AP Calculus AB The Derivative as a Limit Quiz: 10 Questions with Explanations

AP Calculus AB math quiz: The Derivative as a Limit – Maths PDF

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This derivative definition quiz for AP Calculus AB covers the derivative as a limit (Common Core AP CALC AB 2.1, AP CALC AB 2.2, AP CALC AB 2.3, AP CALC AB 2.4), with a free PDF to print.

Test yourself on average and instantaneous rates of change, the limit definition of the derivative, tangent lines and differentiability with 10 multiple-choice questions. Pick one answer for each question, then read the explanation to see why it is right and which common mistake the other choices represent.

AP Calculus AB · Common Core State Standards: AP CALC AB 2.1, AP CALC AB 2.2, AP CALC AB 2.3, AP CALC AB 2.4

Derivative definition quiz: 10 questions

Question 1

Which expression defines \(f^{\prime}(a)\)?

Question 2

Tangent to y = x² at x = 2.

The orange line is tangent to \(y=x^{2}\) at \(x=2\). What is \(f^{\prime}(2)\)?

Question 3

What is the average rate of change of \(f(x)=x^{3}\) on \([1,3]\)?

Question 4

If \(f(x)=x^{2}\), what is \(\lim_{h\to 0}\frac{f(5+h)-f(5)}{h}\)?

Question 5

Graph of f(x) = |x + 1|.

At which \(x\)-value is \(f\) continuous but not differentiable?

Question 6

Which statement is true?

Question 7

A tangent line to \(f\) at \(x=4\) passes through the points \((4,7)\) and \((6,13)\). What is \(f^{\prime}(4)\)?

Question 8

Distance in feet.

Estimate the instantaneous speed at \(t=2\) seconds using the table.

Question 9

For \(f(x)=\sqrt[3]{x}\), what is true at \(x=0\)?

Question 10

If \(f^{\prime}(2)=-3\) and \(f(2)=5\), what is the tangent line at \(x=2\)?

Remember

The derivative is the limit of secant slopes, which are average rates of change.
f′(a) is the slope of the tangent line at x = a; the tangent is y – f(a) = f′(a)(x – a).
Both forms of the definition give the same value when the limit exists.
Differentiable implies continuous, but continuous does not imply differentiable.

How to use this derivative definition quiz

Answer each question, read the explanation that appears, and keep track of your score. A score of 8 out of 10 or more means you are ready for the test.

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