Fundamentals

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

Linear Function

Answer 5

Polynomial of degree 1
f(x) = mx + c

Question 6

Gradient/Slope of a linear function

Answer 6

m -> crosses y-axis at point (0,c)

Question 7

Quadratic Equation

Answer 7

Polynomial of degree 2
ax^2 + bx + c = f(x)

Question 8

Quadratic Curve Shape

Answer 8

If a > 0 -> upward-facing parabola (U)
a < 0 -> downward-facing parabola (n)

Question 9

How to get Quadratic Curve?

Answer 9

1. Shape of parabola (a)
2. y-axis: set f(0) = usually c
3. x-axis: set y in f(x) = 0 and solve using factorisation, completing the square or general formula
4. x-coordinate of vertex: -b/2a
5. y-coordinate of vertex: plug x-coordinate to f(x) and solve for y

Question 10

Polynomial function

Answer 10

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 11

Exponential function

Answer 11

f(x) a^x (a: number and x: variable)

Question 12

a^0

Answer 12

= 1
only point where a^x cuts through y-axis (0,1)

Question 13

e^x

Answer 13

> 1
Increases exponentially and cuts through y-axis at 1

Question 14

Natural Logarithmic Function

Answer 14

Inverse function of e^x aka ln x

Question 15

What is differentiation for?

Answer 15

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

Question 16

Types of critical points

Answer 16

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

Question 17

How to get critical point(s)?

Answer 17

Differentiate f(x), set f’(x) = 0 and solve for x

Question 18

How to determine nature of critical points?

Answer 18

1. Set f’(x) = 0 and solve for x
2. Plug x into f”(x),
   if < 0: x is a local maximum of f
   if > 0: x is a local minimum of f

Question 19

How to sketch curve?

Answer 19

1. x-axis: set f(x) = 0 and solve for x
2. y-axis: plug 0 into f(x) and solve for y
3. critical point(s): set f’(x) = 0 and solve for x
4. nature of critical point(s): differentiate f’(x) and plug x into f”(x)
5. limiting behaviour of curve:
   if polynomial highest power of x is
   even: f(x) -> infinity whether x -> infinite/negative infinite
   odd: f(x) follows x infinite

Question 20

How to determine limiting behaviour of curve as x becomes positively/negatively larger?

Answer 20

Polynomial highest power of x
If even: f(x) -> infinity when x -> infinity or -infinity
odd: f(x) -> infinity when x -> infinity and f(x) -> -infinity when x -> -infinity