Chapter 4: Problem 30
In Exercises \(29-34,\) find all possible functions \(f\) with the given derivative. $$f^{\prime}(x)=2$$
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Multiple Choice If \(a<0,\) the graph of \(y=a x^{3}+3 x^{2}+\) \(4 x+5\) is concave up on (A) \(\left(-\infty,-\frac{1}{a}\right)\) (B) \(\left(-\infty, \frac{1}{a}\right)\) (C) \(\left(-\frac{1}{a}, \infty\right) (D) \)\left(\frac{1}{a}, \infty\right)\( (E) \)(-\infty,-1)$
$$ \begin{array}{l}{\text { Multiple Choice If } f(x)=\cos x, \text { then the Mean Value }} \\ {\text { Theorem guarantees that somewhere between } 0 \text { and } \pi / 3, f^{\prime}(x)=} \\ {\text { (A) }-\frac{3}{2 \pi} \quad \text { (B) }-\frac{\sqrt{3}}{2} \quad(\mathbf{C})-\frac{1}{2} \quad \text { (D) } 0}\end{array} $$
Airplane Landing Path An airplane is flying at altitude \(H\) when it begins its descent to an airport runway that is at horizontal ground distance \(L\) from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function \(y=a x^{3}+b x^{2}+c x+d\) where \(y(-L)=H\) and \(y(0)=0 .\) (a) What is \(d y / d x\) at \(x=0 ?\) (b) What is \(d y / d x\) at \(x=-L ?\) (c) Use the values for \(d y / d x\) at \(x=0\) and \(x=-L\) together with \(y(0)=0\) and \(y(-L)=H\) to show that $$y(x)=H\left[2\left(\frac{x}{L}\right)^{3}+3\left(\frac{x}{L}\right)^{2}\right]$$
Multiple Choice If the linearization of \(y=\sqrt[3]{x}\) at \(x=64\) is used to approximate \(\sqrt[3]{66},\) what is the percentage error? \(\begin{array}{llll}{\text { (A) } 0.01 \%} & {\text { (B) } 0.04 \%} & {\text { (C) } 0.4 \% \text { (D) 1 } \%}\end{array}\) (E) 4\(\%\)
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? \(\begin{array}{lllll}{\text { (A) }-3.875} & {\text { (B) }-3.75} & {\text { (C) }-2.25} & {\text { (D) } 3.75} & {\text { (E) } 3.875}\end{array}\)
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