Introduction to Analysis · Lesson 10

Chapter practice

This lesson mixes the chapter without saying which idea each question is about. The tools are the definition, the four rules with the chain rule, and the single theorem from which everything else in the chapter follows.

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f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

(f g)^{'} = f^{'} g + f g^{'}

(f \circ g)^{'} = (f^{'} \circ g) g^{'}

f^{''} (x)

\frac{f ( b ) - f ( a )}{b - a}

Chapter navigation

Intuition

This lesson mixes the chapter without saying which idea each question is about. The tools are the definition, the four rules with the chain rule, and the single theorem from which everything else in the chapter follows.

A toolbox with five tools in it. Knowing which to reach for is a skill of its own, and these questions practise that rather than any one tool.

What this chapter established

A derivative is a limit of difference quotients. Differentiability implies continuity and not conversely. The rules differentiate anything written down. Rolle gives the Mean Value Theorem, which gives everything a derivative says about a function on an interval, and Taylor is the same theorem with more derivatives.

f(b)-f(a)=f'(c)(b-a)

The tools, and when each is reached for

The definition, when a function is given piecewise or a derivative is claimed not to exist.
The rules and the chain rule, when a formula is to be differentiated.
The Mean Value Theorem, whenever a statement about a whole interval is wanted from a statement about every point.
Darboux, when the question is whether a given function is anyone's derivative.
Taylor, when an approximation has to be estimated rather than merely written down.

Practice

Try it

$f$ is differentiable on $\mathbb{R}$ with $f'(x)\le2$ everywhere and $f(0)=0$ . What can be said about $f(5)$ ?