AP Calculus AB Linearization and L’Hopital’s Rule Test Solutions and Scoring (PDF)

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

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This linearization test answers 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.

Use this answer key to score the linearization test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 4.5, AP CALC AB 4.6, AP CALC AB 4.7

Linearization test answers: worked solutions

Question 1 – Linearization [2 pts]

\(f(16)=4\), \(f^{\prime}(16)=\frac{1}{8}\). \(L(x)=4+\frac{1}{8}(x-16)\).

\(L(17)=4.125\). Final answer: \(\sqrt{17}\approx 4.125\).

Scoring: 1 point for L(x); 1 point for the estimate.

Question 2 – Tangent line estimate [3 pts]

\(L(x)=1+x\).
\(L(0.2)=1.2\) (exact \(1.2214\)).
\(f^{\prime\prime}(x)=e^{x} > 0\): concave up, so the tangent lies below the curve. Underestimate.

The graph of eˣ and its tangent line at x = 0.

Scoring: 1 point per part.

Question 3 – Over or under [2 pts]

\(f^{\prime\prime}(x)=-\frac{1}{x^{2}} < 0\): concave down, so the tangent line lies above the graph and the estimate is too high.

The graph of ln x and its tangent at x = 2.

Scoring: 1 point for the second derivative; 1 point for the conclusion.

Question 4 – Using a table [3 pts]

\(L(4.5)=10-3(0.5)=8.5\).
The table gives \(8.6\), so the estimate is \(0.1\) too low.
The tangent line is below the graph, so concave up.

Scoring: 1 point per part.

Question 5 – Hopital 0 over 0, second form [2 pts]

\(\frac{0}{0}\). Differentiate: \(\frac{3x^{2}}{2x}=\frac{27}{6}=\frac{9}{2}\).

Final answer: \(\frac{9}{2}\).

Scoring: 1 point for identifying 0/0 and differentiating; 1 point for the value.

Question 6 – Hopital infinity form [3 pts]

\(\frac{\infty}{\infty}\). First: \(\frac{2\ln x\cdot\frac{1}{x}}{1}=\frac{2\ln x}{x}\) (still \(\frac{\infty}{\infty}\)). Second: \(\frac{2/x}{1}=\frac{2}{x}\to 0\).

Final answer: \(0\).

Scoring: 1 point for each application; 1 point for the value.

Question 7 – Error estimation [2 pts]

\(A=s^{2}\), \(A^{\prime}=2s=48\). \(\Delta A\approx 48(0.25)=12\).

Final answer: about 12 square inches.

Scoring: 1 point for A′; 1 point for the estimate with units.

Question 8 – Limit by two methods [3 pts]

\(\frac{0}{0}\): \(\frac{\cos x}{1}\to 1\).
\(\frac{0}{0}\): \(\frac{\sec^{2}x}{1}\to 1\).
Substituting gives \(\frac{1}{1}=1\), which is not indeterminate; the limit is simply \(1\).

The graph of sin x / x.

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this linearization test answers

Key facts

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 test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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