AP Calculus AB Linearization and L’Hopital’s Rule Worksheet (PDF)

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

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This linearization practice worksheet 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.

These 24 problems practice local linear approximation, over- and underestimates from concavity, and L’Hopital’s Rule for indeterminate limits (AP Calculus AB, topics 4.5, 4.6, 4.7). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use over- and underestimates, tables of values, repeated applications of the rule and real situations such as measurement error and a pizza area. Work in pencil, show every step, and check the answer key only after a full attempt.

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

Linearization practice worksheet: exercises 1 to 10

Exercise 1 – Linearization of x squared

The tangent to y = x² at x = 3.

Find the linearization of \(f(x)=x^{2}\) at \(a=3\) and use it to estimate \((3.1)^{2}\). Compare with the exact value.

Exercise 2 – Estimating a square root

Tangent to √x at x = 25.

Use a linear approximation of \(f(x)=\sqrt{x}\) at \(a=25\) to estimate \(\sqrt{26}\).

Exercise 3 – Estimating a cube root

Estimate \(\sqrt[3]{28}\) using the linearization of \(f(x)=\sqrt[3]{x}\) at \(a=27\).

Exercise 4 – Estimating a reciprocal

Estimate \(\frac{1}{2.1}\) using the linearization of \(f(x)=\frac{1}{x}\) at \(a=2\).

Exercise 5 – Exponential near zero

Find the linearization of \(f(x)=e^{x}\) at \(a=0\) and estimate \(e^{0.1}\) and \(e^{-0.05}\).

Exercise 6 – Sine near zero

Estimate \(\sin(0.2)\) using the linearization of \(\sin x\) at \(0\).

Exercise 7 – Logarithm near one

Estimate \(\ln(1.05)\) with the linearization of \(\ln x\) at \(a=1\).

Exercise 8 – Tangent near pi over four

The tangent to tan x at x = π/4.

Estimate \(\tan(0.8)\) using the linearization of \(f(x)=\tan x\) at \(a=\frac{\pi}{4}\).

Exercise 9 – Overestimate or underestimate, root

The tangent line lies above y = √x.

The linearization of \(f(x)=\sqrt{x}\) at \(a=9\) is used to estimate \(\sqrt{10}\). Is the estimate an overestimate or an underestimate? Justify.

Exercise 10 – Overestimate or underestimate, cube

The tangent line lies below y = x³.

The linearization of \(f(x)=x^{3}\) at \(a=2\) is used to estimate \((2.1)^{3}\). Over or under?

Exercises 11 and up

Exercise 11 – Using given values

A differentiable function has \(f(3)=7\) and \(f^{\prime}(3)=-2\). Estimate \(f(3.2)\) and \(f(2.9)\).

Exercise 12 – Linear approximation, a quotient

Find the linearization of \(g(x)=\frac{x}{x+1}\) at \(a=1\) and estimate \(g(1.2)\).

Exercise 13 – Hopital with an exponential and a sine

Evaluate \(\lim_{x\to 0}\frac{e^{3x}-1}{\sin 2x}\).

Exercise 14 – Hopital, quadratic

Evaluate \(\lim_{x\to 1}\frac{x^{2}-4x+3}{x^{2}-1}\).

Exercise 15 – Hopital at x equals one

Evaluate \(\lim_{x\to 1}\frac{\ln x}{x^{2}-1}\).

Exercise 16 – Hopital, infinity over infinity

Evaluate \(\lim_{x\to\infty}\frac{x}{e^{x}}\) and \(\lim_{x\to\infty}\frac{\ln x}{x}\).

Exercise 17 – Hopital twice

Evaluate \(\lim_{x\to\infty}\frac{3x^{2}+1}{x^{2}+5x}\) with L’Hôpital’s Rule.

Exercise 18 – Three applications

Evaluate \(\lim_{x\to 0}\frac{\sin x-x}{x^{3}}\).

Exercise 19 – When not to use the rule

Explain why L’Hôpital’s Rule should not be used for \(\lim_{x\to 0}\frac{x+2}{x-1}\) and find the limit.

Exercise 20 – Square garden

Mei wants a square garden with an area of 26 square feet. Use a linear approximation of \(\sqrt{x}\) at \(25\) to estimate the side length, and decide if your estimate is high or low.

Exercise 21 – Change in volume

A spherical balloon has radius 5 cm, which grows to 5.1 cm. Use the linearization of \(V(r)=\frac{4}{3}\pi r^{3}\) at \(r=5\) to estimate the change in volume.

Exercise 22 – Measurement error

A square tile’s side is measured as 10 cm with a possible error of 0.1 cm. Use a linear approximation to estimate the maximum error in the area \(A=s^{2}\).

Exercise 23 – Pizza area

A pizza’s radius grows from 8 in to 8.2 in. Use \(A=\pi r^{2}\) and a linear approximation to estimate the change in area.

Exercise 24 – Temperature estimate

The temperature of an oven in °F is \(T(t)\) with \(T(10)=350\) and \(T^{\prime}(10)=4\) °F per minute. Estimate \(T(12)\), and say what extra information would tell you whether it is an over- or underestimate.

Key facts behind this linearization practice worksheet

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 practice worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the linearization practice worksheet as a PDF with the button below and keep it with your notes.

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