Graphs, lines, polynomials, exponentials and logarithms

Vertical transformationWhere f ( x )=f ( x )−k downwards by “k”

Horizontal transformationWhere f ( x )= f (x−k ) right by “k” units

Stretching/squeezing the functionWhere f ( x )=2 f (x) rises/falls double as quickly

The Quadratic Function

General form

Intercept formf ( x )=(x−g )2

Vertex form

To find the vertex form of a quadratic function:take “(−b/2 )2", and add/subtract it to the formula Find numbers that multiplies to give you “(−b/2 )2” 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 Asymptotes1. 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 LawsWhere a and b are positive, , x & y are real.

1.

2.

3.

4.5.

etc.

1. iff 2. where iff

Logarithms iff

that is:

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

1.

2.

3.

4.

5.

6.

7.

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

1. – 0 + minimum2. + 0 - maximum3. – 0 – or + 0 + not a local

extremaNote, 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 sketching1. Analyze , find domain and

intercepts2. 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

Integrals

Indefinite integrals of basic functions1. x to the n 2. e to the x 3. x denominator

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

Integration by substitution

Based on chain rule:

General indefinite integral formulae

General indefinite integral formulae for substitution

Method of integration by substitution1. Select a substitution to simplify the integrand: one such that u and du (the

derivative of u) are present2. Express the integrand in terms of u and du, completely eliminating x and dx3. Evaluate the new integral4. 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 integral

Error Bounds

For right and left rectangles, f(x) is above the x-axis: |f (b )− f (a )|⋅b−a

n

Properties of a definite integral

1.

2.

3. , where k is a constant

4.

5.

The fundamental theorem of calculus

You do not need to know C

Average value of a continuous function over a period

1b−a∫a

bf (x )dx

More than 2 dimensions

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

Partial derivativesf x ( x , y ) derived with respect to x ||| f xy ( x , y ) derived with respect to x, then y

Maxima and minima1. Express the function as z= f ( x , y )

2. Find f x ( x , y )∧f y( x , y ), and simultaneously equate

3. Find f xx (a ,b) , f xy(a ,b )∧f yy (a , b) (A, B, and C)

4. Find A, and AC−B2 .a. IF AC-B*B>0 & A<0, f(a,b,)

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

is local minimumc. IF AC-B*B<0, f(a,b) is a

saddle pointd. IF AC-B*B=0, test fails

Using Lagrange multipliers1. Write problem in form

a. max/min→ z=f (x , y )

b. g( x , y )=02. Form the function

F (x , y , λ )=f ( x , y )+λg( x , y )3. Derive with respect to x, y and

lambda4. Simultaneously equate answers5. If more than 1 answer, find z

values and deduce which is max/min

If v=f (x , y , z ) , derive with respect to that too, and simultaneously equate with more.

Econometrics

Descriptive Statistics

Mean

X or Y=1n⋅∑ x i=

∑ x i

n

Variance

Sx2=

∑ (x i−x )2

n−1

Standard Deviation

Sx=√Sx2

Sample Covariance

Sxy=1

n−1⋅∑ ( xi−x )( yi− y )

If it is greater than zero, upward sloping.

This is scale dependent.

Sample Correlation

r xy=Sxy

Sx⋅S y=√R2

This is scale independent: between -1 and 1, close to 1 is

upward, 0 is central, -1 is downward sloping.

Finding the regression

Regression formula with one regressorY i=β0+β1X i+u i

Slope

β1=∑ ( x i−x )( y i− y )

∑ [( x i−x )2 ]=

Sxy

( Sx )2

Interceptβ0= y−β1 x

Finding R2

TSS=ESS+SSRSSR=∑ [ y i−(b0+b1 x i) ]2=∑ [ y i− y ]2=∑ ui

2

ESS=∑ ( yi− y )2

TSS=∑( y i− y )2

The Coefficient of Determination = R2

R2= ESSTSS

=1−SSRTSS

This gives the total fit of Y , between 0 (chance) and 1 (perfect prediction)

Standard Errors

Standard Error of the Regression

SER=√ SSRn−2

=√∑ (ui2 )

n−2

Standard error of β1

U i=( xi− x )(u )

Hypothesis Testing1. Define H0

2. Define H1

3. Define Tcrit/Pcrit

a. Note, for Tcrit 2 sided test, half α

4. Find Tact/Pact

Tact

tact=β1−β1,0SE( β1 ) ,

SE( β1)=√ β1

Pact

(2 sided )Pact=2Φ(−|tact|)For one sided, just t

act→−Φ

WANT TO BE LESS THAN ALPHA %

Multiple RegressionY i=β0+β1X1 i+β2X2i+ .. .+βnXni+ui

Omitted Variable BiasOmmitted variables may increase the apparent importance of another variable, damaging the ability to prove causality.

Effect of OVB on β11. Find the variable outside of the

model2. Find Corr(ZY)3. Find Corr (ZX)4. Multiply the signs

5. If positive, there is an upwards

bias (β>β )

Adjusted R2

R2=1−( n−1n−k−1 )( SSRTSS )=1−( n−1

n−k−1 ) (1−R2 )

OLS Wonder Equation

SE( β1)≈S u

Sx i

⋅ 1

√n(1−Rx i onx2 )

A good model for proving causality has a low SE( β1) , a good model for predicting Y has a low R2

Multiple Variable Tests

Reparametrisation

1. Y i=β0+β1X1 i+β2X2i β1>β2

2. Let θ=β1−β2

3. Thus, Y i=β0+(θ+β2)X1 i+β2X2 i

4. Y i=β0+θX1i+ β2 X1i+β2X 2i

5. Now, let X=X1+X2

6. Thus, Y i=β0+θX1i+ β2 X

7. (H0 :θ=0 , H1 :θ>0 : a one sided test). If you can reject H0, then θ>0→B1>B2

F-stat testsHere, the H0 is a joint hypothesis with n restrictions (the number of coefficients equated to 0).

1. Create a restricted regression where H0 is correct/

2. the “fit” changes

a.F=

(SSRR−SSRUR ) /qSSRUR / (n−k−1 )

b.

F=(RUR2 −RR

2 ) /q(1−R

UR2)/(n−k−1)

c. q is restrictions, n is observations and k is variables

3. Compare to Fcrit (using number of restrictions as the numerator and n-k-1 as the denominator)

Non-linear regression modelsModel Equation Derivative Elasticity effectLinear Y=B0+B1 B1 B1(x/y)Polynomial y = 0 + 1x + β β

2x^2 + 3x^3β β1 + 2 2 x β β

+ 3 3 x^2β*(x/y)

Lin-Log Y = B0 +B1ln(x) B1/X B1/y 1% ^ x .01(B1) ^YLog-Lin Ln(Y)=B0+B1X B1Y B1X 1 u ^ x 100B1% ^ YLog-Log Ln(Y)=B0+B1ln(x) B1(y/x) B1 1% ^ x B1 % ^ Y

Elasticity

Elasticity=dydx

⋅xy

Test for Heteroscedasticity H0: no hetero (homoscedastic), H1: Hetero. Look at Prob F in White test (if greater then alpha)