Chapter 4: Problem 33
In Exercises \(29-34,\) find all possible functions \(f\) with the given derivative. $$f^{\prime}(x)=e^{x}$$
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Measuring Acceleration of Gravity When the length \(L\) of a clock pendulum is held constant by controlling its temperature, the pendulum's period \(T\) depends on the acceleration of gravity \(g\) . The period will therefore vary slightly as the clock is moved from place to place on the earth's surface, depending on the change in \(g\) . By keeping track of \(\Delta T\) , we can estimate the variation in \(g\) from the equation \(T=2 \pi(L / g)^{1 / 2}\) that relates \(T, g,\) and \(L .\) (a) With \(L\) held constant and \(g\) as the independent variable, calculate \(d T\) and use it to answer parts \((b)\) and \((c)\) . (b) Writing to Learn If \(g\) increases, will \(T\) increase or decrease? Will a pendulum clock speed up or slow down? Explain. (c) A clock with a 100 -cm pendulum is moved from a location where \(g=980 \mathrm{cm} / \mathrm{sec}^{2}\) to a new location. This increases the period by \(d T=0.001 \mathrm{sec} .\) Find \(d g\) and estimate the value of \(g\) at the new location.
Analyzing Derivative Data Assume that \(f\) is continuous on \([-2,2]\) and differentiable on \((-2,2) .\) The table gives some values of \(f^{\prime}(x)\) $$ \begin{array}{cccc}\hline x & {f^{\prime}(x)} & {x} & {f^{\prime}(x)} \\\ \hline-2 & {7} & {0.25} & {-4.81} \\ {-1.75} & {4.19} & {0.5} & {-4.25} \\\ {-1.5} & {1.75} & {0.75} & {-3.31} \\ {-1.25} & {-0.31} & {1} & {-2}\end{array} $$ $$ \begin{array}{rrrr}{-1} & {-2} & {1.25} & {-0.31} \\ {-0.75} & {-3.31} & {1.5} & {1.75} \\ {-0.5} & {-4.25} & {1.75} & {4.19}\end{array} $$ $$ \begin{array}{cccc}{-0.25} & {-4.81} & {2} & {7} \\ {0} & {-5}\end{array} $$ $$ \begin{array}{l}{\text { (a) Estimate where } f \text { is increasing, decreasing, and has local }} \\ {\text { extrema. }} \\ {\text { (b) Find a quadratic regression equation for the data in the table }} \\ {\text { and superimpose its graph on a scatter plot of the data. }} \\ {\text { (c) Use the model in part (b) for } f^{\prime} \text { and find a formula for } f \text { that }} \\ {\text { satisties } f(0)=0 .}\end{array} $$
Strength of a Beam The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. (a) Find the dimensions of the strongest beam that can be cut from a 12-in. diameter cylindrical log. (b) Writing to Learn Graph \(S\) as a function of the beam's width \(w,\) assuming the proportionality constant to be \(k=1 .\) Reconcile what you see with your answer in part (a). (c) Writing to Learn On the same screen, graph \(S\) as a function of the beam's depth \(d,\) again taking \(k=1 .\) Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of \(k ?\) Try it.
Particle Motion A particle moves from right to left along the parabolic curve \(y=\sqrt{-x}\) in such a way that its \(x\) -coordinate (in meters) decreases at the rate of 8 \(\mathrm{m} / \mathrm{sec} .\) How fast is the angle of inclination \(\theta\) of the line joining the particle to the origin changing when $x=-4 ?
Linearization Show that the approximation of tan \(x\) by its linearization at the origin must improve as \(x \rightarrow 0\) by showing that $$\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$$
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