AP Calculus AB Linearization and L’Hopital’s Rule Test with Point Values (PDF)

Find the linearization of \(f(x)=\sqrt{x}\) at \(a=16\) and use it to estimate \(\sqrt{17}\).

Let \(f(x)=e^{x}\). (a) Write the linearization at \(a=0\). (1 point) (b) Estimate \(e^{0.2}\). (1 point) (c) Is your estimate an over- or underestimate? Justify with \(f^{\prime\prime}\). (1 point)

The tangent line to \(f(x)=\ln x\) at \(x=2\) is used to estimate \(\ln 2.1\). Is the estimate too high or too low? Justify.

A function has \(f(4)=10\) and \(f^{\prime}(4)=-3\); the table gives other values. (a) Use a linear approximation to estimate \(f(4.5)\). (1 point) (b) Compare with the table value. (1 point) (c) Based on the comparison, is \(f\) concave up or concave down near \(x=4\)? (1 point)

Evaluate \(\lim_{x\to 3}\frac{x^{3}-27}{x^{2}-9}\) using L’Hôpital’s Rule.

Evaluate \(\lim_{x\to\infty}\frac{(\ln x)^{2}}{x}\) using L’Hôpital’s Rule. Show each application.

The side of a square window is measured as 24 inches, with a possible error of 0.25 inch. Use a linear approximation to estimate the possible error in its area.

(a) Use L’Hôpital’s Rule to evaluate \(\lim_{x\to 0}\frac{\sin x}{x}\). (1 point) (b) Evaluate \(\lim_{x\to 0}\frac{\tan x}{x}\). (1 point) (c) Explain why L’Hôpital’s Rule cannot be used on \(\lim_{x\to 0}\frac{\cos x}{x+1}\). (1 point)