AP Calculus AB Linearization and L’Hopital’s Rule Worksheet Solutions (PDF)
This linearization practice solutions 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.
This answer key gives a full worked solution for every problem in the linearization worksheet. Compare each step, not only the final answer, because AP graders award credit for a correct setup and clear justification. Final answers are in bold. Redo any problem you missed before moving on to the test.
AP Calculus AB · Common Core State Standards: AP CALC AB 4.5, AP CALC AB 4.6, AP CALC AB 4.7
Linearization practice solutions: exercises 1 to 10
Exercise 1 – Linearization of x squared
\(f(3)=9\) and \(f^{\prime}(3)=6\), so \(L(x)=9+6(x-3)\).
\(L(3.1)=9+6(0.1)=9.6\). The exact value is \(9.61\), so the error is \(0.01\).
Final answer: \(L(x)=9+6(x-3);\ (3.1)^{2}\approx 9.6\).
Exercise 2 – Estimating a square root
\(f(25)=5\), \(f^{\prime}(x)=\frac{1}{2\sqrt{x}}\), so \(f^{\prime}(25)=\frac{1}{10}\).
\(L(26)=5+\frac{1}{10}(1)=5.1\). The calculator gives \(5.0990\).
Final answer: \(\sqrt{26}\approx 5.1\).
Exercise 3 – Estimating a cube root
\(f(27)=3\) and \(f^{\prime}(x)=\frac{1}{3}x^{-2/3}\), so \(f^{\prime}(27)=\frac{1}{3\cdot 9}=\frac{1}{27}\).
\(L(28)=3+\frac{1}{27}\approx 3.037\).
Final answer: \(\sqrt[3]{28}\approx 3.037\).
Exercise 4 – Estimating a reciprocal
\(f(2)=\frac{1}{2}\) and \(f^{\prime}(2)=-\frac{1}{4}\).
\(L(2.1)=\frac{1}{2}-\frac{1}{4}(0.1)=0.475\). The exact value is \(0.47619\).
Final answer: \(0.475\).
Exercise 5 – Exponential near zero
\(f(0)=1\), \(f^{\prime}(0)=1\), so \(L(x)=1+x\).
\(L(0.1)=1.1\) and \(L(-0.05)=0.95\).
Final answer: \(1+x;\ 1.1\text{ and }0.95\).
Exercise 6 – Sine near zero
\(L(x)=\sin 0+\cos 0\cdot x=x\), so \(\sin(0.2)\approx 0.2\). (Exact: \(0.19867\).)
Final answer: \(0.2\).
Exercise 7 – Logarithm near one
\(f(1)=0\), \(f^{\prime}(1)=1\), so \(L(x)=x-1\).
\(L(1.05)=0.05\). (Exact: \(0.04879\).)
Final answer: \(0.05\).
Exercise 8 – Tangent near pi over four
\(f(\frac{\pi}{4})=1\) and \(f^{\prime}=\sec^{2}\frac{\pi}{4}=2\). So \(L(x)=1+2\left(x-\frac{\pi}{4}\right)\).
\(L(0.8)=1+2(0.8-0.7854)=1.0292\). (Exact: \(1.0296\).)
Final answer: \(\tan(0.8)\approx 1.029\).
Exercise 9 – Overestimate or underestimate, root
\(f^{\prime\prime}(x)=-\frac{1}{4}x^{-3/2} < 0\) for \(x > 0\): the graph is concave down.
The tangent line lies above the graph, so \(L(10)=3+\frac{1}{6}\approx 3.1667\) is greater than \(\sqrt{10}\approx 3.1623\).
Final answer: \(\text{overestimate}\).
Exercise 10 – Overestimate or underestimate, cube
\(f^{\prime\prime}(x)=6x > 0\) near \(2\): concave up, so the tangent lies below the graph.
\(L(2.1)=8+12(0.1)=9.2\) and the exact value is \(9.261\).
Final answer: \(\text{underestimate}\).
Exercises 11 and up
Exercise 11 – Using given values
\(L(x)=7-2(x-3)\). Then \(L(3.2)=7-0.4=6.6\) and \(L(2.9)=7+0.2=7.2\).
Final answer: \(6.6\text{ and }7.2\).
Exercise 12 – Linear approximation, a quotient
\(g(1)=\frac{1}{2}\). \(g^{\prime}(x)=\frac{1}{(x+1)^{2}}\), so \(g^{\prime}(1)=\frac{1}{4}\).
\(L(x)=\frac{1}{2}+\frac{1}{4}(x-1)\), so \(L(1.2)=0.55\). (Exact: \(0.5455\).)
Final answer: \(g(1.2)\approx 0.55\).
Exercise 13 – Hopital with an exponential and a sine
Substituting gives \(\frac{0}{0}\). Differentiate top and bottom: \(\frac{3e^{3x}}{2\cos 2x}\).
At \(x=0\): \(\frac{3}{2}\).
Final answer: \(\frac{3}{2}\).
Exercise 14 – Hopital, quadratic
Substituting gives \(\frac{0}{0}\). Differentiate: \(\frac{2x-4}{2x}\), which at \(x=1\) is \(\frac{-2}{2}=-1\).
Final answer: \(-1\).
Exercise 15 – Hopital at x equals one
Substituting gives \(\frac{0}{0}\). Differentiate: \(\frac{1/x}{2x}=\frac{1}{2x^{2}}\).
At \(x=1\) this is \(\frac{1}{2}\).
Final answer: \(\frac{1}{2}\).
Exercise 16 – Hopital, infinity over infinity
- \(\frac{\infty}{\infty}\): \(\lim\frac{1}{e^{x}}=0\).
- \(\frac{\infty}{\infty}\): \(\lim\frac{1/x}{1}=0\).
Final answer: \(0\text{ and }0\).
Exercise 17 – Hopital twice
\(\frac{\infty}{\infty}\). First: \(\frac{6x}{2x+5}\) (still \(\frac{\infty}{\infty}\)). Second: \(\frac{6}{2}=3\).
Final answer: \(3\).
Exercise 18 – Three applications
\(\frac{0}{0}\). Differentiate: \(\frac{\cos x-1}{3x^{2}}\) (still \(\frac{0}{0}\)), then \(\frac{-\sin x}{6x}\) (still \(\frac{0}{0}\)), then \(\frac{-\cos x}{6}\to-\frac{1}{6}\).
Final answer: \(-\frac{1}{6}\).
Exercise 19 – When not to use the rule
Substituting gives \(\frac{2}{-1}=-2\), which is not \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Applying the rule would give \(\frac{1}{1}=1\), which is wrong.
Final answer: \(-2\).
Exercise 20 – Square garden
From the earlier work, \(\sqrt{26}\approx 5.1\). The square root function is concave down, so the tangent line is above the curve.
Final answer: \(\text{about }5.1\text{ ft (a slight overestimate; the exact side is about }5.099\text{ ft)}\).
Exercise 21 – Change in volume
\(V^{\prime}(r)=4\pi r^{2}\), so \(V^{\prime}(5)=100\pi\).
\(\Delta V\approx V^{\prime}(5)\cdot 0.1=10\pi\approx 31.4\).
The exact change is about \(32\).
Final answer: \(\Delta V\approx 10\pi\approx 31.4\text{ cm}^{3}\).
Exercise 22 – Measurement error
\(A^{\prime}(s)=2s\), so \(A^{\prime}(10)=20\). Then \(\Delta A\approx 20(0.1)=2\).
Final answer: \(\text{about }2\text{ cm}^{2}\).
Exercise 23 – Pizza area
\(A^{\prime}(8)=2\pi(8)=16\pi\), so \(\Delta A\approx 16\pi(0.2)=3.2\pi\approx 10.05\).
Final answer: \(\text{about }10.05\text{ in}^{2}\).
Exercise 24 – Temperature estimate
\(T(12)\approx 350+4(2)=358\) °F.
The sign of \(T^{\prime\prime}\) would decide it: concave down (\(T^{\prime\prime} < 0\)) gives an overestimate, concave up an underestimate.
Final answer: \(358\,^{\circ}\text{F}\).
Key facts behind this linearization practice solutions
- 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 practice solutions
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the linearization practice solutions as a PDF with the button below and keep it with your notes.
Download the linearization practice solutions as a PDF with the button below and keep it with your notes.
Download the linearization practice solutions as a PDF with the button below and keep it with your notes.
Download the linearization practice solutions as a PDF with the button below and keep it with your notes.
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