Chapter 4: Problem 53
Multiple Choice Two positive numbers have a sum of 60. What is the maximum product of one number times the square of the second number? (a) 3481 (b) 3600 (c) 27,000 (d) 32,000 (e) 36,000
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Draining Conical Reservoir Water is flowing at the rate of 50 \(\mathrm{m}^{3} / \mathrm{min}\) from a concrete conical reservoir (vertex down) of base radius 45 \(\mathrm{m}\) and height 6 \(\mathrm{m} .\) (a) How fast is the water level falling when the water is 5 \(\mathrm{m}\) deep? (b) How fast is the radius of the water's surface changing at that moment? Give your answer in \(\mathrm{cm} / \mathrm{min.}\)
Oscillation Show that if \(h>0,\) applying Newton's method to $$f(x)=\left\\{\begin{array}{ll}{\sqrt{x},} & {x \geq 0} \\ {\sqrt{-x},} & {x<0}\end{array}\right.$$ leads to \(x_{2}=-h\) if \(x_{1}=h,\) and to \(x_{2}=h\) if \(x_{1}=-h\) Draw a picture that shows what is going on.
Multiple Choice What is the maximum area of a right triangle with hypotenuse 10? \(\begin{array}{llll}{\text { (A) } 24} & {\text { (B) } 25} & {\text { (C) } 25 \sqrt{2}} & {\text { (D) } 48} & {\text { (E) } 50}\end{array}\)
Group Activity In Exercises \(39-42,\) sketch a graph of a differentiable function \(y=f(x)\) that has the given properties. $$\begin{array}{l}{\text { A local minimum value that is greater than one of its local maxi- }} \\ {\text { mum values. }}\end{array}$$
The Linearization is the Best Linear Approximation Suppose that \(y=f(x)\) is differentiable at \(x=a\) and that \(g(x)=m(x-a)+c(m\) and \(c\) constants). If the error \(E(x)=f(x)-g(x)\) were small enough near \(x=a,\) we might think of using \(g\) as a linear approximation of \(f\) instead of the linearization \(L(x)=f(a)+f^{\prime}(a)(x-a) .\) Show that if we impose on \(g\) the conditions i. \(E(a)=0\) ii. \(\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0\) then \(g(x)=f(a)+f^{\prime}(a)(x-a) .\) Thus, the linearization gives the only linear approximation whose error is both zero at \(x=a\) and negligible in comparison with \((x-a)\) .
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