Definitions, Formulas, Tests, and Theorems to Memorize
1) lim f ( x )
xa
exists iff
lim f ( x)  lim f ( x)
x a 
x a 
2) f(x) is continuous at x=a iff f (a)  lim f ( x)
x a
3) Intermediate Value Theorem: if f(x) is continuous on [a,b], and k is between f(a) and f(b), f(c) = k for some c in [a,b].
4) Mean Value Theorem: if f(x) is continuous on [a,b] and differentiable on (a,b), ∃ a c in (a,b) s.t. f '(c) 
5) Definition of the Derivative:
lim
h 0
f ( x  h)  f ( x )
h
6) Average Rate of Change: of f(x) over the interval [a,b] is
7) Average Value: of f(x) over the interval [a,b] is

b
a
lim
or
xa
f (b)  f (a )
ba
f ( x)  f (a)
xa
f (b)  f ( a )
ba
f ( x)dx
ba
8) If (a,b) is on the graph of f(x) and 𝑔(𝑥) = 𝑓 −1 (𝑥) then g '(b) 
1
f '( a )
9) If f’’(x)>0, f(x) is concave up and f’(x) is increasing; if f’’(x)<0, f(x) is concave down and f’(x) is decreasing
10) First Derivative Test: if f’(x) changes from positive to negative at x=c, then f(x) has a local maximum at c
11) First Derivative Test: if f’(x) changes from negative to positive at x=c, then f(x) has a local minimum at c
12) Second Derivative Test: if f’(c)=0 and f’’(c)<0 then f(x) has a local maximum at c
13) Second Derivative Test: if f’(c)=0 and f’’(c)>0 then f(x) has a local minimum at c
14) f(x) has a point of inflection when f’’(x) changes sign
15) Tangent lines and Euler’s Method’s steps are below the curve and are thus an under approximation when f’’(x)>0
16) Tangent lines and Euler’s Method’s steps are above the curve and are thus an over approximation when f’’(x)<0
17) Secant lines are above the curve and thus Trapezoidal Approximations are over approximations when f’’(x)>0
18) Secant lines are below the curve and thus Trapezoidal Approximations are under approximations when f’’(x)<0
19) Area of a trapezoid:
1
h  b1  b2 
2
20) Fundamental Theorem of Calculus: for any constant a,
d x
f (t )dt = f ( x )
dx a
21) FTC Extension: if the bounds are differentiable functions, then
d h( x)
f (t )dt = f  h( x)  h '( x)  f  g ( x)  g '( x)
dx g ( x )
22) Euler’s Method: the change in y can be approximated by
dy
x
dx
23) Logistic Growth: the carrying capacity C = lim P (t ) occurs when
t 
dP
0
dt
1
2
24) Logistic Growth: if the carrying capacity is C, then P(t) has an inflection point and grows fastest when P(t)= C
25) Arc Length (rectangular): of f(x) over the interval [a,b] is
26) L’Hôpital’s Rule: if lim
x a
b
a
1   f '( x) dx
2
f ( x)
f ( x)
f '( x)
0

is of the form
= lim
, if it exists
or  then lim
x

a
x

a
0

g ( x)
g ( x)
g '( x)

27) Nth Term Test: for

a
n0
n
, if lim an  0 , then the series diverges; if lim an  0 then the test is inconclusive
n 

28) P-series Test: the series
1
n
n0
p
n 
converges when p  1 and diverges when p  1

29) Geometric Series Test: the series
 k r 
n
converges when r  1 and diverges when r  1
n 0
30) Ratio Test: examine lim
n 
an1
 L ; the series converges when L  1 and diverges when L  1
an
31) Direct Comparison Test: compare the series to one that is larger and converges or that is smaller and diverges
32) Slope of Tangent Line (Parametric):
dy dy / dt

dx dx / dt
dy
d 2 y derivative of
dx
33) Parametric 2 Derivative:

dx / dt
dx 2
nd
34) Speed (Parametric) : given position vector x(t ), y(t ) , speed =
 x '(t )   y '(t )
2
35) Total Distance = Arc Length (Parametric): for x(t ), y(t ) over [a,b]=
36) Slope of Tangent Line (Polar):
2
  x '(t )   y '(t ) dt
b
2
2
a
dy dy / d

, so first you must write y = r sin  and x = r cos 
dx dx / d
37) Area between a polar curve r(θ) and the origin for      is
1  2
r d
2 