AP Calculus AB Linearization and L’Hopital’s Rule Quiz: 10 Questions with Explanations

AP Calculus AB math quiz: Linearization and L'Hopital's Rule – Maths PDF

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This linearization quiz for AP Calculus AB covers linearization and l'hopital's rule (Common Core AP CALC AB 4.5, AP CALC AB 4.6, AP CALC AB 4.7), with a free PDF to print.

Test yourself on local linear approximation, over- and underestimates from concavity, and L’Hopital’s Rule for indeterminate limits 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 4.5, AP CALC AB 4.6, AP CALC AB 4.7

Linearization quiz: 10 questions

Question 1

Which is the linearization of \(f(x)\) at \(x=a\)?

Question 2

Using the linearization of \(f(x)=x^{2}\) at \(a=5\), what is the estimate of \((5.2)^{2}\)?

Question 3

A concave down graph and its tangent line.

The tangent line at the peak is used to estimate nearby values. What type of estimate is it?

Question 4

If \(f^{\prime\prime}(x) > 0\) near \(a\), the linear approximation at \(a\) is

Question 5

Which limit can be evaluated with L’Hôpital’s Rule?

Question 6

What is \(\lim_{x\to 3}\frac{x^{2}-9}{x-3}\)?

Question 7

Graph of x/eˣ.

What is \(\lim_{x\to\infty}\frac{x}{e^{x}}\)?

Question 8

What is \(\lim_{x\to 0}\frac{\sin 4x}{\tan x}\)?

Question 9

Values at x = 2.

Using the table, estimate \(f(2.1)\) with a linear approximation.

Question 10

Why can’t L’Hôpital’s Rule be used for \(\lim_{x\to 0}\frac{x+1}{x+2}\)?

Remember

Linearization: L(x) = f(a) + f′(a)(x – a), the tangent line at x = a.
Concave up (f″ positive): the tangent line is below the curve, so the estimate is too low.
Concave down (f″ negative): the tangent line is above the curve, so the estimate is too high.
L’Hôpital: for 0/0 or ∞/∞, lim f/g = lim f′/g′ (not the quotient rule).

How to use this linearization 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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