Graphs, lines, polynomials, exponentials and logarithms

Vertical transformation

Where downwards by k

Horizontal transformation

Where right by k units

Stretching/squeezing the function

Where rises/falls double as quickly

The Quadratic Function

General form

Intercept form

Vertex form

To find the vertex form of a quadratic function:

take ", and add/subtract it to the formula Find numbers that multiplies to give you and adds to give you b (roots) transformations

Finding asymptotes

Where

Vertical Asymptotes:

1. After cancelling common factors, where , there is a vertical asymptote

Horizontal Asymptotes

1. If the degree of the degree of , is the horizontal asymptote

2. If the degree of the degree of , is the horizontal asymptote:

a. is the leading coefficient of

b. is the leading coefficient of

3. If degree of degree of , there is no horizontal asymptote

A graph has an exponential shape where . Exponent Laws

Where a and b are positive, , x & y are real.

1.

2.

3.

4.

5.

6.

etc.

1.

2. iff

3. where iff

Logarithms iff

that is:

Log properties

Where b, M and N are positive, and p & x are real numbers

1.

2.

3.

4.

5.

6.

7.

8.

9. iff

Changing base of log etc.

Note, calculator has and

Financial applications

simple interest

The compound interest formula

Continuous compound interest

Computing growth time

Since ,.

Annual percentage yield

, compounded continuously,

Future value of an ordinary annuity

Present value of an ordinary annuity

Derivatives

Slope of a secant between two points

Average rate of change (slope of a secant between x and x+h)

The derivative from first principles

basic differentiation properties

1. Constant

2. Just an x

3. A power of x

4. A constant*a function

5. Sum/difference

Derivatives of logarithmic and exponential functions

1. Base e exponential

2. Base e exponential with constant in power

3. Other exponential

4. Natural log

5. Other log

The product rule

The quotient rule

the chain rule

The general derivative rules

Local extrema

Where the first derivative is 0, and the sign of the first derivative changes around it, it is a local extrema:

1. 0 + minimum

2. + 0 - maximum

3. 0 or + 0 + not a local extrema

Note, where , finding can also identify whether it is a local extrema: where , it is a local minimum; where , it is a local maximum. This test is invalid where .

Graph sketching

1. Analyze , find domain and intercepts

2. Analyze , find partition numbers and critical values and construct a sign chart (to find increasing/decreasing segments and local extrema)

3. Analyze , find partition numbers and construct a sign chart (to find concave up and down segments and to find inflexion points)

4. Sketch : locate intercepts, maxima and minima and inflexion points: if still in doubt, sub points into

Optimization

1. Maximize/minimize on the interval I.

2. Find absolute maxima/minima: at a critical value or at endpoint

IntegralsIndefinite integrals of basic functions

1.

2. x to the n

3. e to the x

4. x denominator

Indefinite integrals of a constant multiplied by a function, or, two functions

1.

Integration by substitution

Based on chain rule:

General indefinite integral formulae

General indefinite integral formulae for substitution

1.

Method of integration by substitution

1. Select a substitution to simplify the integrand: one such that u and du (the derivative of u) are present

2. Express the integrand in terms of u and du, completely eliminating x and dx

3. Evaluate the new integral

4. Re-substitute from u to x.

Note, if this is incomplete (i.e. du is not present) you may multiply by the constant factor and divide, outside of the integral, by its fraction.

The definite integralError Bounds

For right and left rectangles, f(x) is above the x-axis:

Properties of a definite integral

1.

2.

3.

4. , where k is a constant

5.

6.

The fundamental theorem of calculus

You do not need to know C

Average value of a continuous function over a period

More than 2 dimensionsFunctions of several variables

Find the shape of the graph by looking at cross sections (e.g. y=0, y=1, x=0, x=1).

Partial derivatives

derived with respect to x ||| derived with respect to x, then y

Maxima and minima

1. Express the function as

2. Find , and simultaneously equate

3. Find (A, B, and C)

4. Find A, and .

a. IF AC-B*B>0 & A0 & A>0, f(a,b) is local minimum

c. IF AC-B*B 2

b

1

>b

2

= 1 2

q=b

1

-b

2

Yi = 0 + ( +2 )X1i +2X2i

Y

i

=b

0

+(q+b

2

)X

1i

+b

2

X

2i

n(x) =

n(x)=

Yi = 0 +X1i +2X1i +2X2i

Y

i

=b

0

+qX

1i

+b

2

X

1i

+b

2

X

2i

X = X1 + X2

X=X

1

+X

2

Yi = 0 +X1i +2X

Y

i

=b

0

+qX

1i

+b

2

X

H0 : = 0,H1 : > 0

H

0

:q=0,H

1

:q>0

> 0 B1 > B2

q>0B

1

>B

2

d(x)

d(x)

F =SSRR SSRUR( ) / qSSRUR / n k 1( )

F=

SSR

R

-SSR

UR

()

/q

SSR

UR

/n-k-1

()

F =RUR2 RR

2( ) / q(1 RUR

2 ) / (n k 1)

F=

R

UR

2

-R

R

2

()

/q

(1-R

UR

2

)/(n-k-1)

Elasticity = dydxxy

Elasticity=

dy

dx

x

y

y = a b

y=

a

b

a

a

n(x)

n(x)

b

b

d(x)

d(x)

n(x)>

n(x)>

d(x)

d(x)

f (x) = bx, x 1, x > 0

f(x)=b

x

,x1,x>0

a 1,b 1

a1,b1

axay = ax+y

a

x

a

y

=a

x+y

ax

ay= axy

a

x

a

y

=a

x-y

(ax )y = axy

(a

x

)

y

=a

xy

(ab)x = axbx

(ab)

x

=a

x

b

x

(ab)x = a

x

bx

(

a

b

)

x

=

a

x

b

x

ax = ay

a

x

=a

y

x = y

x=y

x 0,ax = bx

x0,a

x

=b

x

a = b

a=b

y = loga x

y=log

a

x

x = ay

x=a

y

a = bc : logb a = c

a=b

c

:log

b

a=c

b 1

b1

logb1= 0

log

b

1=0

logb b =1

log

b

b=1

logb bx = x

log

b

b

x

=x

blogb x = x, x > 0

b

log

b

x

=x,x>0

logb MN = logb M + logb N

log

b

MN=log

b

M+log

b

N

logbMN= logb M logb N

log

b

M

N

=log

b

M-log

b

N

logb Mp = p logb M

log

b

M

p

=plog

b

M

logb M = logb N

log

b

M=log

b

N

M = N

M=N

"log"= log10

"log"=log

10

"ln"= loge

"ln"=log

e

ln xlnb

= logb x

lnx

lnb

=log

b

x

ex lnb = bx

e

xlnb

=b

x

I = PR t

I=PRt

A = P(1+ i)n = P(1+ rm)mt

A=P(1+i)

n

=P(1+

r

m

)

mt

A = Pert

A=Pe

rt

A = P(1+ i)n

A=P(1+i)

n

lnA = n ln(P(1+ i))

lnA=nln(P(1+i))

APY = (1+ rm)m 1

APY=(1+

r

m

)

m

-1

APY = er 1

APY=e

r

-1

FV = PMT ((1+ i)n 1i

)

FV=PMT(

(1+i)

n

-1

i

)

PV = PMT (1 (1+ i)n

i)

PV=PMT(

1-(1+i)

-n

i

)

f (a) f (b)a b

f(a)-f(b)

a-b

f (a+ h) f (a)h

,h 0

f(a+h)-f(a)

h

,h0

limh0

f (x + h) f (x)h

lim

h0

f(x+h)-f(x)

h

f (x) = c f '(x) = 0

f(x)=cf'(x)=0

f (x) = ax2 + bx + c

f(x)=ax

2

+bx+c

f (x) = x f '(x) =1

f(x)=xf'(x)=1

f (x) = xn f '(x) =