25 cardsNSW HSC

NSW HSC Mathematics Advanced Core Concepts

Essential formulas, calculus rules, trigonometry exact values, and statistics for the NSW HSC.

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Derivative of

x^{n}

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Derivative of

x^{n}

\frac{d}{d x} (x^{n}) = n x^{n - 1}

for any real number

n

Derivative of

ln (x)

and

ln [f (x)]

\frac{d}{d x} ln (x) = \frac{1}{x}

and

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

Derivative of

e^{k x}

and

e^{f (x)}

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

and

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

Product Rule for Differentiation

(u \cdot v)^{'} = u^{'} v + u v^{'}

Quotient Rule for Differentiation

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

Chain Rule for Composite Functions

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

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

Gradient of Normal to Curve at

(x_{1}, y_{1})

m_{normal} = - \frac{1}{m _{tangent}} = - \frac{1}{f ^{'} ( x _{1} )}

Derivative of

sin (k x)

cos (k x)

tan (k x)

(sin k x)^{'} = k cos (k x)

(cos k x)^{'} = - k sin (k x)

(tan k x)^{'} = k sec^{2} (k x)

Exact Trig Values:

sin (π /6)

cos (π /6)

tan (π /6)

sin (3 0^{\circ}) = \frac{1}{2}

cos (3 0^{\circ}) = \frac{3}{2}

tan (3 0^{\circ}) = \frac{1}{3}

Exact Trig Values:

sin (π /4)

cos (π /4)

tan (π /4)

sin (4 5^{\circ}) = \frac{1}{2}

cos (4 5^{\circ}) = \frac{1}{2}

tan (4 5^{\circ}) = 1

Fundamental Pythagorean Identity

sin^{2} (x) + cos^{2} (x) = 1

, and

1 + tan^{2} (x) = sec^{2} (x)

Indefinite Integral of

x^{n}

(

n \neq = - 1

)

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Integral of

\frac{1}{x}

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Reverse Chain Rule / Standard Logarithmic Form

\int \frac{f ^{'} ( x )}{f ( x )} d x = ln ∣ f (x) ∣ + C

Trapezoidal Rule for Area Approximation

\int_{a}^{b} f (x) d x \approx \frac{b - a}{2 n} [f (x_{0}) + 2 f (x_{1}) + \dots + 2 f (x_{n - 1}) + f (x_{n})]

Area Between Two Curves

f (x) \geq g (x)

A = \int_{a}^{b} [f (x) - g (x)] d x

Volume of Solid of Revolution (x-axis)

V = π \int_{a}^{b} [y (x)]^{2} d x

Exponential Growth and Decay Equation

\frac{d N}{d t} = k N ⟹ N (t) = N_{0} e^{k t}

(

k > 0

growth,

k < 0

decay).

Probability Density Function (PDF) Requirements

f (x) \geq 0

for all

x

, and

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

Expected Value of Continuous Random Variable

X

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

Variance of Continuous Random Variable

X

V a r (X) = E (X^{2}) - [E (X)]^{2} = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}

Standard Normal Distribution Z-score

z = \frac{x - μ}{σ}

, transforming

X \sim N (μ, σ^{2})

into

Z \sim N (0, 1)

Empirical Rule for Normal Distribution

Approx. 68% within

μ \pm σ

, 95% within

μ \pm 2 σ

, 99.7% within

μ \pm 3 σ

Stationary Point Nature Test (Second Derivative)

f^{'} (x_{0}) = 0

with

f^{''} (x_{0}) > 0 ⟹

Local Minimum; with

f^{''} (x_{0}) < 0 ⟹

Local Maximum.

Compound Interest and Superannuation Formula

A = P (1 + \frac{r}{n})^{n t}

; Future value of ordinary annuity

F V = M \frac{( 1 + i ) ^{n} - 1}{i}

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