91Ó°ÊÓ

Finding roots with Newton's method For the given function f and initial approximation \(x_{0},\) use Newton's method to approximate a root of \(f .\) Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. $$f(x)=\cos ^{-1} x-x ; x_{0}=0.75$$

Use the guidelines of this section to make a complete graph of \(f\). $$f(x)=x^{4}-6 x^{2}$$

Increasing and decreasing functions Find the intervals on which \(f\) is increasing and the intervals on which it is decreasing. $$f(x)=4-x^{2}$$

Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$f(x)=e^{x}$$

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow-1} \frac{x^{4}+x^{3}+2 x+2}{x+1}$$

Linear approximation Find the linear approximation to the following functions at the given point a. $$f(x)=4 x^{2}+x ; a=1$$

Of all boxes with a square base and a volume of \(8 \mathrm{m}^{3},\) which one has the minimum surface area? (Give its dimensions.)

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{2 x+5}$$

Use the guidelines of this section to make a complete graph of \(f\). $$f(x)=x^{4}+4 x^{3}$$

Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square-based box with the greatest volume under these conditions?