Summary GR 12 IEB CORE MATHS: CALCULUS/CUBIC FUNCTIONS

Luca Truen Differential Calculus Matric 2021


Limits Cubic graphs
"The limit as x is approaching 2"
- when limit is approaching x: sub - have 1 y-intercept
value into x - have 1-3 x-intercepts
- when limit is approaching h: sub - local max or min
value into h - point of inflection as a stationary point
NB only sub when ans will ≠ 0
"in terms of h" = do not need to sub STANDARD FORM: y= ax3 + bx2 + cx + d
- a>o starts concave up .




- a<0 starts concave down
1st Principles
- d = y-intercept
F'(x) = lim f(x+h) - f(x) Stationary points/MIN + MAX Points:
h-0 h - f'(x) = 0
- determine the derivative + gradient - double check!!
- h represent increase of x from A-B - gives the local min and max
- sub 0 into h once equation will not = 0
NB: algebraic simplification

Remainder and
Derivatives factor theorem
The derivative of a function measure the rate at Cubic polynomial can be the
which the dependent variable changes product of a near and quadratic
1st derivative = gradient of the tangent function.
- find 1st derivative by using power rule - can factorise using highest CF,
- multiply coefficient by exponent grouping or sum/difference
- -1 from exponent - P(x) = (divisor)(quotient) +
- drop constant remainder
- ax^n = an.x^n-1 - gives roots of cubic = x-intercepts
NB: exponent + algebraic fraction rules - set f(x)=0
Different notations: Dy = m = derivative - factorise: CF and group
dx
Second derivative = points of inflection and concavity:
cubic
1. Set f''(x) = 0
- f''(x) > 0 @ stationary point = LOCAL MIN + CONCAVE UP
- f''(x) < 0 @ stationary point = LOCAL MAX + " DOWN
- f''(x) = 0 @ stationary point = POINT OF INFLECTION