Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Calculus Cheat Sheet

Limits
Definitions
Precise Definition : We say lim f ( x ) = L if
Limit at Infinity : We say lim f ( x ) = L if we
x ®a

x ®¥

for every e > 0 there is a d > 0 such that
whenever 0 < x - a < d then f ( x ) - L < e
...


“Working” Definition : We say lim f ( x ) = L

There is a similar definition for lim f ( x ) = L

if we can make f ( x ) as close to L as we want
by taking x sufficiently close to a (on either side
of a) without letting x = a
...


x ®a

Right hand limit : lim+ f ( x ) = L
...

Left hand limit : lim- f ( x ) = L
...

There is a similar definition for lim f ( x ) = -¥
x ®a

except we make f ( x ) arbitrarily large and
negative
...

Relationship between the limit and one-sided limits
lim f ( x ) = L Þ lim+ f ( x ) = lim- f ( x ) = L
lim+ f ( x ) = lim- f ( x ) = L Þ lim f ( x ) = L
x ®a

x ®a

x ®a

x ®a

x ®a

x ®a

lim f ( x ) ¹ lim- f ( x ) Þ lim f ( x ) Does Not Exist

x ®a +

x ®a

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 é f ( x ) ± g ( x ) ù = lim f ( x ) ± lim g ( x )
û x®a
x ®a ë
x ®a
3
...
lim ê
provided lim g ( x ) ¹ 0
ú=
x ®a
x ®a g ( x )
ë
û lim g ( x )
x ®a
n

n
5
...
lim é n f ( x ) ù = n lim f ( x )
û
x ®a ë
x®a

Basic Limit Evaluations at ± ¥
Note : sgn ( a ) = 1 if a > 0 and sgn ( a ) = -1 if a < 0
...
lim e x = ¥ &
x®¥

2
...
If r > 0 and x r is real for negative x
b
then lim r = 0
x ®-¥ x
3
...
n even : lim x n = ¥
x ®± ¥

6
...
n even : lim a x + L + b x + c = sgn ( a ) ¥
n

x ®± ¥

8
...
n odd : lim a x n + L + c x + d = - sgn ( a ) ¥

Visit http://tutorial
...
lamar
...


x ®-¥

© 2005 Paul Dawkins

Calculus Cheat Sheet

Evaluation Techniques
Continuous Functions
L’Hospital’s Rule
f ( x) 0
f ( x) ± ¥
If f ( x ) is continuous at a then lim f ( x ) = f ( a )
x ®a
If lim
= or lim
=
then,
x ®a g ( x )
x ®a g ( x )
0
±¥
Continuous Functions and Composition
f ( x)
f ¢( x)
lim
= lim
a is a number, ¥ or -¥
f ( x ) is continuous at b and lim g ( x ) = b then
x ®a g ( x )
x ®a g ¢ ( x )

(

)

x ®a

lim f ( g ( x ) ) = f lim g ( x ) = f ( b )
x ®a

x ®a

Polynomials at Infinity
p ( x ) and q ( x ) are polynomials
...


(

(

)

Piecewise Function

)

-1
1
=(18)( 6 ) 108
Combine Rational Expressions
1æ 1
1ö
1 æ x - ( x + h) ö
lim ç
- ÷ = lim ç
÷
h ®0 h x + h
x ø h ®0 h ç x ( x + h ) ÷
è
è
ø
1 æ -h ö
1
-1
= lim ç
=- 2
÷ = lim
h ®0 h ç x ( x + h ) ÷
h®0 x ( x + h )
x
è
ø
=

)

x 2 3 - 42
3 - 42
3x 2 - 4
3
x
lim
= lim 2 5
= lim 5 x = x ®-¥ 5 x - 2 x 2
x ®-¥ x
x ®- ¥
2
x -2
x -2
ì x 2 + 5 if x < -2
lim g ( x ) where g ( x ) = í
x ®-2
î1 - 3x if x ³ -2
Compute two one sided limits,
lim- g ( x ) = lim- x 2 + 5 = 9
x ®-2

x ®-2

x ®-2+

x ®-2

lim g ( x ) = lim+ 1 - 3 x = 7

One sided limits are different so lim g ( x )
x ®-2

doesn’t exist
...


Some Continuous Functions
Partial list of continuous functions and the values of x for which they are continuous
...
Polynomials for all x
...
cos ( x ) and sin ( x ) for all x
...
Rational function, except for x’s that give
8
...

3p p p 3p
3
...

x ¹ L , - , - , , ,L
2
2 2 2
4
...

9
...
e x for all x
...
ln x for x > 0
...

Then there exists a number c such that a < c < b and f ( c ) = M
...
math
...
edu for a complete set of Calculus notes
...

h ®0
h
If y = f ( x ) then all of the following are
equivalent notations for the derivative
...

df
dy
f ¢ ( a ) = y ¢ x =a =
=
= Df ( a )
dx x =a dx x =a

Interpretation of the Derivative
2
...
m = f ¢ ( a ) is the slope of the tangent

change of f ( x ) at x = a
...
If f ( x ) is the position of an object at
time x then f ¢ ( a ) is the velocity of

equation of the tangent line at x = a is
given by y = f ( a ) + f ¢ ( a )( 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
...


( f ± g )¢ = f ¢ ( x ) ± g ¢ ( x )

3
...
ç
èg

d
(c) = 0
dx
d n
6
...

f ( g ( x )) = f ¢ ( g ( x )) g¢ ( x )
dx
This is the Chain Rule
5
...
math
...
edu for a complete set of Calculus notes
...

n
n -1
d
d
1
...

cos é f ( x ) ù = - f ¢ ( x ) sin é f ( x ) ù
ë
û
ë
û
ë
û
ë
û
dx
dx
d f ( x)
d
e
= f ¢ ( x ) e f ( x)
tan é f ( x ) ù = f ¢ ( x ) sec 2 é f ( x ) ù
2
...

ë
û
ë
û
dx
dx
d
f ¢( x)
d
7
...

ln é f ( x ) ù =
ë
û
dx
dx
f ( x)
f ¢( x)
d
d
tan -1 é f ( x ) ù =
8
...

sin é f ( x ) ù = f ¢ ( x ) cos é f ( x ) ù
ë
û
ë
û
ë
û
dx
1 + é f ( x )ù
dx
ë
û

)

(

(

(
(

)

(

)

)

)

(

(

)

)

Higher Order Derivatives
The Second Derivative is denoted as
The nth Derivative is denoted as
d2 f
dn f
f ¢¢ ( x ) = f ( 2) ( x ) = 2 and is defined as
f ( n ) ( x ) = n and is defined as
dx
dx
¢
f ¢¢ ( x ) = ( f ¢ ( x ) )¢ , i
...
the derivative of the
f ( n ) ( x ) = f ( n -1) ( x ) , i
...
the derivative of
first derivative, f ¢ ( x )
...


(

)

Implicit Differentiation
¢ if e 2 x -9 y + x3 y 2 = sin ( y ) + 11x
...
The “trick” is to
differentiate as normal and every time you differentiate a y you tack on a y¢ (from the chain rule)
...

e 2 x -9 y ( 2 - 9 y¢ ) + 3 x 2 y 2 + 2 x3 y y¢ = cos ( y ) y¢ + 11
2e

2 x -9 y

- 9 y¢e

( 2 x y - 9e x
3

2 x -9 y

2 -9 y

+ 3x y + 2 x y y¢ = cos ( y ) y¢ + 11
2

2

3

- cos ( y ) ) y¢ = 11 - 2e2 x -9 y - 3x 2 y 2

Þ

11 - 2e 2 x -9 y - 3x 2 y 2
y¢ = 3
2 x y - 9e2 x -9 y - cos ( y )

Increasing/Decreasing – Concave Up/Concave Down
Critical Points
x = c is a critical point of f ( x ) provided either
1
...
f ¢ ( c ) doesn’t exist
...
If f ¢ ( x ) > 0 for all x in an interval I then
f ( x ) is increasing on the interval I
...
If f ¢ ( x ) < 0 for all x in an interval I then
f ( x ) is decreasing on the interval I
...
If f ¢ ( x ) = 0 for all x in an interval I then

Concave Up/Concave Down
1
...

2
...

Inflection Points
x = c is a inflection point of f ( x ) if the
concavity changes at x = c
...

Visit http://tutorial
...
lamar
...


© 2005 Paul Dawkins

Calculus Cheat Sheet

Absolute Extrema
1
...


Extrema
Relative (local) Extrema
1
...


2
...

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 )
...
a £ c, d £ b , 2
...
max
...
f ( d ) is the abs
...
in [ a, b]
...

1
...

2
...

3
...

4
...
max
...
min
...


2
...

1st Derivative Test
If x = c is a critical point of f ( x ) then x = c is
1
...
max
...

2
...
min
...

3
...

2nd Derivative Test
If x = c is a critical point of f ( x ) such that
f ¢ ( c ) = 0 then x = c
1
...

2
...

3
...

Finding Relative Extrema and/or
Classify Critical Points
1
...

2
...


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 a < c < b such that f ¢ ( c ) =

f (b) - f ( a )

...


Visit http://tutorial
...
lamar
...


f ( xn )
f ¢ ( xn )

© 2005 Paul Dawkins

Calculus Cheat Sheet

Related Rates
Sketch picture and identify known/unknown quantities
...
e
...
Plug in known quantities and solve for the unknown quantity
...
A 15 foot ladder is resting against a wall
...
Two people are 50 ft apart when one
The bottom is initially 10 ft away and is being
starts walking north
...
01 rad/min
...
How fast
4
between them changing when q = 0
...
Using
Pythagorean Theorem and differentiating,
x 2 + y 2 = 152 Þ 2 x x¢ + 2 y y¢ = 0

After 12 sec we have x = 10 - 12 ( 1 ) = 7 and
4

so y = 152 - 7 2 = 176
...

7
7 ( - 1 ) + 176 y¢ = 0 Þ y¢ =
ft/sec
4
4 176

We have q ¢ = 0
...
and want to find
x¢
...
5 so plug in q ¢ and solve
...
5 ) tan ( 0
...
01) =
50
x¢ = 0
...
Solve constraint for
one of the two variables and plug into first equation
...

Ex
...
Determine point(s) on y = x 2 + 1 that are
500 ft of fence material and one side of the
closest to (0,2)
...
Determine dimensions that
will maximize the enclosed area
...
Solve constraint for x and plug
into area
...

A¢ = 500 - 4 y Þ y = 125
nd
By 2 deriv
...
max
...
Finally, find x
...

Visit http://tutorial
...
lamar
...


2

constraint is y = x 2 + 1
...

2
x2 = y - 1 Þ f = x2 + ( y - 2)
= y -1 + ( y - 2) = y 2 - 3 y + 3
Differentiate and find critical point(s)
...
min
...

x 2 = 3 - 1 = 1 Þ x = ± 12
2
2
2

The 2 points are then

(

1
2

)

(

, 3 and 2

1
2

)

,3
...
Divide [ a, b ] into n subintervals of

is a function, F ( x ) , such that F ¢ ( x ) = f ( x )
...


Indefinite Integral : ò f ( x ) dx = F ( x ) + c

*
i

Then

¥

where F ( x ) is an anti-derivative of f ( x )
...

i
b

®¥

*
i

=1

Fundamental Theorem of Calculus
Variants of Part I :
Part I : If f ( x ) is continuous on [ a, b ] then
d u( x)
x
f ( t ) dt = u ¢ ( x ) f éu ( x ) ù
ë
û
g ( x ) = ò f ( t ) dt is also continuous on [ a, b ]
dx ò a
a
d b
d x
f ( t ) dt = -v¢ ( x ) f év ( x ) ù
and g ¢ ( x ) =
f ( t ) dt = f ( x )
...
e
...

a

ò f ( x ) ± g ( x ) dx = ò f ( x ) dx ± ò g ( x ) dx
b
b
b
ò a f ( x ) ± g ( x ) dx = ò a f ( x ) dx ± ò a g ( x ) dx

Properties

ò cf ( x ) dx = c ò f ( x ) dx , c is a constant
b
b
ò a cf ( x ) dx = c ò a f ( x ) dx , c is a constant

a

b

ò a f ( x ) dx = 0
b

ò c dx = c ( b - a )
ò f ( x ) dx £ ò f ( x )
a

b

ò a f ( x ) dx = -òb f ( x ) dx
b

c

a

a

dx

b

a

b

a

a

c

ò f ( x ) dx = ò f ( x ) dx + ò f ( x ) dx for any value of c
...
math
...
edu for a complete set of Calculus notes
...

( )
ò a f ( g ( x ) ) g ¢ ( x ) dx = ò g (a ) f ( u ) du
b

u Substitution : The substitution u = g ( x ) will convert

g b

using

du = g ¢ ( x ) dx
...

Ex
...
Choose u and dv from
a

integral and compute du by differentiating u and compute v using v = ò dv
...


ò xe

u=x

ò xe

-x

-x

dx

Ex
...
n odd
...

2
2
cosines using sin x = 1 - cos x , then use
the substitution u = cos x
...
m odd
...

to sines using cos 2 x = 1 - sin 2 x , then use
the substitution u = sin x
...
n and m both odd
...
or 2
...
n and m both even
...

and/or half angle formulas to reduce the
4
...

Trig Formulas : sin ( 2 x ) = 2sin ( x ) cos ( x ) , cos 2 ( x ) =

Ex
...


x sec5 xdx = ò tan 2 x sec 4 x tan x sec xdx
= ò ( sec2 x - 1) sec 4 x tan x sec xdx
= ò ( u 2 - 1) u 4 du

n odd
...

m even
...

n odd and m even
...
or 2
...
Each integral will be
dealt with differently
...
math
...
edu for a complete set of Calculus notes
...

a 2 - b 2 x 2 Þ x = a sin q
b

a
b 2 x 2 - a 2 Þ x = b sec q

cos 2 q = 1 - sin 2 q
Ex
...
Because we have an indefinite
integral we’ll assume positive and drop absolute
value bars
...
From
substitution we have sin q = 32x so,

From this we see that cot q =

òx

4 - 9x = 2 cos q
...
So,

+c

P( x )

ò Q( x) dx where the degree of P ( x ) is smaller than the degree of

Q ( x )
...
Integrate the partial fraction decomposition (P
...
D
...

Factor in Q ( x )

Term in P
...
D Factor in Q ( x )

ax + b

ax 2 + bx + c

Ex
...
F
...


=

A

x -1

+ Bx +C =
x2 + 4

A( x2 + 4) + ( Bx + C ) ( x -1)
( x -1)( x 2 + 4 )

Set numerators equal and collect like terms
...

A+ B = 7
C - B = 13
4A - C = 0
A=4
B=3
C = 16

An alternate method that sometimes works to find constants
...
Chose nice values of x and plug in
...
This won’t always work easily
...
math
...
edu for a complete set of Calculus notes
...


Area Between Curves : The general formulas for the two main cases for each are,
y = f ( x) Þ A = ò

b
a

éupper function ù
ë
û

- élower
ë

function ù dx
û

& x = f ( y) Þ A = ò

d
c

é right function ù
ë
û

- éleft
ë

function ù dy
û

If the curves intersect then the area of each portion must be found individually
...


b

A = ò f ( x ) - g ( x ) dx
a

d

A = ò f ( y ) - g ( y ) dy

c

b

a

c

A = ò f ( x ) - g ( x ) dx + ò g ( x ) - f ( x ) dx

c

Volumes of Revolution : The two main formulas are V = ò A ( x ) dx and V = ò A ( y ) dy
...

Rings
Cylinders
2
2
A = 2p ( radius ) ( width / height )
A = p ( outer radius ) - ( inner radius)

(

)

Limits: x/y of right/bot ring to x/y of left/top ring
Limits : x/y of inner cyl
...

Horz
...
Axis use f ( y ) ,
Horz
...
Axis use f ( x ) ,
g ( x ) , A ( x ) and dx
...


g ( y ) , A ( y ) and dy
...


Ex
...
Axis : y = a £ 0

Ex
...
Axis : y = a £ 0

outer radius : a - f ( x )

outer radius: a + g ( x )

radius : a + y

inner radius : a - g ( x )

inner radius: a + f ( x )

radius : a - y
width : f ( y ) - g ( y )

width : f ( y ) - g ( y )

These are only a few cases for horizontal axis of rotation
...
For vertical axis of rotation ( x = a > 0 and x = a £ 0 ) interchange x and
y to get appropriate formulas
...
math
...
edu for a complete set of Calculus notes
...
The three basic formulas are,
b

b

L = ò ds

b

SA = ò 2p y ds (rotate about x-axis)

a

SA = ò 2p x ds (rotate about y-axis)

a

a

where ds is dependent upon the form of the function being worked with as follows
...
With parametric and polar you will always need to substitute
...

Integral is called convergent if the limit exists and has a finite value and divergent if the limit
doesn’t exist or has infinite value
...

Infinite Limit
1
...


ò

¥

t

f ( x ) dx = lim ò f ( x ) dx
t ®¥

a

2
...


Discontinuous Integrand
b

b

b

1
...
at a: ò f ( x ) dx = lim ò f ( x ) dx
+
a

t ®a

3
...
Discont
...


Comparison Test for Improper Integrals : If f ( x ) ³ g ( x ) ³ 0 on [ a, ¥ ) then,
¥

¥

¥

a

Useful fact : If a > 0 then

òa

1

xp

a

2
...
then ò f ( x ) dx divg
...
If ò f ( x ) dx conv
...


dx converges if p > 1 and diverges for p £ 1
...
math
...
edu for a complete set of Calculus notes

---