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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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 .

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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

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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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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.

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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

.

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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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