Precalculus Comprehensive Study Notes: Algebra, Functions, Trigonometry, Sequences, Geometry, and Calculus

Algebraic Manipulation and Scientific Notation

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers using powers of ten.

• Definition: A number is written in scientific notation as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

• Example: The distance from the Earth to the Sun is approximately 149,600,000 km. In scientific notation: $1.496 \times 10^8$ km or $1.496 \times 10^{11}$ m.

Solving for a Variable

To make a variable the subject of a formula, use algebraic manipulation to isolate it.

• Example: Make $x$ the subject of $y = \frac{x}{x+2y} + 4p$.

• Steps:

  1. Multiply both sides by $(x+2y)$ to clear the denominator.

  2. Expand and collect like terms.

  3. Isolate $x$ on one side.

• Result: $x = \frac{8py - 2ry}{r - 4p - 1}$

Percentage Rate Calculation

• Formula: $\text{Rate} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}$

• Example: If an item costs $24.50 and the total bill after tax is $27.32, the tax rate is$\frac{27.32 - 24.50}{24.50} = 0.1151 = 11.5\%$.

Indices and Logarithms

Index Laws and Simplification

Index laws are used to simplify expressions involving powers.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $a^m / a^n = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Example: $\left(\frac{x^3 u^3 y^5}{y^3 u^5}\right)^{1/3} = \sqrt[3]{\frac{x^3 y^5}{y^3 u^5}} = x^{1} y^{2/3} u^{-2/3}$

Logarithmic Equations

• Definition: $\log_b a = c$ means $b^c = a$.

• Solving: $5^{1-x} = 25 \implies \log_5(5^{1-x}) = \log_5 25 \implies 1-x = 2 \implies x = -1$

Inequalities and Interval Notation

Solving Quadratic Inequalities

• Example: $3 - x^2 \leq -1 \implies x^2 \geq 4 \implies x \geq 2$ or $x \leq -2$

Interval Notation

• Definition: Interval notation expresses the set of solutions to inequalities.

• Example: $[-2, 0)$ means all $x$ such that $-2 \leq x < 0$.

Geometry: Volume, Area, and Heron's Formula

Volume of a Cone

• Formula: $V = \frac{1}{3} \pi r^2 h$

• Example: For $r = 2$ cm, $h = 3$ cm: $V = \frac{1}{3} \pi (2)^2 (3) = 4\pi \approx 12.566$ cm$^3$

Heron's Formula for Area of a Triangle

• Formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$

• Example: For sides 3, 1.5, 2: $s = 3.25$, $A = 1.332$ cm$^2$

Sequences and Series

Arithmetic Sequences

• General Term: $a_n = a_1 + (n-1)d$

• Sum of First $n$ Terms: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$

• Example: $a_1 = 3$, $d = 2.5$, $a_{12} = 30.5$, $S_{15} = 307.5$

Geometric Sequences

• General Term: $a_n = a_1 r^{n-1}$

• Sum of First $n$ Terms: $S_n = a_1 \frac{1 - r^n}{1 - r}$

• Sum to Infinity (|r| < 1): $S = \frac{a_1}{1 - r}$

• Example: $a_1 = 25$, $r = \frac{1}{5}$, $S = 31.25$

Exponential and Logarithmic Models

Value Depreciation

• Model: $P_t = P_0 e^{-kt}$

• Example: $P_5 = 21000 e^{-0.15 \times 5} = 9919.70$

Radioactive Decay

• Model: $N_t = N_0 e^{-kt}$

• Example: $N_0 = 500$, $N_t = 16$, $k = 0.13863$

• Solving for $t$: $t = \frac{\ln(16/500)}{-0.13863} \approx 24.83$ days

Trigonometry

Area of a Sector

• Formula: $A = \frac{1}{2} r^2 \theta$ (with $\theta$ in radians)

• Example: $r = 2$ cm, $\theta = 3$ rad: $A = 6$ cm$^2$

Cosine Rule

• Formula: $c^2 = a^2 + b^2 - 2ab \cos C$

• Example: $b^2 = a^2 + c^2 - 2ac \cos B$

Finding Angles and Radians

• Conversion: $1$radian$= \frac{180}{\pi}$ degrees

• Example: $\cos \alpha = \frac{2.5}{4}$, $\alpha = 0.9$ rad

Solving Trigonometric Equations

• Example: $\cos \theta = 0.5 \implies \theta = \frac{\pi}{3}$ or $\theta = 2\pi - \frac{\pi}{3}$

Vectors

Vector Operations

• Subtraction: $\vec{c} = \vec{b} - 2\vec{a}$

• Magnitude: $|\vec{a}| = \sqrt{a_1^2 + a_2^2}$

• Unit Vector: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

• Example: $\vec{a} = -2\hat{i} - 4\hat{j}$, $|\vec{a}| = \sqrt{20} = 2\sqrt{5}$, $\hat{a} = \frac{-\sqrt{5}}{5} \hat{i} - \frac{2\sqrt{5}}{5} \hat{j}$

Resultant Force

• Magnitude: $|\vec{F}| = \sqrt{(F_1)^2 + (F_2)^2}$

• Angle: $\tan \alpha = \frac{F_2}{F_1}$

Conic Sections

Circle

• Standard Form: $(x-h)^2 + (y-k)^2 = r^2$

• Centre: $(h, k)$

• Radius: $r$

• Example: $(x+2)^2 + (y-4)^2 = 50$ has centre $(-2, 4)$ and radius $\sqrt{50}$

Ellipse

• Standard Form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Completing the Square: Used to rewrite general quadratic forms into standard form.

Polynomials and Partial Fractions

Polynomial Long Division

• Process: Divide the terms of the dividend by the divisor, subtract, and repeat.

• Example: $(2x^2 + x^2 - 2x - 1) \div (x-1)$

Factorization

• Example: $2x^2 + x^2 - 2x - 1 = (x-1)(2x^2 + 3x + 1)$

Partial Fractions

• Decomposition: $\frac{x+2}{(x^2)(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3}$

• Find $A$ and $B$ by substituting suitable $x$ values.

Calculus: Differentiation and Integration

Differentiation

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Product Rule: $\frac{d}{dx}[u v] = u'v + uv'$

• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$

• Example: $f(x) = -2x^2 - \sqrt{x}$, $f'(x) = -4x - \frac{1}{2\sqrt{x}}$

Stationary Points and Second Derivative Test

• Stationary Points: Where $f'(x) = 0$

• Nature: If $f''(x) < 0$, maximum; if $f''(x) > 0$, minimum

• Example: $f'(x) = -6x$, $f''(1) = -6 < 0$ (max), $f''(-1) = 6 > 0$ (min)

Integration

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• Substitution: Change of variable to simplify the integral.

• Integration by Parts: $\int u dv = uv - \int v du$

• Example: $\int \frac{2}{x^2} - 3x dx = -2x^{-1} - \frac{3}{4}x^4 + C$

Summary Table: Key Formulas

Topic	Formula	Example
Scientific Notation	$a \times 10^n$	$1.496 \times 10^8$ km
Arithmetic Sequence	$a_n = a_1 + (n-1)d$	$a_{12} = 30.5$
Geometric Series (Sum to Infinity)	$S = \frac{a_1}{1 - r}$	$S = 31.25$
Sector Area	$A = \frac{1}{2} r^2 \theta$	$A = 6$ cm$^2$
Circle Equation	$(x-h)^2 + (y-k)^2 = r^2$	Centre $(-2,4)$, $r=\sqrt{50}$
Derivative (Power Rule)	$\frac{d}{dx} x^n = n x^{n-1}$	$f'(x) = -4x - \frac{1}{2\sqrt{x}}$
Integration (Power Rule)	$\int x^n dx = \frac{x^{n+1}}{n+1} + C$	$\int x^2 dx = \frac{x^3}{3} + C$