Ken’s Cheat Sheet for AP Calculus
f ( x  h)  f ( x )
f ( x)  lim
h
h0
d
(u n )  nu n 1  u '
dx
d
(sin u )  cos  u   u '
dx
d
(cot u )   csc 2  u   u '
dx
d u
(e )  e u  u '
dx
d
1
( Arc sin u ) 
u '
dx
1 u2
d
 f  g   f  g  g  f 
dx
d
f ( g ( x))  f ( g ( x)) g ( x)
dx
d
(cos u )   sin  u   u '
dx
d
(sec u )  sec  u  tan  u   u '
dx
d u
(a )  a u  u ' ln a
dx
d
1
( Arc tan u ) 
u '
dx
1 u2
d  N  DN ' ND '

dx  D 
D2
dy dy du
Chain Rule


dx du dx
d
(tan u )  sec 2  u   u '
dx
d
(csc u )   csc  u  cot  u   u '
dx
d
1
(ln u )   u '
dx
u
d
1
( Arc sec u ) 
u '
dx
| u | u2 1
 a dx  ax  C
x n 1
 x dx  n  1  C, n  1
au
u
a

u
'

C

ln a
 sec  u  tan  u   u '  sec u  C
 u  u '  ln u 
e
u
n
 u '  eu  C
 sin  u   u ' dx   cos u  C
 cos  u   u '  sin u  C
 csc  u  cot  u   u '   csc u  C
 tan  u   u '  ln sec  u   C
 cot  u   u '  ln sin  u   C
 csc  u   u '  ln csc  u   cot  u   C  ln x dx  x ln x  x  C
Avg .Value of A function =
 V (t )dt  displacement
1 b
f ( x)dx
b  a a
1
a
u
C
 u ' ln a  a u  C
 sec  u   u '  tan u  C
 csc  u   u '   cot u  C
 sec  u   u '  ln sec  u   tan  u   C
2
2
 f  x dx  F (b)  F (a)  NET change in F
 V (t ) dt  Distance Traveled speed  velocity
b
a
t
S (t )  S0   V ( x) dx
"Postion is determined by adding the change in position to the displacement"
0
If Velocity and Acceleration have the same sign, then speed is increasing.
Fundamental Theorem of Calculus Part I
If F ( x) 
Fundamental Theorem of Calculus Part II
Formal Definition of Continuity

v
u
f (t )dt , then F '( x)  f (v)v '- f (u)u '
 f  x  dx  F (b)  F (a), where F (x ) is the antiderivative of f ( x)
b
a
lim f ( x )  f (c ) If and Only If f is continuous at x = c.
x c
“When looking for a limit at x = c, substitution works if and only if the function is continuous at x = c”
Average Rate of Change = Slope of a Secant Line
Instantaneous Rate of Change = Slope of a Tangent Line
For a limit to exist, the left and right hand limits must agree (be equal)
Washer Method…. V    R 2  r 2 dx....
V     R  r  dx
“Look for extremes at critical numbers and endpoints.”
2
If a function is differentiable at point x  a , it is continuous at that point. The converse is false, in other
words, continuity does not imply differentiability
Intermediate Value Theorem: If a function is continuous on [a,b], then it takes on every value between f (a ) and f (b) .
Extreme Value Theorem: “Every Closed Endpoint is an Extreme”
If f is continuous over a closed interval, then f has a maximum and minimum value over that interval
Mean Value Theorem (for derivatives)
If f is continuous on
“For every Secant line, there is a parallel Tangent”
a, b and differentiable on a, b
then at some point between a and b :
f (b)  f (a )
 f  ( c)
ba
Mean Value Theorem (for definite integrals) “The function passes through its average value”
1 b
If f is continuous on a, b then at some point c in  a, b , f  c  
f  x  dx
b  a a
Rolle’s Theorem “A well behaved function that is not one to one will have a horizontal tangent”
If f is continuous on a, b and differentiable on a, b such that f ( a )  f (b) , then there is at least
one number c in the open interval a, b such that f (c)  0 .
L’Hôpital’s Rule
lim
If
x a
f ( x)
f ( x)
f ( x)
f ( x)
0

is of the form
, and if lim
exists, then lim
.
or
 lim
x

a
x

a
x

a
0

g ( x)
g ( x)
g ( x)
g ( x)
x 2  2x
lim
0
x   x3  3
x3  2x
lim

x   x2  8
2 x 2  3x  2
2
lim

5
x   10 x  5 x 2
Use Ratio of Leading Terms
Q:
“What do we do with derivatives?” A1: We set them equal to zero and sign test them.
A2: We integrate them (find the area under them)
Definition of Concavity
If f '  x  is increasing, then f ( x) is concave up.
If f '  x  is decreasing, then f ( x) is concave down.
Rate of increase is INCREASING…CONCAVE UP
Rate of increase is DECREASING….CONCAVE DOWN.
1st Derivative shows: maximum ( + to -) and minimum (-0 to +) values, increasing and decreasing intervals,
slope of the tangent line to the curve, and velocity
2nd Derivative shows: inflection points, concavity, and acceleration
2nd Derivative Test for Extremes
f (c)  0, and f ' '(c)  0, then f has a local maximum at x = c.
f (c)  0, and f ' '(c)  0, then f has a local minimum x = c.
2nd Derivative Test for Concavity
If f ( x)  0 in a, b , then f is concave upward in a, b .
If f ( x)  0 in a, b , then f is concave downward in a, b .
st
1 Derivative Test for Increasing and Decreasing
If f (c)  0 then
f ( x) is INCREASING at x = c.
If f (c)  0 then
f ( x) is DECREASING at x = c.
Definition of Critical Number.
c is a critical number of f if and only if c is in the domain of f and f '(c)  0 or f '(c) fails to exist.
Definition of Inflection Point
c is an inflection point of f if and only if f changes concavity at x  c and f has a tangent line at x  c.
Exponential Growth Formula:
“The rate of growth is proportional to the amount present.”
K
ln( what happened )
how long it took
dP
 kP "math" for exponential growth.
dt
P(t )  P0 e kt
When separating variables…Find C as soon as possible. Separate…..Integrate…..Find C…….Get Y alone.
The definite integral of the rate of change of f yields the net change in f over the given interval.
Ex: The integral from a to b of the rate of change of Population yields the change in Population from a to b.
Integral means: Area “under” the curve
number of rectangles)
Anti-derivative
Accumulation
Net change Summation (often an infinite
Inverse points have reciprocal slopes