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Chapter 3: Problem 5
Suppose \(f\) is a one-to-one function with \(f(2)=8\) and \(f^{\prime}(2)=4 .\) What is the value of \(\left(f^{-1}\right)^{\prime}(8) ?\)
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Find the slope of the curve \(5 \sqrt{x}-10 \sqrt{y}=\sin x\) at the point \((4 \pi, \pi)\).
An airliner passes over an airport at noon traveling \(500 \mathrm{mi} / \mathrm{hr}\) due west. At \(1: 00 \mathrm{p} . \mathrm{m} .,\) another airliner passes over the same airport at the same elevation traveling due north at \(550 \mathrm{mi} / \mathrm{hr} .\) Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 2: 30 p.m.?
Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$
The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{6}+h\right)-\frac{\sqrt{3}}{2}}{h}$$
Use any method to evaluate the derivative of the following functions. $$f(z)=z^{2}\left(e^{3 z}+4\right)-\frac{2 z}{z^{2}+1}$$
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