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VCE Mathematical Methods: Core Formulas, Calculus & Statistics

Formulas, theorems, calculus rules, and statistical distributions for Victorian Certificate of Education (VCE) Mathematical Methods Units 3 & 4.

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Composite Function Existence Condition

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Composite Function Existence Condition

The composite function

f (g (x))

exists if and only if the range of the inner function is a subset of the domain of the outer function:

ran (g) \subseteq dom (f)

Inverse Function Existence Condition

f^{- 1} (x)

exists if and only if

f

is a one-to-one function (passes horizontal line test).

dom (f^{- 1}) = ran (f)

and

ran (f^{- 1}) = dom (f)

Logarithm Rules and Base

e

lo g_{e} (x y) = lo g_{e} x + lo g_{e} y

;

lo g_{e} (\frac{x}{y}) = lo g_{e} x - lo g_{e} y

;

lo g_{e} (x^{n}) = n lo g_{e} x

;

e^{l o g_{e} x} = x

for

x > 0

Circular Function General Solutions:

sin (θ) = c

sin (θ) = sin (α)

, then

θ = 2 k π + α

θ = (2 k + 1) π - α

k \in Z

Circular Function General Solutions:

cos (θ) = c

cos (θ) = cos (α)

, then

θ = 2 k π \pm α

k \in Z

Circular Function General Solutions:

tan (θ) = c

tan (θ) = tan (α)

, then

θ = k π + α

k \in Z

Derivative of Composite Functions (Chain Rule)

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Derivative of Products and Quotients

Product:

(uv)^{'} = u^{'} v + u v^{'}

. Quotient:

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Derivatives of Exponential and Logarithmic Functions

\frac{d}{d x} (e^{k x}) = k e^{k x}

;

\frac{d}{d x} (lo g_{e} (x)) = \frac{1}{x}

;

\frac{d}{d x} (lo g_{e} (f (x))) = \frac{f ^{'} ( x )}{f ( x )}

Derivatives of Circular Functions

\frac{d}{d x} (sin (k x)) = k cos (k x)

;

\frac{d}{d x} (cos (k x)) = - k sin (k x)

;

\frac{d}{d x} (tan (k x)) = k sec^{2} (k x)

Stationary Points and Nature Criteria

f^{'} (x) = 0

: Local min if

f^{''} (x) > 0

; local max if

f^{''} (x) < 0

. If

f^{''} (x) = 0

with sign change of

f^{''}

, it is a point of inflection.

Average Rate of Change vs Instantaneous Rate of Change

Average rate over

[a, b] = \frac{f ( b ) - f ( a )}{b - a}

(secant gradient). Instantaneous rate at

x = a

f^{'} (a)

(tangent gradient).

Linear Approximation (Tangent Line)

f (x) \approx f (a) + f^{'} (a) (x - a)

for

x

close to

a

Fundamental Theorem of Calculus

\int_{a}^{b} f (x) d x = F (b) - F (a)

where

F^{'} (x) = f (x)

Average Value of a Continuous Function

f_{avg} = \frac{1}{b - a} \int_{a}^{b} f (x) d x

Area Between Two Curves

f (x)

and

g (x)

Area = \int_{a}^{b} ∣ f (x) - g (x) ∣ d x

Discrete Random Variable: Expectation and Variance

E (X) = μ = \sum x p (x)

;

Var (X) = σ^{2} = E (X^{2}) - [E (X)]^{2} = \sum (x - μ)^{2} p (x)

Linear Transformation of Random Variables

E (a X + b) = a E (X) + b

;

Var (a X + b) = a^{2} Var (X)

;

sd (a X + b) = ∣ a ∣ sd (X)

Binomial Distribution: Mean and Variance

X \sim Bi (n, p)

Pr (X = x) = (x n) p^{x} (1 - p)^{n - x}

;

E (X) = n p

;

Var (X) = n p (1 - p)

Continuous Random Variable: PDF Properties

f (x) \geq 0

for all

x

;

\int_{- \infty}^{\infty} f (x) d x = 1

;

Pr (a \leq X \leq b) = \int_{a}^{b} f (x) d x

Continuous Random Variable: Expectation and Variance

E (X) = \int_{- \infty}^{\infty} x f (x) d x

;

Var (X) = \int_{- \infty}^{\infty} x^{2} f (x) d x - [E (X)]^{2}

Normal Distribution Standardisation (Z-Score)

Z = \frac{X - μ}{σ} \sim N (0, 1)

with mean

0

and standard deviation

1

Normal Distribution 68-95-99.7 Empirical Rule

Pr (μ - σ \leq X \leq μ + σ) \approx 0.68

;

Pr (μ - 2 σ \leq X \leq μ + 2 σ) \approx 0.95

;

Pr (μ - 3 σ \leq X \leq μ + 3 σ) \approx 0.997

Sample Proportion Distribution (

\hat{P}

)

Expected value

E (\hat{P}) = p

; Standard deviation

sd (\hat{P}) = \frac{p ( 1 - p )}{n}

Approximate 95% Confidence Interval for Proportion

[\overset{p}{^} - 1.96 \frac{p ^ ( 1 - p ^ )}{n}, \overset{p}{^} + 1.96 \frac{p ^ ( 1 - p ^ )}{n}]

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