Weaknesses

Question 1

R

Answer 1

Real numbers on the line

Question 2

Z

Answer 2

Set of integers
{…,-3,-2,-1,0,1,2,3,…}

Question 3

N

Answer 3

Positive Integers (natural numbers)
1,2,3,4,5,…

Question 4

Composition Function

Answer 4

h(x) = f(g(x))
or k = g(f(x))

Question 5

Power Rules

Answer 5

a^x . a^y = a^x+y
(a^x) . y = a^x . y
a^0 = 1
a^-n = 1/a^n
a^1/2 = square root of a
a^m/n = (a^1/n)^m

Question 6

Linear Function

Answer 6

Polynomial of degree 1
f(x) = mx + c
gradient/slope = m -> crosses y-axis at point (0,c)

Question 7

Quadratic Equation and Curve

Answer 7

Polynomial of degree 2
ax^2 + bx + c = 0
1. shape: if a > 0 = parabola that opens upward (U)
if a < 0 = parabola that opens downward (n)
2. y-axis: set x = 0 in f(x) -> f(0) (usually just c)
3. x-axis: set y = f(x) = 0 find solution (thru factorisation, completing the square or general formula)

Question 8

Polynomial Function

Answer 8

Has non-negative powers or positive integer exponent of variable
e.g. y = -2x^2+4x+5 or y = 4x^3+2x^2-4x+8

Question 9

Demand and Supply Function

Answer 9

Linear
Demand curve: downward sloping
Supply curve: upward sloping
Demand function: expressing quantity demanded in terms of p
Supply function: solve demand function using p for q
Inverse demand function: expressing p (price) in terms of q (quantity)
Inverse supply function: solve inverse demand function using q for p
Equilibrium price (p): quantity demanded = quantity supplied
Equilibrium quantity (q): solve qD or qS with p*

If =/ linear
turn qD and qS quadratic and solve = 0 for equilibrium price

Question 10

Exponentials

Answer 10

f(x) = a^x (where a is a number)
a^0 = 1, (0,1) only point where a^x crosses y-axis
f(x) = e^x > 1 so increases exponentially and cuts y-axis at 1

Question 11

Natural Logarithm

Answer 11

Inverse function of e^x
ln x
ln(a.b) = ln a + ln b
ln(a/b) = ln a - ln b
ln(a^b) = b . ln a

Question 12

What is differentiation for?

Answer 12

1. Show how fast a quantity is changing
2. FInd maximum/minimum value of a function

Question 13

f’(x) rules

Answer 13

1. d/dx (x^k) = kx^k-1
2. d/dx (e^x) = e^x
3. d/dx (ln x) = 1/x
4. d/dx (sin x) = cos x
5. d/dx (cos x) = -sin x
6. Sum rule: h(x) = f(x) + g(x) -> h’(x) = f’(x) + g’(x)
7. Product rule: h(x) = f(x) x g(x) -> h’(x) = f’(x) x g(x) + f(x) x g’(x)
8. Quotient rule: h(x) = f(x)/g(x) -> h’(x) = f’(x) x g(x) - f(x) x g’(x) / g(x)^2
9. Chain rule: f(x) = s(r(x)) -> f’(x) = s’(r(x))r’(x)
   f(x) = (ax+b)^n -> f’(x) = an(ax+b)^n-1
   f(x) = ln(g(x)) -> f’(x) = g’(x) / g(x)

Question 14

Critical Point

Answer 14

1. Local maximum (CP >= f(x))
2. Local minimum (CP <= f(x))
3. Inflexion point (CP = 0)

Question 15

Nature of critical point

Answer 15

f”(x) or d^2f/dx^2
if f’(a) = 0
and f”(a) < 0 -> x = a: local maximum of f
and f”(a) > 0 -> x = a: local minimum of f

Question 16

Curve sketching

Answer 16

y = f(x)
1. Cross x-axis -> set f(x) = 0 for (x, 0)
2. Cross y-axis -> set f(0) for y
3. Critical points and nature -> set f’(x) = 0 and determine if local maximum/minimum or inflexion point
4. Curve limiting behaviour for large positive/negative values of x
-> if polynomial highest power of x is even:
f(x) approaches infinity as x approaches infinity or negative infinity
if n is odd: f(x) appraoches infinity as x approaches infinity and vice versa (f(x) and x approaches negative infinity)
if exponential with a power, exponent dominates

Question 17

Marginal Cost & Marginal Revenue

Answer 17

MC: TC’(q)
MR: TR’(q)

Question 18

Maximum Profit

Answer 18

Set Profit’(q) = TR’(q) - TC’(q) = 0
Meaning MC = MR

Question 19

Monopoly Revenue

Answer 19

TR(q) = q x (p^D)(q)

Question 20

Power to fraction

Answer 20

a^-1 = 1/a
a^-m = 1/a^m
a^1/2 = square root of a