Pure Mathematics 3 (Edexcel IAL) - Calculus and Functions Study Notes

1. Algebraic Fractions

Simplifying Algebraic Fractions

Algebraic fractions are expressions where both the numerator and denominator are polynomials. Simplifying these fractions involves factorizing and cancelling common factors.

• Factorization: Factorize both numerator and denominator fully before cancelling.

• Cancelling: Only cancel factors that appear in both numerator and denominator.

• Example: Simplify $\frac{x^2 - 2x - 3}{4x^2 - 36}$ Factorize: $x^2 - 2x - 3 = (x-3)(x+1)$, $4x^2 - 36 = 4(x^2 - 9) = 4(x-3)(x+3)$ Cancel $(x-3)$: $\frac{x+1}{4(x+3)}$

Multiplying and Dividing Fractions

• To multiply: Multiply numerators together and denominators together, then simplify.

• To divide: Invert the second fraction and multiply.

• Example: $\frac{9x^2 - 4}{3x^2 - 2x} \div \frac{3x^2 - x - 2}{x^2 + x}$ Invert and multiply: $\frac{9x^2 - 4}{3x^2 - 2x} \times \frac{x^2 + x}{3x^2 - x - 2}$ Factorize and cancel common factors.

Adding and Subtracting Fractions

• Find the Lowest Common Denominator (LCD).

• Rewrite each fraction with the LCD, then add or subtract numerators.

• Example: $\frac{3x}{x^2 - 4x + 3} + \frac{5}{x^2 - 4}$ Factorize denominators, find LCD, rewrite, and combine.

2. Functions

Notation

• Function notation: $f(x)$ denotes a function named $f$ with variable $x$.

• Alternative notations: $y = f(x)$, $f: x \mapsto 3x + 7$.

Domain, Range, and Graph

The domain is the set of all possible input values ($x$) for which the function is defined. The range is the set of all possible output values ($y$).

• To find the domain, consider restrictions such as division by zero or square roots of negative numbers.

• To find the range, sketch the graph or solve for $x$ in terms of $y$.

• Example: For $f(x) = x^2 - 2x + 3$, domain $x \in [-2, 4]$, range $y \in [-7, 5]$ (excluding $y = -7$ if $x = -2$ is not included).

Defining Functions

• Some mappings are not functions unless the domain is restricted.

• Example: $x \mapsto \sqrt{x}$ is not a function for $x < 0$ unless domain is $x \geq 0$.

Composite Functions

A composite function is formed by applying one function to the result of another, denoted $f(g(x))$.

• Order matters: $f(g(x)) \neq g(f(x))$ in general.

• Example: If $f(x) = 3x - 2$, $g(x) = x^2 + 1$, then $f(g(x)) = 3(x^2 + 1) - 2 = 3x^2 + 1$.

Inverse Functions and Their Graphs

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.

• To find the inverse: Replace $f(x)$ with $y$, swap $x$ and $y$, solve for $y$.

• Example: $f(x) = 3x - 2$ $y = 3x - 2$ Swap: $x = 3y - 2$ Solve: $y = \frac{x + 2}{3}$ So $f^{-1}(x) = \frac{x + 2}{3}$

Modulus Functions

The modulus function $|x|$ returns the non-negative value of $x$. Graphically, it reflects negative parts of a function above the $x$-axis.

• Example: To sketch $y = |x^2 - 3x - 4|$, first sketch $y = x^2 - 3x - 4$, then reflect any part below the $x$-axis above it.

Standard Graphs

• Common graphs include $y = x^2$, $y = \sqrt{x}$, $y = \frac{1}{x}$, $y = |x|$, and their transformations.

• Transformations include translations, reflections, stretches, and compressions.

3. Trigonometry (Overview from Table of Contents)

Trigonometric functions and identities are essential in calculus for solving equations and modeling periodic phenomena.

• Key functions: $\sin x$, $\cos x$, $\tan x$

• Identities: Pythagorean, double angle, sum and difference, etc.

• Graphs: Know the shapes and key points of $\sin x$, $\cos x$, $\tan x$.

4. Differentiation (Preview from Table of Contents)

Differentiation is the process of finding the derivative, which represents the rate of change of a function.

• Chain Rule: Used for composite functions.

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

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

5. Exponentials and Logarithms (Preview from Table of Contents)

Exponentials and logarithms are used to model growth and decay, and to solve equations involving powers.

• Exponential function: $y = e^{ax}$

• Logarithmic function: $y = \ln x$

• Properties: $\ln(ab) = \ln a + \ln b$, $\ln(a^k) = k \ln a$

6. Numerical Methods (Preview from Table of Contents)

Numerical methods are used to approximate solutions to equations that cannot be solved algebraically.

• Root finding: Methods such as iteration and bisection.

• Convergence: Conditions under which iterative methods approach the correct solution.

Additional info:

• Some content previewed here (e.g., trigonometry, differentiation, exponentials, numerical methods) is inferred from the table of contents and not fully detailed in the provided images. For a complete study, refer to the full notes or textbook.