Calculus Cheat Sheet - Lamar University
Calculus Cheat Sheet
Limits
Definitions
Precise Definition : We say lim f x L if xa
for every 0 there is a 0 such that
whenever 0 x a then f x L .
Limit at Infinity : We say lim f x L if we x
can make f x as close to L as we want by
taking x large enough and positive.
"Working" Definition : We say lim f x L xa
if we can make f x as close to L as we want
There is a similar definition for lim f x L x
except we require x large and negative.
by taking x sufficiently close to a (on either side of a) without letting x a .
Infinite Limit : We say lim f x if we xa
can make f x arbitrarily large (and positive)
Right hand limit : lim f x L . This has xa
the same definition as the limit except it
by taking x sufficiently close to a (on either side of a) without letting x a .
requires x a .
Left hand limit : lim f x L . This has the xa
There is a similar definition for lim f x xa
except we make f x arbitrarily large and
same definition as the limit except it requires xa.
negative.
Relationship between the limit and one-sided limits
lim f x L lim f x lim f x L
xa
xa
xa
lim f x lim f x L lim f x L
xa
xa
xa
lim f x lim f x lim f x Does Not Exist
xa
xa
x a
Properties
Assume lim f x and lim g x both exist and c is any number then,
x a
x a
1.
lim
xa
cf
x
c
lim
xa
f
x
2.
lim
xa
f
x
g
x
lim
xa
f
x
lim
xa
g
x
4.
lim
xa
f g
x x
lim
xa
lim
xa
f g
x x
provided
lim g x 0
xa
5.
lim
xa
f
xn
lxima
f
xn
3.
lim
xa
f
x
g
x
lim
xa
f
x
lim g x
xa
6.
lim
xa
n
f
x
n
lim
xa
f
x
Basic Limit Evaluations at
Note : sgn a 1 if a 0 and sgn a 1 if a 0 .
1. lim ex & lim ex 0
x
x
5. n even : lim xn x
2. lim ln x & lim ln x
x
x0
3.
If
r
0
then
lim
x
b xr
0
4. If r 0 and xr is real for negative x
then lim x
b xr
0
6. n odd : lim xn & lim xn
x
x
7. n even : lim a xn b x c sgn a x
8. n odd : lim a xn b x c sgn a x
9. n odd : lim a xn c x d sgn a x
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? 2005 Paul Dawkins
Calculus Cheat Sheet
Evaluation Techniques
Continuous Functions
L'Hospital's Rule
If f x is continuous at a then lim f x f a xa
If
lim
xa
f x g x
0 0
or
lim
xa
f x g x
then,
Continuous Functions and Composition
f x is continuous at b and lim g x b then xa
lim f g x f lim g x f b
xa
xa
Factor and Cancel
lim
x2
x2
x2
4x 12 2x
lim
x2
x
2 xx
x 2
6
lim
x2
x
x
6
8 2
4
Rationalize Numerator/Denominator
lim
x9
3 x x2 81
lim
x9
3 x x2 81
3 3
x x
lim
9x
lim
1
x9 x2 81 3 x x9 x 9 3 x
1
18
6
1 108
Combine Rational Expressions
lim
h0
1 h
x
1
h
1 x
lim
h0
1 h
x x h xx h
lim
h0
1 h
x
h x
h
lim
h0
x
1 x
h
1 x2
lim
xa
f x g x
lim xa
f x gx
a is a number,
or
Polynomials at Infinity
p x and q x are polynomials. To compute
lim
x
px qx
factor largest power of x in
q x out
of both p x and q x then compute limit.
lim
x
3x2 5x
2
4 x2
lim
x
x2 x2
3
4 x2
5 x
2
lim
x
3
4 x2
5 x
2
3 2
Piecewise Function
lim g x
x 2
where
g
x
x2 1
5 3x
if x 2 if x 2
Compute two one sided limits,
lim g x lim x2 5 9
x 2
x 2
lim g x lim 1 3x 7
x 2
x 2
One sided limits are different so lim g x x 2
doesn't exist. If the two one sided limits had
been equal then lim g x would have existed x 2
and had the same value.
Some Continuous Functions
Partial list of continuous functions and the values of x for which they are continuous.
1. Polynomials for all x. 2. Rational function, except for x's that give
division by zero.
7. cos x and sin x for all x. 8. tan x and sec x provided
3. n x (n odd) for all x. 4. n x (n even) for all x 0 . 5. ex for all x.
x
,
3 2
,
2
,
2
,
3 2
,
9. cot x and csc x provided
6. ln x for x 0 .
x , 2 , , 0, , 2 ,
Intermediate Value Theorem
Suppose that f x is continuous on [a, b] and let M be any number between f a and f b .
Then there exists a number c such that a c b and f c M .
Visit for a complete set of Calculus notes.
? 2005 Paul Dawkins
Calculus Cheat Sheet
Derivatives
Definition and Notation
If
y
f
x
then the derivative is defined to be
f x lim h0
f
x h
h
f
x .
If y f x then all of the following are
equivalent notations for the derivative.
f
x
y
df dx
dy dx
d dx
f
x
Df
x
If y f x all of the following are equivalent
notations for derivative evaluated at x a .
f a
y xa
df dx
xa
dy dx
xa
Df
a
If y f x then,
Interpretation of the Derivative
2. f a is the instantaneous rate of
1. m f a is the slope of the tangent
change of f x at x a .
line to y f x at x a and the
3. If f x is the position of an object at
equation of the tangent line at x a is
given by y f a f a x a .
time x then f a is the velocity of
the object at x a .
Basic Properties and Formulas
If f x and g x are differentiable functions (the derivative exists), c and n are any real numbers,
1. c f c f x
2. f g f x g x
3. f g f g f g ? Product Rule
4.
f g
fg f g2
g
? Quotient Rule
5.
d dx
c
0
6.
d dx
xn
n xn1 ? Power Rule
7.
d dx
f
g x
f
g x g x
This is the Chain Rule
d dx
x
1
d dx
sin
x
cos
x
d dx
cos
x
sin
x
d dx
tan
x
sec2
x
d dx
sec
x
sec
x
tan
x
Common Derivatives
d dx
csc
x
csc
x
cot
x
d dx
cot
x
csc2
x
d
dx
sin1 x
1 1 x2
d
dx
cos1 x
1 1 x2
d
dx
tan1 x
1 1 x2
d dx
a
x
a
x
ln
a
d dx
e
x
e
x
d dx
ln
x
1 x
,
x0
d dx
ln
x
1 x
,
x0
d dx
log
a
x
x
1 ln
a
,
x0
Visit for a complete set of Calculus notes.
? 2005 Paul Dawkins
Calculus Cheat Sheet
Chain Rule Variants
The chain rule applied to some specific functions.
1.
d dx
f xn
n f xn1 f x
5.
d dx
cos f x
f xsin f x
2.
d dx
e f x
f xe f x
6.
d dx
tan f x
f xsec2 f x
3.
d
dx
ln f
x
f x f x
7.
d dx
sec
f
(
x)
f ( x) sec
f (x) tan
f (x)
4.
d
dx
sin
f
x
f
x cos
f
x
8.
d dx
tan1 f x
1
f
f
x x
2
Higher Order Derivatives
The Second Derivative is denoted as
The nth Derivative is denoted as
f
x
f
2 x
d2 f dx2
and is defined as
f
n
x
dn f dxn
and is defined as
f x f x , i.e. the derivative of the
f n x f n1 x , i.e. the derivative of
first derivative, f x .
the (n-1)st derivative, f n1 x .
Implicit Differentiation
Find y if e2x9 y x3 y2 sin y 11x . Remember y y x here, so products/quotients of x and y
will use the product/quotient rule and derivatives of y will use the chain rule. The "trick" is to differentiate as normal and every time you differentiate a y you tack on a y (from the chain rule).
After differentiating solve for y .
e2x9 y 2 9 y 3x2 y2 2x3 y y cos y y 11 2e2 x9 y 9 ye2 x9 y 3x2 y2 2x3 y y cos y y 11
2x3 y 9e2x9 y cos y y 11 2e2x9 y 3x2 y 2
y
11 2e2 x9 y 3x2 y 2
2x3 y 9e2x9 y cos y
Increasing/Decreasing ? Concave Up/Concave Down
Critical Points
x c is a critical point of f x provided either Concave Up/Concave Down
1. f c 0 or 2. f c doesn't exist.
1. If f x 0 for all x in an interval I then
f x is concave up on the interval I.
Increasing/Decreasing
1. If f x 0 for all x in an interval I then
f x is increasing on the interval I.
2. If f x 0 for all x in an interval I then f x is concave down on the interval I.
2. If f x 0 for all x in an interval I then f x is decreasing on the interval I.
3. If f x 0 for all x in an interval I then
Inflection Points
x c is a inflection point of f x if the
concavity changes at x c .
f x is constant on the interval I.
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? 2005 Paul Dawkins
Calculus Cheat Sheet
Absolute Extrema
1. x c is an absolute maximum of f x if f c f x for all x in the domain.
2. x c is an absolute minimum of f x if f c f x for all x in the domain.
Extrema Relative (local) Extrema 1. x c is a relative (or local) maximum of
f x if f c f x for all x near c.
2. x c is a relative (or local) minimum of
f x if f c f x for all x near c.
Fermat's Theorem
If f x has a relative (or local) extrema at
x c , then x c is a critical point of f x .
1st Derivative Test
If x c is a critical point of f x then x c is 1. a rel. max. of f x if f x 0 to the left
of x c and f x 0 to the right of x c .
Extreme Value Theorem
If f x is continuous on the closed interval
2. a rel. min. of f x if f x 0 to the left of x c and f x 0 to the right of x c .
a,b then there exist numbers c and d so that, 3. not a relative extrema of f x if f x is
1. a c, d b , 2. f c is the abs. max. in
the same sign on both sides of x c .
a,b , 3. f d is the abs. min. in a,b .
Finding Absolute Extrema To find the absolute extrema of the continuous
function f x on the interval a,b use the
following process.
1. Find all critical points of f x in a,b . 2. Evaluate f x at all points found in Step 1.
2nd Derivative Test
If x c is a critical point of f x such that
f c 0 then x c
1. is a relative maximum of f x if f c 0 .
2. is a relative minimum of f x if f c 0 .
3. may be a relative maximum, relative
minimum, or neither if f c 0 .
3. Evaluate f a and f b .
4. Identify the abs. max. (largest function value) and the abs. min.(smallest function value) from the evaluations in Steps 2 & 3.
Finding Relative Extrema and/or Classify Critical Points
1. Find all critical points of f x .
2. Use the 1st derivative test or the 2nd derivative test on each critical point.
Mean Value Theorem
If f x is continuous on the closed interval a,b and differentiable on the open interval a,b
then there is a number
acb
such that
f c
f
b
b
f a
a
.
Newton's Method
If
x n
is the nth guess for the
root/solution of
f
x 0
then (n+1)st
guess is
xn 1
xn
f xn f xn
provided f xn exists.
Visit for a complete set of Calculus notes.
? 2005 Paul Dawkins
Calculus Cheat Sheet
Related Rates Sketch picture and identify known/unknown quantities. Write down equation relating quantities and differentiate with respect to t using implicit differentiation (i.e. add on a derivative every time you differentiate a function of t). Plug in known quantities and solve for the unknown quantity.
Ex. A 15 foot ladder is resting against a wall.
The bottom is initially 10 ft away and is being
pushed towards the wall at
1 4
ft/sec.
How fast
is the top moving after 12 sec?
Ex. Two people are 50 ft apart when one starts walking north. The angle changes at
0.01 rad/min. At what rate is the distance between them changing when 0.5 rad?
x is negative because x is decreasing. Using Pythagorean Theorem and differentiating,
x2 y2 152 2x x 2 y y 0
After 12 sec we have
x
10
12
1 4
7
and
so y 152 72 176 . Plug in and solve for y .
7
1 4
176 y 0 y 4
7 ft/sec 176
We have 0.01 rad/min. and want to find
x . We can use various trig fcns but easiest is,
sec
x 50
sec
tan
x 50
We know 0.5 so plug in and solve.
sec
0.5
tan
0.5
0.01
x 50
x 0.3112 ft/min Remember to have calculator in radians!
Optimization
Sketch picture if needed, write down equation to be optimized and constraint. Solve constraint for
one of the two variables and plug into first equation. Find critical points of equation in range of
variables and verify that they are min/max as needed.
Ex. We're enclosing a rectangular field with 500 ft of fence material and one side of the field is a building. Determine dimensions that
Ex. Determine point(s) on y x2 1 that are closest to (0,2).
will maximize the enclosed area.
Maximize A xy subject to constraint of
x 2 y 500 . Solve constraint for x and plug
into area.
x 500 2 y A y 500 2 y
500 y 2 y2
Differentiate and find critical point(s). A 500 4 y y 125
By 2nd deriv. test this is a rel. max. and so is the answer we're after. Finally, find x.
x 500 2125 250
The dimensions are then 250 x 125.
Minimize f d 2 x 02 y 22 and the
constraint is y x2 1. Solve constraint for
x2 and plug into the function.
x2 y 1 f x2 y 22
y 1 y 22 y2 3y 3
Differentiate and find critical point(s).
f 2y3
y
3 2
By the 2nd derivative test this is a rel. min. and
so all we need to do is find x value(s).
x2
3 2
1
1 2
x 1 2
The 2 points are then
,1 3
22
and
,1 3
22
.
Visit for a complete set of Calculus notes.
? 2005 Paul Dawkins
Calculus Cheat Sheet
Integrals
Definitions
Definite Integral: Suppose f x is continuous Anti-Derivative : An anti-derivative of f x
on a,b . Divide a,b into n subintervals of
is a function, F x , such that F x f x .
width x and choose xi* from each interval.
Then
b f x dx lim n f
a
n i1
xi* x .
Indefinite Integral : f x dx F x c
where F x is an anti-derivative of f x .
Fundamental Theorem of Calculus
Part I : If f x is continuous on a,b then
g
x
x
a
f
t dt
is also continuous on
a,b
Variants of Part I :
d dx
ux a
f
t dt
ux
f
u x
and
gx
d x
dx a
f
t dt
f
x .
Part II : f x is continuous ona,b , F x is
an anti-derivative of f x (i.e. F x f x dx )
d dx
b
vx
f
t dt
v x
f
v x
d
dx
ux f t dt u x f u( x) v x f v(x)
v x
then
b
a
f
x dx
F
b
F
a.
Properties
f x g x dx f x dx g x dx
cf x dx c f x dx , c is a constant
b
a
f
x
g
x dx
b
a
f
x dx
b
a
g
x dx
b
a
cf
x dx
c b a
f
x dx
,
c
is
a
constant
a
a
f
x dx
0
b
a
c
dx
c
b
a
b
a
f
x dx
a
b
f
x dx
b
a
f
x dx
b
a
f
x
dx
b
a
f
x
dx
c
a
f
x
dx
b
c
f
x dx
for any value of c.
If
f x gx
on a x b then
b
a
f
x dx
b
a
g
x dx
If
f
x 0
on
a
xb
then
b
a
f
x dx 0
If
m
f
x
M
on
a
x
b
then
mb a
b
a
f
x dx
M
b a
k dx k x c
xn
dx
1 n1
x n 1
c, n
1
x1 dx
1
x
dx
ln
x
c
Common Integrals
cos u du sin u c sin u du cos u c sec2 u du tan u c
1 ax b
dx
1
a
ln
ax
b
c
ln u du u ln u u c
eu du eu c
sec u tan u du sec u c csc u cot udu csc u c csc2 u du cot u c
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tan u du ln sec u c
sec u du ln secu tan u c
1 a2
u2
du
1
a
tan 1
u a
c
1 du sin1 a2 u2
u a
c
? 2005 Paul Dawkins
Calculus Cheat Sheet
Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.
u Substitution : The substitution
u
g x will convert
b
a
f
g
x
g
x
dx
gb ga
f
u
du
using
du g x dx . For indefinite integrals drop the limits of integration.
Ex. 2 5x2 cos x3 dx 1
2 5x2 cos 1
x3
dx
8 1
5 3
cos
u
du
u x3
du 3x2dx
x2dx
1 3
du
x 1 u 13 1 :: x 2 u 23 8
5 3
sin u
8 1
5 3
sin 8
sin
1
Integration by Parts :
u dv uv v du and
b u dv uv
a
b a
b v du .
a
Choose u and dv from
integral and compute du by differentiating u and compute v using v dv .
Ex. xex dx
u x dv e x du dx v ex
xe x dx xe x ex dx xe x ex c
Ex. 5 ln x dx 3
u ln x
dv dx
du
1 x
dx
v x
5 3
ln
x
dx
x
ln
x
5 3
5 dx
3
x ln x x
5 3
5ln 5 3ln 3 2
Products and (some) Quotients of Trig Functions
For sinn x cosm x dx we have the following :
For tann x secm x dx we have the following :
1. n odd. Strip 1 sine out and convert rest to 1. n odd. Strip 1 tangent and 1 secant out and
cosines using sin2 x 1 cos2 x , then use the substitution u cos x .
convert the rest to secants using tan2 x sec2 x 1, then use the substitution
2. m odd. Strip 1 cosine out and convert rest
u sec x .
to sines using cos2 x 1 sin2 x , then use the substitution u sin x .
2. m even. Strip 2 secants out and convert rest to tangents using sec2 x 1 tan2 x , then
3. n and m both odd. Use either 1. or 2.
use the substitution u tan x .
4. n and m both even. Use double angle
3. n odd and m even. Use either 1. or 2.
and/or half angle formulas to reduce the
4. n even and m odd. Each integral will be
integral into a form that can be integrated.
dealt with differently.
Trig
Formulas
:
sin 2x
2sin xcos x ,
cos2
x
1 2
1 cos2x ,
sin 2
x
1 2
1 cos2x
Ex. tan3 x sec5 x dx
tan3 x sec5 xdx tan2 x sec4 x tan x sec xdx
sec2 x 1 sec4 x tan x sec xdx
u2 1u4du
u sec x
1 7
sec7
x
1 5
sec5
x
c
Ex.
sin5 cos3
x x
dx
sin5 cos3
x x
dx
sin4 xsin cos3 x
x
dx
(sin2 x)2 sin cos3 x
x
dx
(1cos2 x)2 sin x cos3 x
dx
u cos x
(1u2 )2 u3
du
12u2 u4 u3
du
1 2
sec2
x
2 ln
cos
x
1 2
cos2
x
c
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? 2005 Paul Dawkins
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