Semester 1

NUMBERS AND SETS

NUMBERS AND SETS

LINEAR FUNCTIONS

LINEAR FUNCTIONS

NON-LINEAR FUNCTIONS

NON-LINEAR FUNCTIONS

LIMITS AND INEQUALITIES

LIMITS AND INEQUALITIES

What are limits?

The value that the function approaches as it goes to an x value

e.g. shown on picture

How do you write out and calculate:

• approaching from the left

• approaching from the right

E.g. if limit was 4 use a table and do:

Approaching from the left (below):

3.9, 3.99, 3.999 > input that into the function & see what number it is closest to

Approaching from the right (above):

4.1, 4.01, 4.001 > input that into the function & see what number it is closest to

How do you work out the value of the function?

Should say: e.g. ‘2 if x=2’ so f(2) = 2 - DONT PUT INTO EQUATION

If just has the equation, then put the value into equation to find the limit

What are the 3 features that are required for a function to be continuous at a specific point (a limit)?

(continuity = drawing a graph at that point without lifting up your pen)

1. The value of the function approaching from the left (below) = value of function approaching from right (above) - so basically if limit exists > if not limit doesn’t exist

2. You can plug x = a into a function and get a REAL NUMBER

3. The limit = the function value at point a > if not there is a ‘jump’ or mismatch

What happens if any one of those features fails?

A function is discontinuous at point a

*REMEMBER*

Behaviour of y = value of the function

e.g. if function increasing, value of y increasing

*REMEMBER*

Behaviour of y = value of the function

e.g. if function increasing, value of y increasing

What types of inequalities are these:

x > y

x > y

A < z < B

x > y = strict inequality (x strictly greater than y)

x > y = weak inequality (x is greater than or equal to y)

A < z < B = double inequality (z is in the right half open interval) [A, B) - includes A but not B so the right half is ‘open’

AC > BC A = 2, B = 1

In terms of the direction of an inequality, what happens if you times with a:

• positive number

• negative number

Positive = preserves direction of inequality

e.g. C = 3, AC > BC bc 6 > 3

Negative = reverses direction of inequality

e.g. if C = -3, AC < BC bc -6 < -3

What is the formula for a rectangular hyperbola?

y = (C / x + B) + A

CALCULUS (DIFFERENTIATION)

CALCULUS (DIFFERENTIATION)

What is a line joining 2 points on a curve called?

A chord

What is the slope of a straight line (chord) defined as?

△y / △x

What is a tangent?

A line which touches a curve at one point

What is the slope of a tangent defined as?

How can we find the slope of a tangent at any point along the curve (without graphing the function)?

E.g. if want the slope of a tangent at point P:

• If we have a chord (PQ)

• As we move point Q around the curve, nearer to point P

• = slope of the chord will become a better approximation of the slope of a tangent at point P

Find the meaning of this notation:

Find the notation for these meanings:

What does differentiating from first principles mean?

Finding the slope of a tangent (by looking at the slope between 2 points that get infinitely close together)

= derivative of a function

What is the formula for finding the slope of a tangent?

Differentiating from first principles:

• Describe the 7 steps

• For each step, explain what is going on

1. Get your formula and & work out your answer shorthand already (extra)

2. Write out the formula for f’(x)

3. Work out f(x+h)

4. Substitute f(x+h) and f(x) into formula

5. Get rid of brackets + cancel out common terms

6. Factor + simplify (divide by h)

7. Take the limit (h=0) to get final answer & check answer the same as step 1

What are the 7 rules of differentiation?

(SPAMPF Q)

1. Sums of functions rule

2. Power rule

3. Additive constant rule

4. Multiplicative constant rule

(more complicated rules)

5. Product rule

6. Function of a function rule (chain rule)

7. Quotient rule

(5+6 are the same thing just 5 is 2 brackets with different things and 6 is same thing in both brackets)

1. Sums of functions rule

if f(x) contains different functions within it

e.g. f(x) = g(x) + h(x)

then each function within the function needs to be differentiated

2. Power rule

for the power, you always minus 1

and whatever the power is, you times it by the constant

square root = fraction > √x = x^1/2 OR ³√x = x^1/3

*1 minus the fraction*

3. Additive constant rule

constant disappears in the derivative

4. Multiplicative constant rule

any constant that multiplies an x term remains unchanged when differentiated (just times by power)

5. Product rule

different things in each bracket

dy/dx = v(du/dx) + u (dv/dx)

U WITH A V, V WITH A U

because of things in front = cross multiplication

6. Function of a function rule (chain rule)

same thing in each bracket

dy/dx = dy/du x du/dx

over u = main one outside of brackets

over x = equation inside brackets

7. Quotient rule

OPTIMISATION

OPTIMISATION

Is x or y:

• independent

• dependent

x = independent (changes)

y = dependent (measure)

When a function of x/y is differentiated - what does this give us?

The derivative of the function = gradient function for a line/curve

• This means you can calculate the gradient of the tangent (to the curve) at any point along its length

What is a stationary point?
What is the necessary condition for a stationary point?

Necessary condition = dy/dx = 0

A point on the graph of the function where the function's derivative is zero/slope is 0

• This means the function (dependent variable, usually price) has stopped increasing/decreasing (it is flat)

What are 3 different types of stationary points?

1. Turning points = maximums (turns from positive to negative) - OPTIMA of the function

2. Turning points = minimums (turns from negative to positive) - OPTIMA of the function

3. Points of inflection

FOR A QUADRATIC FUNCTION:

If we want to find the coordinates of a stationary points what are the steps to finding it?

1. Differentiate the function (dy/dx or f’(x))

2. FOC dy/dx = 0

• Work out formula to get x value/values (quadratic or cubic formula)

3. Put x coordinate back into original formula to get your y coordinate

= e.g. (1.5, -0.75) = OPTIMA

FOR A CUBIC FUNCTION:

If we want to find the coordinates of a stationary points what are the steps to finding it?

1. Differentiate the function (dy/dx or f’(x))

2. FOC dy/dx = 0

• Factorise or use quadratic formula to get 2 roots

3. Put 2 x coordinates back into original formula to get your y coordinates

How do we classify the stationary point for both:

• quadratic function

• cubic function

Find the second derivative (differentiate the differentiated function) - SLOPE OF THE SLOPE

d²y/dx² or f’’(x)

• Put your x coordinate/coordinates into f’’(x)

• For quadratic - should get 1 output

• For cubic - should get 2 (same number - one min, one max, e.g. 24 and -24)

Can classify it by:

• If dy/dx = 0 and d²y/dx² < 0 then the point is a maximum for y (negative number)

• If dy/dx = 0 and d²y/dx² > 0 then the point is a minimum for y (positive number)

e.g. if we graphed (5,25), the 2nd derivative (-2) confirms this point is a (local and global) maximum of the function

When are turning points described as:

• local

• global

Local turning points (cubic function)

• local maximum or minimum

• when there are other points on the curve which have higher/lower values of y

Global turning points (quadratic function)

• global AND local maximum or minimum

• no other higher/lower points on curve

What are points of inflection?

POI is where d²y/dx² = 0 (2nd derivative = 0)

Where curve changes curvature

If d²y/dx² = 0 at a particular point, then this implies the slope of the original function is nor increasing or decreasing at that point > so the slope of the original function meets a (local) max or min value

How can you classify points of inflection?

When dy/dx = 0 and d²y/dx² = 0

If d³y/dx³ > 0 = upward flowing around stationary point

If d³y/dx³ < 0 = downward flowing around stationary point

What is the difference between the 2 types of POI:

• stationary

• non stationary

Stationary = concave > convex (gradient decreasing then increasing)

f’(x) = 0 AND f’’(x) = 0

Non-stationary = convex > concave (gradient increasing then decreasing)

f’(x) = a number and f’’(x) = 0

What does it mean if f’’’(x), so the 3rd derivative = 0

You’ve got something wrong - go check it again

d³y/dx³ should always equal a positive or negative number for a point of inflection

Econ Applications

Econ Applications

What are features of total costs (q)?

• Positive fixed costs = graph starts above zero (when p = 0) > TC has positive intercept

• TC increase with output (TR’(q) > 0 bc MC increase after making 1 output)

• quadratic TC > steepness of curve increases with output (TC’’(q) > 0) as productivity decreases with output rising (curve convex from below)

What is the formula for TC (q)

TC (q) = TVC (q) + TFC

What is the formula for TR (q)

What does TR always have to be a function of?

TR (q) = P(q) x Q

Q - Should always be TR (q) = 2q² - 2q

*NO P’S*

What price, TR (q) and MR (q) do these both set:

• Monopoly

• Perfect competition

Monopoly - firm acts as a price setter, pricing in relation to market demand

e.g. q = 600 - 1/2p

so TR (q) = 1200q - 2q²

so MR (q) = TR’(q) = 1200 - 2q = CURVE

Perfect competition - firm acts as a price taker, adopting the fixed market defined price

e.g. £320

so TR (q) = 320q

so MR (q) = TR’(q) = 320 = LINEAR

monopoly = more equation

perf comp = less equation/singular number

If, e.g. for a monopoly, you get given a demand function, e.g.

q = 600q - 1/2p

to work out TR (q) what do you need to do?

Convert into an inverse demand function (p = q)

e.g.

q = 600 - 1/2p > p = 1200 - q

TR (q) = (1200 - 2q) x q

TR (q) = 1200q - 2q²

What is the formula for π (q)

(total profit)

π (q) = TR (q) - TC (q)

What is the formula for MR (q)

MR (q) = TR’(q)

(slope of TR curve OR first derivative of TR)

What is the formula for MC (q)

MC = TC’(q)

(slope of TC curve OR first derivative of TC)

Are marginal costs independent of fixed costs?

YES 

• Marginal costs only change due to the change in quantity

• Fixed costs do not change with quantity 

Which is why in an equation of TVC + TFC, e.g.

AVC = 500q² + 5q and FC = 200

TC (q) = 500q² + 5q + 200

but MC (q) = 1000q + 5 

(getting rid of the 200 - the TFC)

What is the formula for AC (q)

AC (q) = TC (q) / q

e.g. TC (q) = 5q² + 5q + 2000

then

AC (q) = 5q² + 5q + 2000 / q

= 5q + 5 + 2000q^-1

How do you find the point where AC are at its minimum?

AC (min) = MC

(the q where they both are)

REMEMBER:

q² / q = q (2-1)

q^0 / q = q^-1 (0-1)

REMEMBER:

q² / q = q (2-1)

q^0 / q = q^-1 (0-1)

REMEMBER:

1/2p = 600 - q

can be written as:

p = 2(600 - q)

OR

p = (600 - q) / ½ or 0.5

What are the steps to finding:

1) the profit maximising level of output

2) the maximum profit a firm can achieve

3) checking if MC = MR at profit maximising level of output

4) selling price which is charged at profit maximising level of output

1) Identify if its a monopoly or perfect competition

2) Write expressions for TR (q), TC (q) and π (q)

(remember put brackets around to reduce mistakes)

3) Order conditions:

• FOC: π’(q) = 0 (slope of the function at the optimum point = 0)

quadratic > linear - can just solve = 0 (only 1 q)

cubic > quadratic - use quadratic formula (2 q’s)

quadratic formula = q=(-b±√(b²-4ac))/(2a)

• SOC: π’’(q) < 0 for a max

linear - already get answer to compare to 0 (GLOBAL)

quadratic > linear - input quantity’s found into π’’(q) then compare to 0 (2 LOCALS)

What are the steps to finding:

1) the profit maximising level of output

2) the maximum profit a firm can achieve

3) checking if MC = MR at profit maximising level of output

4) selling price which is charged at profit maximising level of output

1) Get ORIGINAL π (q) function (from step 2)

2) Input MAX PROFIT quantity ONLY to find maximum profit

3) answer = £ to 2.d.p.

What are the steps to finding:

1) the profit maximising level of output

2) the maximum profit a firm can achieve

3) checking if MC = MR at profit maximising level of output

4) selling price which is charged at profit maximising level of output

1) Write expressions for MC (q) and MR (q)

2) Input profit max quantity into equations

3) Check MC = MR > IF DON’T YOU HAVE ANSWER WRONG GO BACK AND CHECK

What are the steps to finding:

1) the profit maximising level of output

2) the maximum profit a firm can achieve

3) checking if MC = MR at profit maximising level of output

4) *extra step if asked* - selling price which is charged at profit maximising level of output

1) Use price equation

2) Input profit max quantity

3) Answer with £ to 2.d.p