Mathematics 31

Unit 1: Precalculus and Limits

The algebra needed for calculus and the central idea that begins it: the limit. Algebra and composition of functions, transformations, equivalent forms, and a rigorous treatment of limits, continuity and limit theorems — including the trigonometric limits.

• Operations and composition of functions
• Solution sets of inequalities
• Simplifying and solving with trigonometric identities

• Translation, reflection and dilatation
• Parallel, perpendicular, tangent, normal and secant lines
• Solving systems of equations

• Factoring, rationalizing and simplifying

• The concept of a limit and continuity
• Evaluating limits with the limit theorems
• Geometric series and trigonometric limits

Unit 2: Derivatives

The derivative as the limit of a slope. From the definition and the tangent-as-limit idea, through the power/sum/difference theorems, the product, quotient and chain rules, implicit and higher-order differentiation, to the derivatives of the trigonometric functions.

• The tangent slope as a limit
• Power, sum and difference theorems

• The product, quotient and chain rules

• Differentiating implicitly

• Second and higher derivatives

• Differentiating trigonometric functions

Unit 3: Applications of Derivatives

The derivative as a tool. Using the first and second derivatives to sketch curves and locate maxima, minima and inflection points and to solve optimization problems; and using the chain rule to relate rates of change through time.

• First and second derivative analysis
• Optimization problems

• Related rates via the chain rule

Unit 4: Integrals

Integration as the inverse of differentiation and as accumulated area. Antiderivatives and the family of curves, separable first-order differential equations, area under a curve as a limit of rectangle sums, the definite integral and the Fundamental Theorem of Calculus, and the displacement-velocity-acceleration chain of motion.

• Antiderivatives and families of curves
• Separable first-order differential equations

• Area as a limit of rectangle sums

• The definite integral and the FTC
• Evaluating integrals and area between curves

• Kinematics with derivatives and antiderivatives

Unit 5: Electives: Growth, Numerical and Integration Methods

Elective strands extending the calculus toolkit: the calculus of exponential and logarithmic functions built on e, numerical methods for roots and integrals, and further techniques of integration (substitution, parts, partial fractions).

• Differentiating and antidifferentiating exp/log functions

• Modelling growth, decay and return to equilibrium

• Approximating roots with Newton-Raphson

• Riemann sums and the quadrature rules

• Substitution and trigonometric substitution

• Integration by parts
• Integration by partial fractions

Unit 6: Electives: Volumes and Applied Calculus

Elective strands applying calculus: volumes of revolution by the disc method, and modelling in the physical sciences and engineering, the biological sciences, and business and economics.

• The disc method

• Differential equations in physical models

• Growth, decay and the logistic equation

• Optimizing revenue, profit and cost

Unit 7: Electives: Calculus Theorems and Proof

The elective proof strand: the nature of proof for limit, derivative and integral theorems, and the great existence theorems — intermediate value, Rolle's, mean value and the Fundamental Theorem of Calculus.

• Proof and the value theorems