13. When you see…
Find the interval where the
slope of f (x) is increasing
You think…
14. Slope of f (x) is increasing
Find the derivative of f ’(x) = f “ (x)
Set numerator and denominator = 0
to find critical points
Make sign chart of f “ (x)
Determine where it is positive
22. Find inflection points
Express f “ (x) as a fraction
Set numerator and denominator = 0
Make a sign chart of f “ (x)
Find where it changes sign
( + to - ) or ( - to + )
37. When you see…
Find the absolute
minimum of f(x) on [a, b]
You think…
38. Find the absolute minimum of f(x)
a) Make a sign chart of f ’(x)
b) Find all relative maxima
c) Plug those values into f (x)
d) Find f (a) and f (b)
e) Choose the largest of c) and d)
39. When you see…
Show that a piecewise
function is differentiable
at the point a where the
function rule splits
You think…
40. Show a piecewise function is
differentiable at x=a
First, be sure that the function is continuous at
x =a .
Take the derivative of each piece and show that
lim− f ′( x ) = lim f ′( x )
x →a x →a +
43. When you see…
Given v(t), find how far a
particle travels on [a, b]
You think…
44. Given v(t), find how far a particle
travels on [a,b]
b
v ()
∫t dt
Find a
45. When you see…
Find the average
velocity of a particle
on [a, b]
You think…
46. Find the average rate of change on
[a,b]
b
∫) s(s(
v(
t dt
b) )
−a
Find a
b−
a
=
b−
a
47. When you see…
Given v(t), determine if a
particle is speeding up at
t=a
You think…
48. Given v(t), determine if the particle is
speeding up at t=a
Find v (k) and a (k).
Multiply their signs.
If positive, the particle is speeding up.
If negative, the particle is slowing down
50. Given v(t) and s(0), find s(t)
s()=
t ∫ v ()dt + C
t
Plug in t = 0 to find C
51. When you see…
Show that Rolle’s
Theorem holds on [a, b]
You think…
52. Show that Rolle’s Theorem holds on
[a,b]
Show that f is continuous and differentiable
on the interval
If f () f ( , then find some c in [b]
a = b) a,
such that f ′c)0.
(=
53. When you see…
Show that the Mean
Value Theorem holds
on [a, b]
You think…
54. Show that the MVT holds on [a,b]
Show that f is continuous and differentiable
on the interval.
Then find some c such that
f (−(
b) f a )
f ′c)
(= b − .a
56. Find the domain of f(x)
Assume domain is (−∞, ∞ ).
Domain restrictions: non-zero denominators,
Square root of non negative numbers,
Log or ln of positive numbers
62. Find f ‘( x) by definition
f ( +h ) ()
x −f x
() h→
f ′ x =lim
0 h
or
f () ()
x −f a
() x → x −a
f ′ x =lim
a
63. When you see…
Find the derivative of
the inverse of f(x) at x = a
You think…
64. Derivative of the inverse of f(x) at x=a
Interchange x with y.
dy
Solve for dx implicitly (in terms of y).
Plug your x value into the inverse relation
and solve for y.
dy
Finally, plug that y into your dx .
65. When you see…
y is increasing
proportionally to y
You think…
66. . y is increasing proportionally to y
dy
=
ky
dt
translating to
y=
Ce kt
67. When you see…
Find the line x = c that
divides the area under
f(x) on [a, b] into two
equal areas
You think…
68. Find the x=c so the area under f(x) is
divided equally
c b
∫ f (x )dx =∫ f (x )dx
a c
75. When you see…
The line y = mx + b is
tangent to f(x) at (a, b)
You think…
76. y = mx+b is tangent to f(x) at (a,b)
.
Two relationships are true.
The two functions share the same
slope ( m = f ′( x ) )
and share the same y value at x1 .
80. Area using right Riemann sums
A = base[ x1 + x 2 + x3 + ... + x n ]
81. When you see…
Find area using
midpoint rectangles
You think…
82. Area using midpoint rectangles
Typically done with a table of values.
Be sure to use only values that are
given.
If you are given 6 sets of points, you can
only do 3 midpoint rectangles.
84. Area using trapezoids
base
A= [ x0 + 2 x1 + 2 x 2 + ... + 2 x n − 1 + x n ]
2
This formula only works when the base is the
same.
If not, you have to do individual trapezoids
88. Meaning of the integral of f(t) from a to x
The accumulation function –
accumulated area under the
function f ( x )
starting at some constant a
and ending at x
89. When you see…
Given a base, cross
sections perpendicular to
the x-axis that are
squares
You think…
90. Semi-circular cross sections
perpendicular to the x-axis
The area between the curves typically
is the base of your square.
b
So the volume is ∫ (base )dx
2
a
91. When you see…
Find where the tangent
line to f(x) is horizontal
You think…
98. Approximate f(0.1) using tangent line
to f(x) at x = 0
Find the equation of the tangent line to f
using y − y1 = m( x − x1 )
where m = f ′( 0) and the point is ( 0, f ( 0) ) .
Then plug in 0.1 into this line.
Be sure to use an approximation ( ≈) sign.
99. When you see…
Given the value of F(a)
and the fact that the
anti-derivative of f is F,
find F(b)
You think…
100. Given F(a) and the that the
anti-derivative of f is F, find F(b)
Usually, this problem contains an antiderivative
you cannot take. Utilize the fact that if F (x )
is the antiderivative of f,
b
then ∫F (x )dx =F (b) −F (a ) .
a
Solve for F (b ) using the calculator
to find the definite integral
103. When you see…
b b
Given ∫ f (x )dx , find ∫ [f (x )+ k ]dx
a a
You think…
104. Given area under a curve and vertical
shift, find the new area under the curve
b b b
∫ [ f ( x ) + k ] dx = ∫ f ( x )dx + ∫ kdx
a a a
105. When you see…
Given a graph of f '( x )
find where f(x) is
increasing
You think…
106. Given a graph of f ‘(x) , find where f(x) is
increasing
Make a sign chart of f ′( x )
Determine where f ′( x ) is positive
107. When you see…
Given v(t) and s(0), find the
greatest distance from the
origin of a particle on [a, b]
You think…
108. Given v(t) and s(0), find the greatest distance from
the origin of a particle on [a, b]
Generate a sign chart of v( t ) to find
turning points.
Integrate v( t ) using s ( 0 ) to find the
constant to find s( t ) .
Find s(all turning points) which will give
you the distance from your starting point.
Adjust for the origin.
109. When you 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 E(t) gallons/min on
[t1 , t2 ] , find
110. a) the amount of water in
the tank think…
You at m minutes
111. Amount of water in the tank at t minutes
t2
g + ∫( F (t ) − E ( t ) )dt
t
112. b) the rate the water
amount is changing
at m
You think…
113. Rate the amount of water is
changing at t = m
m
d
∫ ( F ( t ) − E ( t ) )dt = F ( m ) − E ( m )
dt t
114. c) the time when the
water is at a minimum
You think…
115. The time when the water is at a minimum
F ( m ) − E ( m ) = 0,
testing the endpoints as well.
116. When you see…
Given a chart of x and f(x)
on selected values between
a and b, estimate f '( x ) where
c is between a and b.
You think…
117. Straddle c, using a value k greater
than c and a value h less than c.
f (k ) −f (h )
so f ′c) ≈
(
k− h
118. When you see…
dy
Given dx , draw a
slope field
You think…
119. Draw a slope field of dy/dx
Use the given points
dy
Plug them into dx ,
drawing little lines with the
indicated slopes at the points.
120. When you see…
Find the area between
curves f(x) and g(x) on
[a,b]
You think…
121. Area between f(x) and g(x) on [a,b]
b
A = ∫[ f ( x ) − g ( x )]dx
a
,
assuming f (x) > g(x)
122. When you see…
Find the volume if the
area between the curves
f(x) and g(x) is
rotated about the x-axis
You think…
123. Volume generated by rotating area between
f(x) and g(x) about the x-axis
∫ [( f ( x ) ) ]
b
− ( g ( x ) ) dx
2 2
A=
a
assuming f (x) > g(x).
