Calculus I · Unit 2B · quiz

Derivative Theorems and Shape Concept Quiz

Concept Quiz: Theorems and Shape

• Why are critical numbers only candidates for extrema? • State every hypothesis of the Mean Value Theorem. • What sign change in $f'$ identifies a local maximum? • How does concavity differ from increasing or decreasing behavior? • Why can a vertical asymptote not be an inflection point? • When is the Second Derivative Test inconclusive?

After the explanation

Use the section idea

Reading lens

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?

Structured concept quiz

Derivative Theorems and Shape Concept Quiz

Write a response before revealing the model. These conceptual items use an honest attempt-and-reveal rubric rather than pretending an open response has one machine-provable wording.

Interactive checktheorems-shape-concept-quiz-01

Why are critical numbers only candidates for extrema?