Weeks 6–13: Differential Calculus

Functions

Understanding functions is foundational in calculus. Functions describe relationships between variables and are characterized by their domain, range, and behavior.

• Definition: A function is a rule that assigns to each element in a set (domain) exactly one element in another set (range).

• Types of Functions: Includes linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric, and piecewise functions.

• Domain and Range: The domain is the set of input values; the range is the set of possible output values.

• Symmetry: Functions may be even (symmetric about the y-axis) or odd (symmetric about the origin).

• Composite Functions: The composition of two functions and is .

• Transformations: Includes shifts, stretches, compressions, and reflections.

• Piecewise Functions: Defined by different expressions over different intervals.

• Example:

Limits

Limits are central to calculus, providing a rigorous way to describe the behavior of functions as inputs approach specific values.

• Definition: The limit of as approaches is if $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $a$.

• Epsilon-Delta Proofs: Formal proofs for limits, required for linear functions in this course.

• Notation:

• Example:

Continuity

A function is continuous if its graph can be drawn without lifting the pen. Continuity is essential for many calculus concepts.

• Definition: is continuous at if .

• Polynomial Functions: All polynomials are continuous everywhere.

• Example: is continuous for all real .

Inverse Functions

Inverse functions reverse the effect of the original function. They are important in solving equations and understanding function behavior.

• Definition: If is one-to-one, its inverse satisfies .

• Logarithmic and Exponential Functions: and .

• Inverse Trigonometric Functions: , , .

• Example: If , then .

Derivatives

The derivative measures the rate of change of a function. It is a fundamental concept in calculus, used to analyze and model change.

• Definition: The derivative of at is .

• Differentiation Formulae: Includes power rule, product rule, quotient rule, and chain rule.

• Trigonometric Functions: , .

• Log/Exp Derivatives: , .

• Implicit Differentiation: Used when is defined implicitly by an equation.

• Related Rates: Application of derivatives to problems involving changing quantities.

• Linear Approximations: Using tangent lines to approximate function values.

• Differentials:

• Example: ,

Applications of Differentiation

Derivatives are used to solve real-world problems, including optimization, motion, and curve analysis.

• Critical Points: Where or is undefined.

• Optimization: Finding maximum and minimum values.

• Inflection Points: Where the concavity of changes.

• Example: To maximize , set .

Logarithmic Differentiation

Logarithmic differentiation is useful for functions involving products, quotients, or powers.

• Technique: Take the natural logarithm of both sides, differentiate, and solve for .

• Example: For ,

Derivatives of Inverse Trig Functions

Inverse trigonometric functions have specific derivative formulas.

• Formulas:

• Example:

Anti-derivatives

Anti-derivatives, or indefinite integrals, reverse the process of differentiation.

• Definition: If , then is an anti-derivative of .

• Notation:

• Example:

Weeks 1–3: Precalculus Review

Set Theory and Number Systems

Set theory and number systems provide the language and structure for mathematics.

• Set Theory: Study of collections of objects.

• Number Systems: Includes natural numbers (), integers (), rationals (), reals (), and complex numbers ().

• Example: is the set of all real numbers.

Interval Notation and Indices

Interval notation is used to describe subsets of real numbers; indices refer to exponents.

• Interval Notation: (closed), (open), , .

• Indices: means multiplied by itself times.

• Example: is all real numbers with .

Expanding and Factorising

Expanding and factorising are algebraic techniques for manipulating expressions.

• Expanding: Multiplying out brackets, e.g., .

• Factorising: Writing expressions as products, e.g., .

Quadratic Equations & Nature of Roots

Quadratic equations are second-degree polynomials; their roots can be real or complex.

• General Form:

• Quadratic Formula:

• Nature of Roots: Determined by the discriminant

• Example: If , two real roots; , one real root; , two complex roots.

Rationalising Fractions

Rationalising removes radicals from denominators.

• Technique: Multiply numerator and denominator by a suitable expression.

• Example:

Solving Inequalities

Inequalities are solved using algebraic manipulation and graphical methods.

• Linear Inequalities:

• Quadratic Inequalities:

• Example: Solve

Absolute Value: Equations, Inequalities, Graphs

Absolute value measures distance from zero; equations and inequalities involving absolute value require special techniques.

• Definition: if , if

• Example: or or

Logarithms and Exponential Functions

Logarithms and exponentials are inverse functions, essential in calculus.

• Exponential Function:

• Logarithm: is the inverse of

• Properties: ,

• Example:

Trigonometry: Radian Measure, Reciprocal Functions

Trigonometric functions are periodic and are measured in radians for calculus.

• Radian Measure: radians = 180 degrees

• Reciprocal Functions: , ,

• Example: ,

The Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials.

• Formula:

• Example:

Weeks 4–5: Logic and Proof Techniques

Logic and Truth Tables

Logic is the basis for mathematical reasoning; truth tables show the truth values of logical statements.

• Propositions: Statements that are either true or false.

• Truth Tables: Used to analyze logical connectives (AND, OR, NOT, IMPLIES).

• Example: The truth table for (AND) is true only when both and are true.

Proof Techniques

Several proof techniques are used in mathematics to establish the truth of statements.

• Direct Proof: Prove the statement directly from definitions and known facts.

• Proof by Contrapositive: Prove "if not Q, then not P" instead of "if P, then Q".

• Proof by Contradiction: Assume the statement is false and derive a contradiction.

• Proof by Cases: Consider all possible cases.

• Proof by Mathematical Induction: Prove for , then assume true for , prove for .

• Example: Prove by induction.

Sigma Notation Review

Sigma notation is used to represent sums compactly.

• Notation: means add for from 1 to .

• Example:

Non-examinable Proofs / Results

• Proof of Property (1) of Limits: Found in Calculus Ch 2 notes, Limits and Continuity, pg 20.

• Proof that a polynomial is continuous everywhere: Found in Calculus Ch 2 notes, Limits and Continuity, pg 41.

• Derivation of : Found in Calculus Ch 4 notes, Derivatives, pgs 41 & 42.

Additional info: Hyperbolic functions (Ch 6.7) are not covered in this course.