calc ab exam study guide

7/27/2019 Calc Ab Exam Study Guide

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Calculus AB

Derivative FormulasDerivative Notation:

For a function f(x), the derivative would be )(' xf

Leibniz's Notation:

For the derivative ofy in terms ofx, we write d yd x

For the second derivative using Leibniz's Notation: d yd x

2

2

Product Rule:

y f x g x

dy

dxf x g x g x f x

=

= +

( ) ( )

' ( ) ( ) ' ( ) ( )

y x x

dy

dxx x x x

dy

dxx x x x

=

= +

= +

2

2

2

2

2

sin

sin cos ( )

sin cos

Quotient Rule:

y f xg x

dy

dx

f x g x g x f x

g x

=

=

( )( )

' ( ) ( ) ' ( ) ( )

( ( )) 2

y

x

x

dx

x x x x

x

dy

dx

x x x

dy

x

=

=

=

sin

cos ( ) sin

cos sin

3

3 2

6

4

3

3

Chain Rule:

y f x

dy

dxn f x f x

n

n

=

=

( ( ))

( ( )) ( ' ( ))1

y x

dy

dxx x

dy

dx

x x

= +

= +

= +

( )

( )

( )

2 3

2 2

2 2

1

3 1 2

6 1


2/14

Natural Log

y f x

dy

dx f xf x

=

=

ln( ( ))

( )' ( )

1 y x

dy

dx xx

dydx

xx

= +

=+

= +

ln( )2

2

2

1

1

12

2 1

Power Rule:

y x

dy

dxax

a

a

=

= 1

y x

dy

dxx

=

=

2

10

5

4

Constant with a Variable Power:

y a

dy

dxa a f

f x

f x

=

=

( )

( ) ln ' ( )x

y

dy

dx

x

x

=

=

2

2 1ln

y

dy

dxx

x

x

=

=

3

3 3 2

2

2

ln

Variable with a Variable power

Take ln of both sides!y f x g x=( )

( )

y x

y x

y x x

y

d y

d xx x

xx

dy

dxx x x

xx

x

x

x

=

=

=

= +

= +

sin

sin

sin

ln ln

ln sin ln

cos ln sin

[cos ln sin ]

1 1

1


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Implicit Differentiation:Is done when the equation has mixed variables:

x x y y

deriva tive x xy yd y

d x

x yd y

d x

y xd y

d xy

dy

dxx xy

dy

dx

x xy

y x y

2 2 3 4

3 2 2 3

2 2 3 3

3

2 2 3

5

2 2 3 4

3 4 2 2

2 2

3 4

+ + =

+ + + =

+ =

=

+

[ ] 0

Trigonometric Functions:

d

dxx xsin cos=

d

dx x xcos sin=

d

dxx xtan sec= 2

d

dxx xsec sec tan= x

Inverse Trigonometric Functions:

y x

dy

dx x

=

=

arcsin1

11

2

y x

dy

dx x

=

=+

arctan1

11

2

y x

dy

dx xx

=

=

arcsin 4

8

31

14

y x

dy

dx xx

=

=+

arctan

1

13

6

2


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Substitution

When integrating a product in which the terms are somehow related, we

usually letu= the part in the parenthesis, the part under the radical, thedenominator, the exponent, or the angle of the trigonometric function

x x dx u x

du x dx

x x dx

u du

u C

x C

2 2

2 1 2

1 2

3 2

2 3 2

1 1

2

1

22 1

1

21

2

2

31

31

+ = +

=

= +

=

= +

= + +

;

( )

( )

/

/

/

/

cos ;

co s

co s

sin

sin

2 2

2

1

22 2

1

2

1

2

1

22

x dx u x

du dx

x dx

u du

u C

x C

=

=

=

=

= +

= +

Integration by Parts

When taking an integral of a product, substitute foru the term whosederivative would eventually reach 0 and the other term fordv.

The general form: uv (pronounced "of dove")vdu Example:

x e dx

u x dv e dx

du dx v e

x e e dx

xe e C

x

x

x

x x

x x

= =

= =

=

= +

1


6/14

Example 2:

Cxxxxx

xxxxx

xvdxdu

dxxdvxu

xxxx

xvdxxdu

dxxdvxu

xx

++

==

==

==

==

sin2cos2sin

cos2cos2[sin

cos2

sin2

sin2sin

sin2

cos

cos

2

2

2

2

2

Inverse Trig Functions

Formulas:

Ca

x

xa

+=

arcsin

22

1

Ca

x

a

xa

+=

+

arctan1

122

Examples:

Cx

xva

x

+=

==

3arcsin

;3;29

1

Cx

xvax

+=

==+

4arctan

41

;4;16

12

More examples:

C

x

x

xva

x

+=

==

2

3

arcsin3

1

94

3

3

1

3;2;294

1

2

Cx

Cx

x

xvax

+=

+=

+

==+

43arctan

121

4

3arctan

4

1

3

1

169

3

3

1

3;4169

1

2

2


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Trig Functions

+=+=

+=+=

+=+=

CxdxxCxdxx

CxxxCx

dxx

CxdxxCxdxx

sinlncotcoslntan

sectansec2

2sin2cos

tanseccossin 2

Properties of Logarithms

Form

logarithmic form exponential form

y = logax ay = x

Log properties

y = log x3 => y = 3 logx

log x + log y = log xy

log x log y = log (x/y)

Change of Base Law

This is a useful formula to know.

a

xor

a

xxy a

ln

ln

log

loglog =

Properties of Derivatives

1st

Derivative shows: maximum and minimum values, increasing and decreasingintervals, slope of the tangent line to the curve, and velocity

2nd

Derivative shows: inflection points, concavity, and acceleration

- Example on the next page -


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Example:

Find everything about this function23632 23 += xxxy

2,3

)2)(3(60

)6(60

3666

2

2

=

=

=

=

x

xx

xx

xxdx

dy

1stderivative finds max, min, increasing,

decreasing

max min(-2, 46) (3, -79)

increasing decreasing( , -2] [3, ) (-2, 3)

2

1

)12(60

6122

2

=

=

=

x

x

xdx

yd

2ndderivative finds concavity and inflection points

inflection pt( , -16 )concave up concave down

( , ) (1/2, )


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Miscellaneous

Newtons Method

Newtons Method is used to approximate a zero of a function

)('

)(

cf

cfc where c is the 1stapproximation

Example:

If Newtons Method is used to approximate the real root ofx3 + x 1 = 0, then a first approximation of x1 = 1 would lead to athirdapproximation of x3:

13)('

1)(2

3

+=

+=

xxf

xxxf

3

2

686.86

59

)4

3('

)4

3(

4

3

750.43

)1(')1(1

xor

f

f

xorff

==

==

Separating Variables

Used when you are given the derivative and you need to take the integral.We separate variables when the derivative is a mixture of variables

Example:

If 49ydx

dy= and if y = 1 when x = 0, what is the value of y when

x =3

1?

Cxy

dx

y

dy

dxy

dyy

dx

dy

+=

=

==

9

39

99

3

4

4

4

Continuity/Differentiable Problems

f(x) is continuous if and only if both halves of the function have the sameanswer at the breaking point.f(x) is differentiable if and only if the derivative of both halves of thefunction have the same answer at the breaking point


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Example:

3,96

)(

3,2

>

=

xx

xf

xx

66

)('

)3(62

=

=

=

xf

inplugx

- At 3, both halves = 9, therefore,f(x) is continuous- At 3, both halves of the derivative = 6, therefore,f(x) isdifferentiable

Useful Information

- We designate position asx(t) ors(t)- The derivative of positionx(t) is v(t), or velocity- The derivative of velocity, v(t), equals acceleration, a(t).- We often talk about position, velocity, and acceleration when were

discussing particles moving along the x-axis.

- A particle is at rest when v(t) = 0.- A particle is moving to the right when v(t) > 0 and to the left when

v(t) < 0

- To find the average velocity of a particle: b

a

dttvab

)(1

Average Value

Use this formula when asked to find the average of something

b

a

dxxfab

)(1

Mean Value Theorem

NOT the same average value.

According to the Mean value Theorem, there is a number, c, between aandb, such that the slope of the tangent line at c is the same as the slopebetween the points (a,f(a)) and (b,f(b)).

ab

afbfcf

=

)()()('


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Growth Formulas

Double Life Formula: dtyy /0 )2(=

Half Life Formula: htyy /0 )2/1(=

Growth Formula: kteyy 0=

y = ending amount y0 = initial amount t = timek = growth constant d = double life time h = half life time

Useful Trig. Stuff

Double Angle Formulas:xxx cossin22sin =

xxx 22 sincos2cos =

Identities:

xx

xx

xx

22

22

22

csccot1

sectan1

1cossin

=+

=+

=+

cotsin

costan

cos

sin

cscsin

1sec

cos

1

==

==

Integration Properties

Area

f(x) is the equation on top b

a

dxxgxf )]()([

Volume

f(x) always denotes the equation on top

About the x-axis: About the y-axis:

b

a

b

a

dxxgxf

dxxf

]))(())([(

)]([

22

2

b

a

b

a

dxxgxfx

dxxfx

)]()([2

)]([2

about line y = -1 about the line x = -1

+b

a

dxxf 2]1)([ +b

a

dxxfx )]()[1(2

Examples:


12/14

]2,0[)( 2xxf =

x-axis: y-axis:

=2

0

42

0

22 )( dxxdxx =2

0

2

0

32 2][2 dxxdxxx

about y = -1In this formulaf(x) ory is the radius of the shaded region. Whenwe rotate about the line y = -1, we have to increase the radius by 1.That is why we add 1 to the radius

++=+2

0

242

0

22 )12(]1[ dxxxdxx

about x = -1In this formula,x is the radius of the shaded region. When werotate about the line x = -1, we have the increased radius by 1.

+=+2

0

232

0

2 )(2])[1(2 dxxxdxxx

Trapeziodal Rule

Used to approximate area under a curve using trapezoids.

Area [ ])()(2....)(2)(2)(2

1210 nn xfxfxfxfxfn

ab+++++

where n is the number of subdivisions

Example:f(x) = x2+1. Approximate the area under the curve from [0,2] usingtrapezoidal rule with 4 subdivisions

[ ]

[ ]

[ ] 750.416

76)4/76(

4

1

5)4/13(2)2(2)4/5(214

1

)2()5.1(2)1(2)5(.2)0(

8

02

4

2

0

===

++++=

++++

=

=

=

=

fffffA

n

b

a

Riemann Sums


13/14

Used to approximate area under the curve using rectangles.

a) Inscribed rectangles: all of the rectangles are below the curve

Example:f(x) =x2

+ 1 from [0,2] using 4 subdivisions(Find the area of each rectangle and add together)

I=.5(1) II=.5(5/4) III=.5(2) IV=.5(13/4)Total Area = 3.750

b) Circumscribed Rectangles: all rectangles reach above the curve

Example:f(x) =x2 + 1 from [0,2] using 4 subdivisions

I=.5(5/4) II=.5(2) III=.5(13/4) IV=.5(5)Total Area = 5.750

Reading a Graph

When Given the Graph of f(x)

Make a number line because you are more familiar with number line.

This is the graph off(x).Make a number line.- Wheref(x) = 0 (x-int)

is where there are

possible max and mins.- Signs are based on if the

graph is above or belowthe x-axis (determinesincreasing anddecreasing)

min maxx = -4, 3 x = 0

increasing decreasing(-4,0) (3,6] [-6,-4] (0,3)

- contd on next page -


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calc ab exam study guide

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