Precalculus and Introductory Calculus: Comprehensive Study Guide

Numbers, Ratios, and Indices

Scientific Notation and Large Numbers

Scientific notation is a way of expressing very large or very small numbers using powers of ten. It is commonly used in science and mathematics to simplify calculations and representation.

• Example: $4 \times 10^5$ means 400,000.

• Application: Calculating the number of items in large groups, such as the number of sardines in schools to match the mass of a whale.

Ratios and Proportions

Ratios compare quantities of the same kind and can be used to divide a total amount into parts.

• Definition: A ratio of 2:3:5 means for every 2 parts of the first item, there are 3 of the second and 5 of the third.

• Example: If a 1.5 kg bag contains rice, wheat, and corn in a 2:3:5 ratio, you can calculate the mass of each component by dividing the total mass in the given ratio.

Indices and Surds

Indices (exponents) and surds (roots) are used to express repeated multiplication and roots of numbers.

• Index Laws: $a^m \times a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$

• Surds: An expression involving roots that cannot be simplified to remove the root (e.g., $\sqrt{2}$).

• Example: Simplifying $\frac{\sqrt{27}a^5b^3}{6\sqrt{3}a^2(b^2)^{-\frac{1}{2}}}$ using index laws.

Equations and Inequalities

Solving Exponential Equations

Exponential equations involve variables in the exponent. They are solved by expressing both sides with the same base and equating exponents.

• Example: Solve $2^{2x} - 10(2^x) + 16 = 0$ by substituting $y = 2^x$.

Solving Inequalities

Inequalities express a range of possible values for a variable. Solutions are often given in interval notation.

• Example: Solve $\frac{1}{x-9} \leq 1$ and express the solution in interval notation.

Factorisation and Simplification

Factorising expressions involves writing them as a product of simpler expressions. Simplification often involves cancelling common factors.

• Example: Factorise and simplify $\frac{x^2-16}{x^2+2x+8}$.

Geometry and Measurement

Volume of Solids

The volume of a cone is calculated using the formula:

• $V = \frac{1}{3}\pi r^2 h$

• Example: Find the volume of a cone with base diameter 0.4 m and height 75 cm (convert all units to the same system before calculating).

Circle Geometry

Areas of sectors and segments of circles can be found using the central angle and radius.

• Area of a sector: $A = \frac{\theta}{360^\circ} \pi r^2$ (if $\theta$ is in degrees)

• Example: Find the area remaining after removing a sector from a circle.

Trigonometry in Triangles

Trigonometric ratios and the Pythagorean theorem are used to solve for unknown sides and angles in right and non-right triangles.

• Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

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

• Example: Find the height of a kite using the string length and horizontal distance.

Functions and Graphs

Quadratic and Polynomial Functions

Quadratic functions are polynomials of degree 2 and can be written in standard, vertex, or factored form.

• Standard form: $y = ax^2 + bx + c$

• Vertex form: $y = a(x-h)^2 + k$

• Completing the square: Used to convert standard form to vertex form.

• Polynomial long division: Used to divide polynomials.

Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas, represented by quadratic equations in two variables.

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

• Centre: $(h, k)$

Exponential and Logarithmic Models

Exponential functions model growth and decay processes. The general form is $P = P_0 e^{kt}$.

• Example: Population growth, spread of gossip, or zombie apocalypse models.

• Doubling time: If a quantity doubles every period, $P = P_0 \times 2^n$ where $n$ is the number of periods.

Vectors

Vector Operations

Vectors have both magnitude and direction. They can be added, subtracted, and multiplied by scalars.

• Unit vector: A vector of length 1 in the direction of a given vector $\vec{v}$ is $\frac{\vec{v}}{|\vec{v}|}$.

• Example: Find $\vec{c} = \vec{b} - 2\vec{a}$ and a unit vector in the direction of $\vec{b}$.

Trigonometric Functions and Identities

Exact Values and Identities

Trigonometric functions have exact values for certain angles and can be related using identities.

• Example: If $\cos(\theta) = -\frac{\sqrt{3}}{4}$, find the exact values of $\sin(\theta)$.

Calculus: Differentiation and Integration

Differentiation

Differentiation finds the rate of change of a function. The first derivative gives the slope, and the second derivative gives information about concavity.

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

• Nature of stationary points: Use $f''(x)$ to determine if a point is a maximum, minimum, or point of inflection.

• Example: For $f(x) = (1-x^3)\sin(3x)$, find stationary points and their nature.

Applications of Differentiation

Differentiation is used to solve real-world problems, such as finding the maximum height of a projectile.

• Example: The height of a ball: $h(t) = h_0 + v_0 t - \frac{gt^2}{2}$; set $h'(t) = 0$ to find the time of maximum height.

Integration

Integration is the reverse process of differentiation and is used to find areas under curves and solve accumulation problems.

• Indefinite integrals: $\int f(x) dx$ gives a family of functions.

• Definite integrals: $\int_a^b f(x) dx$ gives the area under $f(x)$ from $x=a$ to $x=b$.

• Integration by substitution: Used when the integrand is a composite function.

• Example: $\int x \sin(x^2) dx$ (use $u = x^2$).

Summary Table: Key Formulas and Concepts

Topic	Key Formula/Concept	Example
Scientific Notation	$a \times 10^n$	$3.2 \times 10^4$
Volume of Cone	$V = \frac{1}{3}\pi r^2 h$	$r=0.2$ m, $h=0.75$ m
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$	$x^2-4x+3=0$
Sine Rule	$\frac{a}{\sin A} = \frac{b}{\sin B}$	Triangle with $a=7$, $A=30^\circ$
Exponential Growth	$P = P_0 e^{kt}$	$P_0=4$, $k=0.75$
Derivative	$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$	$f(x) = x^2$
Definite Integral	$\int_a^b f(x) dx$	$\int_0^1 x dx$