Calculus I · 2B · review

Derivative Theorems and Shape Review

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

Course progress35 of 57 core pages35 of 57

Section overview

Theorems, extrema, and curve shape

What this section is building

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

Notice

Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.

Decide

List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.

Avoid

Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.

Review

Summary

Critical numbers locate candidates. The closed interval method finds absolute extrema. Rolle's and Mean Value Theorems guarantee interior derivative behavior under continuity and differentiability hypotheses. Signs of $f'$ determine monotonicity; signs of $f"$ determine concavity.

Exercise

Find absolute extrema of $x^4-4x^2$ on $[-3,2]$ .

Answer reveal

Exercise 1 answer

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Exercise

Verify Rolle's Theorem for $f(x)=\sin x$ on $[0,\pi]$ .

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Exercise 2 answer

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Exercise

Apply the Mean Value Theorem to $f(x)=\sqrt x$ on $[1,4]$ .

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Exercise 3 answer

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Exercise

Analyze increasing and decreasing intervals of $x^4-4x^3$ .

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Exercise 4 answer

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Exercise

Classify all critical points of $x^4-4x^2$ .

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Exercise 5 answer

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Exercise

Find concavity and inflection points of $x^3-6x^2+9x$ .

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Exercise 6 answer

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Exercise

Perform a complete curve analysis of $(x^2-1)/(x^2+1)$ .

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Exercise 7 answer

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Exercise

Explain why $f"(c)=0$ is not sufficient for an inflection point.

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Exercise 8 answer

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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.

Review

Summary

Exercise

Find absolute extrema of $x^4-4x^2$ on $[-3,2]$ .

Answer reveal

Exercise 1 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Verify Rolle's Theorem for $f(x)=\sin x$ on $[0,\pi]$ .

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Apply the Mean Value Theorem to $f(x)=\sqrt x$ on $[1,4]$ .

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Analyze increasing and decreasing intervals of $x^4-4x^3$ .

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Classify all critical points of $x^4-4x^2$ .

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Find concavity and inflection points of $x^3-6x^2+9x$ .

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Perform a complete curve analysis of $(x^2-1)/(x^2+1)$ .

Answer reveal

Exercise 7 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Explain why $f"(c)=0$ is not sufficient for an inflection point.

Answer reveal

Exercise 8 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

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.

Review

Related resources

Original instruction with traceable references.

Review

Related resources

Original instruction with traceable references.