Level 7: Limits and the derivative

Change and motion · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In level 6 you closed in on the speed at an instant by shrinking time intervals. This level turns that idea into the real tools of calculus. You learn what a limit is and when a function is continuous, define the derivative as a limit and work it out from first principles, discover the power rule that makes it quick, read the derivative off a graph, and use it to describe motion: position, velocity and acceleration.

Lesson	You will be able to
Limits	Find limits from tables, pictures and algebra, use one-sided limits and the limit laws, and spot limits that do not exist
Continuity	Say exactly what continuous means, classify holes, jumps and asymptotes, and use the intermediate value theorem to find roots
The derivative from first principles	Use \(f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\) to differentiate \(x^2\), \(x^3\), \(\frac1x\) and \(\sqrt{x}\), and see where derivatives fail
The power rule	Prove the power rule, differentiate any polynomial and power, and find tangent and normal lines
The derivative as a function	Sketch \(f'\) from the graph of \(f\) and back again, and match a function with its derivatives
Velocity and acceleration	Use \(s\), \(s'\) and \(s''\) to analyse a throw on Earth and on Mars, braking cars and particles that turn round

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