Calculus I · Unit 2A · review

Derivative Foundations Review

Foundation Review

Summary

• $f'(a)$ is the limit of average rates over shrinking nonzero intervals. • It is both an instantaneous rate and a tangent slope. • The derivative function maps each input to its tangent slope. • Derivative units are output units per input unit. • Differentiability implies continuity, but continuity does not guarantee differentiability.

Exercise

Use the limit definition to find $f'(2)$ for $f(x)=x^2+1$ .

Exercise

Use the limit definition to find $g'(1)$ for $g(x)=1/x$ .

Exercise

Find the tangent and normal lines to $y=x^2$ at $x=-1$ .

Exercise

A drug-response function $R(d)$ is measured in beats per minute and dose $d$ in milligrams. Interpret $R'(5)=-1.8$ .

Exercise

Estimate $f'(4)$ from $f(3.8)=9.1$ and $f(4.2)=10.7$ .

Exercise

Give one example of a function continuous but not differentiable at $0$ , and explain why.

Exercise

Sketch a possible derivative graph for a function that decreases, flattens to a horizontal tangent, then increases.

Exercise

Determine where $f(x)=|x^2-1|$ is not differentiable.

After the explanation

Use the section idea

Reading lens

Watch a secant slope stabilize into a tangent slope and then generalize from one point to a derivative function.

Mental model

A derivative exists when shrinking two-point slopes settle to one finite local slope.

Decision

Choose whether the task asks for a value at one point, a full derivative function, or an estimate from data.

Common trap

Confusing the graph's height with its slope or assuming continuity automatically gives differentiability.

Check yourself

Can you move among a limit definition, tangent slope, graph estimate, and units without changing the meaning?

Source & rights

Original instruction with traceable references.

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