Formula Sheet for Mid-semester Exam

Week 1

Chapter 1.2 – Graphs and Lines

General form of a line:y=mx+b

“m” is the gradient of the line (m= riserun )

Where “b” is the y intercept

−bm

is x intercept

Point Gradient Formulay− y1=m(x−x1)

(x1 , y1) is a point on the line

m is the gradient of the line (m= riserun )

Chapter 2.1 – Functions

Vertical transformationWhere f ( x )=f ( x )−kThe function is shifted downwards by “k” units

Horizontal transformationWhere f ( x )=f (x−k )The function is shifted to the right by “k” units

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

As such, where f ( x )= f (x)2

, the function rises/falls half as quickly

ReflectionWhere , the graph is reflected about the x axis

Where x and y are inverted (i.e. ) the graph is perpendicular to how the graph will have been

Shapes of graphs:f ( x )=mx+b line

parabola

hyperbola

“v” shaped graph

Chapter 2.3 – Quadratic Functions

The Quadratic Function

General form

Intercept form

vertex form

To find the x-intercepts of a quadratic function:Either: factorize to intercept form, then equate each bracket to zeroOr: use the quadratic formula:

To find the vertex form of a quadratic function:Either: use the quadratic formula to find the x-intercepts, and use the midpoint between the two intercepts:

Or: find where Or: complete the square

e.g.

1. take “ ", and add/subtract it to the formula

a.

b.

c.

2. Find numbers that multiplies to give you “ ” and adds to give you “b”: these are the roots of the equation.

a. Multiplies for 16, sums to -8 = -4

b.

c.3. Analyze the transformations:

a. Here, we have a parabola, inverted, shifted upwards by 7 units and across 4 units to the right.

Week 2

Chapter 2.4 – Polynomials and rational function

Degree of a polynomialIs the highest “power” in the polynomial chain:

e.g. , degree of the polynomial is “20”

Y-interceptsWhere x=0

X-InterceptsWhere y=0Note, where the polynomial has been factorized, equating each bracket to zero will give the x-intercepts (/roots)

e.g.

Roots are: ( ,0)(6,0)(1,0)

Finding asymptotes

Where Vertical Asymptotes:

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

Chapter 2.5 – Exponential functionsNote: graphs shift as per regular functions, see Chapter 2.1 – Functions.

A graph has an exponential shape where .

Properties of

1. All graphs go through for any base b2. The graph is continuous3. The axis is a horizontal asymptote

4. Where , increases as increases

5. Where decreases as increases

Exponent LawsWhere a and b are positive, , x & y are real.

1.

2.

3.

4.

5.

Further:

1. iff 2. where iff

Common bases10 &

Note, growth/decay formulae are often in the form , c & k are constants, t is time.

Chapter 2.6 – Logarithmic Functions

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

Chapter 3.1 – simple interest, where I is interest, P is principle, r is the annual simple interest rate, and t is the time in

years.Note: do not forget to add the principle again when working out future value, since this formula only works out interest

Chapter 3.1 – compound interest

The compound interest formula

: where A is the future value at the end of n periods, P is the principle, r is the annual the annual nominal rate of interest, m is the amount of compounding periods per year, i is the interest rate per compounding period, n is the total number of compounding periods.Note: make sure “i” and “n” are in the same units of time.

Continuous compound interest, r is the annual compounding rate, t is time in years.

Computing growth time

Since , .

Annual percentage yield

, or, if compounded continuously,

Chapter 3.4 – AnnuitiesStrategy (make into flowchart for final notes?):

1. Make a timeline of payments2. If single payment: either simple or compound interest3. If multiple:

a. Payments into an account increasing in value (FV)b. Payments being made out of an account decreasing in value (PV)c. All amortization is PV.

Future value of an ordinary annuity

, where FV is the future value, PMT is the periodic payment, i is the rate per period, n is the number of periods/payments.

Present value of an ordinary annuity

Week 3

Chapter 10.4 – the derivative

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

note: it is most probable that the h on the denominator will goThis will fail if the line is non-differentiable at a point, e.gg where the graph:

is not continuous has a sharp corner has a vertical tangent

Chapter 10.5 – basic differentiation properties1. Constant

2. Just an x

3. A power of x

4. A constant*a function

5. Sum/difference

Chapter 11.2 – 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

Chapter 11.3 – product/quotient rule

The product rule

The quotient rule

Chapter 11.4 – the chain rule

easy to do via substitutionthe above formula means: derivative of whole thing times derivative of bracket

The general derivative rules

1.

2.

3.

Week 4Note, in this chapter, sign charts make stuff easier

Chapter 12.1 – first derivatives and graphs

The first derivative gives the slope of a graph at a point: a positive first derivative will give an upward slope, and vice versa. A “0” or undefined first derivative gives a partition value. It is also a critical value when it appears in both the domain of f’(x) and f(x), e.g. an asymptote is a partition value but not a critical value.

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 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 .

Chapter 12.2 – second derivatives and graphsThe second derivative describes the concavity of a graph (where , the concavity is positive

, and (/the slope) is increasing; where , the concavity is negative and (/the slope) is decreasing.

Point of inflexionA point of inflexion is where the concavity of the graph changes (and, as such, the sign of the

derivative, too, will change. This occurs where (or, if it is a vertical point of inflexion, undefined) the line is continuous and the sign of the second derivative changes about that point.

Graph sketching1. 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

Chapter 12.6 – optimization 1. Introduce variables, look for relationships among variables, and construct a mathematical

model of the form Maximize/minimize on the interval I.

2. Find critical values of .3. Find absolute maxima/minima: this will occur at a critical value or at an endpoint of an interval

a. Check that the function is continuous over an interval

b. Evaluate at the endpoints of the interval

c. Find the critical values of d. The absolute maximum is the largest value found in step “b” or “c”.

Chapter 4.1 – Systems of linear equations in two variablesSimultaneous equations of two lines: isolate a variable and substitute.

Week 5

Chapter 13.1 – antiderivatives and indefinite integralsAntiderivative is symbolized by , and may be accompanied by any constant .

Indefinite integral

is a family of antiderivatives

Indefinite integrals of basic functions

1. x to the power of n

2. e to the power of x

3. x as a denominator

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

1. 2.

Chapter 13.2 – integration by substitution

Based on the chain rule: (derivative of outside function multiplied by the derivative of the inside function)

Thus

General indefinite integral formulae

1.

2.

3.

Integration by substitutionSometimes it is hard to recognize the form of the function to be integrated (that is: to see which of the above formulae apply to it). So, we substitute the messy part for “u” and integrate with respect to “u”, rather than x.

Where

General indefinite integral formulae

1.

2.

3.

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 inverse: e.g. Where integral of u is 1, du=dx. (and find x as “u-k”)

Chapter 13.4 – the definite integral

is the definite integral of from x=a to x=b. Worked out by subtracting where x=a from where x=b. Note, these are not absolute values: above x axis is positive, below is negative: opposite if other direction.

Properties of a definite integral

1.

2.

3. , where k is a constant

4.

5.

Error Bounds

Where f(x) is above the x-axis: |f (b )−f (a )|⋅b−a

n

The fundamental theorem of calculus

Average value of a continuous function over a period1b−a∫a

bf (x )dx

Week 6

Chapter 15.1 – functions of several variablesSubstitute (x,y,z,…,et.) into the equation given.Find the shape of the graph by looking at cross sections (e.g. y=0, y=1, x=0, x=1).

Chapter 15.2 – partial derivativesDerivatives with respect to a certain symbol: watch for signs, all other symbols count as constantsf x ( x , y ) derived with respect to x ||| f xy ( x , y ) derived first with respect to x, then y

Chapter 15.3 – maxima and minima1. Express the function as z=f ( x , y )

2. Find f x ( x , y )∧f y( x , y ), and simultaneously equate them to find critical values

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

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

Chapter 15.4 – maxima and minima using Lagrange multipliers1. Write problem in form

a. max/min→ z=f (x , y )

b. g( x , y )=0

2. 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.