Chapter 4: Problem 40
\(y=\left\\{\begin{array}{ll}{3-x,} & {x<0} \\ {3+2 x-x^{2},} & {x \geq 0}\end{array}\right.\)
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sign of \(f^{\prime}\) Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(\)b\()<$$f$$(\)a\()\). Show that \(f^{\prime}\) is negative at some point between \(a\) and \(b\).
Multiple Choice A particle is moving around the unit circle (the circle of radius 1 centered at the origin). At the point \((0.6,\) 0.8\()\) the particle has horizontal velocity \(d x / d t=3 .\) What is its vertical velocity \(d y / d t\) at that point? \(\begin{array}{lllll}{\text { (A) }-3.875} & {\text { (B) }-3.75} & {\text { (C) }-2.25} & {\text { (D) } 3.75} & {\text { (E) } 3.875}\end{array}\)
Expanding Circle The radius of a circle is increased from 2.00 to 2.02 \(\mathrm{m} .\) (a) Estimate the resulting change in area. (b) Estimate as a percentage of the circle's original area.
Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$\begin{array}{l}{\text { A local minimum value that is greater than one of its local maxi- }} \\ {\text { mum values. }}\end{array}$$
Writing to Learn You have been asked to determine whether the function \(f(x)=3+4 \cos x+\cos 2 x\) is ever negative. (a) Explain why you need to consider values of \(x\) only in the interval \([0,2 \pi] . \quad\) (b) Is f ever negative? Explain.
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