Calculus I · 2A · review

Derivative Foundations Review

Review derivative foundations review with mixed practice and links back to the exact lessons behind each skill.

Course progress11 of 49 core pages11 of 49

Section overview

Derivative meaning and foundations

What this section is building

Review derivative foundations review with mixed practice and links back to the exact lessons behind each skill.

Notice

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

Decide

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

Avoid

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

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Foundation Review

Summary

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Foundation Review

Related resources

Original instruction with traceable references.

Foundation Review

Related resources

Original instruction with traceable references.