the
derivative of
f(g(x))
104
Find the derivative of f(g(x))
Think . . .
Chain Rule
105
You
think…
When y=
ou
see…
=
Given =
;
, find
106
Given area under a curve and vertical shift, find the new area under the
curve
107
You
think…
When y=
ou
see…
Giv=
en a
graph of
find where fx) is
increasin=
g
108
Given a graph of f ‘(x) , find where f(x) is increasing
109
You
think…
When y=
ou
see…
Given
v(t) and sa, b]
110
Given <= span style=3D'font-family:"Times New Roman";font-size:64%;text-shadow:auto'>= v(= t) and s(0)= , find the greatest distance from= = the origin of a particle on [= a, <= /span>= b]
111
When y=
ou
see…
Given a
water tank with g gallons initially being filled at the rate of F(t) gallons/min and emptied at =
the rate of Et) gallons/min on
<=
/span>
=
;
, find
112
You
think…
a)the amount=
of
water in
=
the tank at m minutes
113
114
Amount of water in the tank at t minutes
init=
ial
gallons
115
You
think…
<=
span
style=3D'font-size:138%'>b) the rate the water
amount is chang=
ing
at m
116
Rate the amount of water is=
changing
at t =3D m
117
You
think…
c) t=
he
time when the =
;
water is at a minimum
118
The time when the water is at a minimum
119
You
think…
When y=
ou
see…
Giv=
en a
chart of x and fx) on selected values between a and b, estimate
where c is between =
a and=
b.
120
121
You
think…
When y=
ou
see…
Given , draw a =
slope
field
122
Draw a slope field of dy/dx
123
You
think…
When y=
ou
see…
Fin=
d the
area between curves=
f(x) and gx) on [a,b]
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124
Area between f(x) and g(x) on [a,b]
125
You
think…
When y=
ou
see…
Find the volume if the area =
between the curves f(xg(x)
with a representative rectangle =
perpendicular to the axis of rotation
126
Volume generated by rotating area between =
f(x) and g(x) with a representati=
ve
rectangle <=
i>perpendicular to the axis of rotation
127
You
think…
When y=
ou
see…
Find the volume if the area =
between the curves f(xg(x)
with a representative rectangle parallel to the axis of rotati=
on
128
Volume generated by rotating area between =
f(x) and g(x) with a representati=
ve
rectangle <=
i>parallel to the axis of rotation
Reme=
mber: Always . . . Big - sm=
all
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