Level 8: Calculus

Change and motion · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

Level 7 built the derivative. This level finishes the toolkit and puts it to work. You learn the rules for products, quotients and functions inside functions, and the derivatives of \(e^x\), \(\ln x\), \(\sin x\) and \(\cos x\), each with a reason. Then you use derivatives to sketch any curve, find the best possible answer to real problems, and link rates that change together. Finally you meet the second half of calculus, areas and the fundamental theorem, and take a first step into differential equations, where calculus becomes the language of science.

Lesson	You will be able to
The product, quotient and chain rules	Prove each rule (the product rule from a growing rectangle) and differentiate mixed functions
Derivatives of \(e^x\), \(\ln x\), \(\sin x\) and \(\cos x\)	Explain why \(e^x\) is its own derivative, prove \(\frac{\sin h}{h} \to 1\), and differentiate exponentials, logarithms and trig functions
Curve sketching	Use \(f'\) and \(f''\) to find turning points, concavity, inflection points and asymptotes, and sketch cubics and rational functions
Optimization	Find the largest box, the cheapest can and the shortest path, checking critical points and endpoints
Related rates	Link rates with the chain rule: sliding ladders, growing balloons, filling cones
Areas under curves and the fundamental theorem	Use Riemann sums, prove the area under \(x^2\) from 0 to 1 is exactly \(\frac13\), and integrate with antiderivatives
Differential equations: a first look	Solve \(\frac{dy}{dt} = ky\) and Newton's law of cooling, read slope fields and use Euler's method

Previous: Level 7: limits and the derivative · Change and motion topic page · Onwards: the U of T calculus course, MAT137