Calculus I · Unit 2A · quiz

Derivative Foundations Concept Quiz

Concept Quiz: Foundations

• What must happen to secant slopes for $f'(a)$ to exist? • Distinguish $f'(a)$ from $f'(x)$ . • Why does differentiability imply continuity? • Can a vertical tangent have a finite derivative? Explain. • If $f'(2)=-5$ , what are the point and slope needed for the tangent equation? • What are the units of $A'(r)$ if $A$ is square meters and $r$ is meters?

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?

Structured concept quiz

Derivative Foundations 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 checkfoundations-concept-quiz-01

What must happen to secant slopes for $f'(a)$ to exist?