Chapter 4: Problem 16
\(x^{4}+x-3=0\)
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Multiple Choice Which of the following conditions would enable you to conclude that the graph of \(f\) has a point of inflection at \(x=c ?\) (A) There is a local maximum of \(f^{\prime}\) at \(x=c\) . (B) \(f^{\prime \prime}(c)=0 .\) (C) \(f^{\prime \prime}(c)\) does not exist. (D) The sign of \(f^{\prime}\) changes at \(x=c\) . (E) \(f\) is a cubic polynomial and \(c=0\)
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.
Moving Shadow A man 6 ft tall walks at the rate of 5 \(\mathrm{ft} / \mathrm{sec}\) toward a streetlight that is 16 \(\mathrm{ft}\) above the ground. At what rate is the length of his shadow changing when he is 10 \(\mathrm{ft}\) from the base of the light?
Motion along a Circle A wheel of radius 2 ft makes 8 revolutions about its center every second. (a) Explain how the parametric equations \(x=2 \cos \theta, \quad y=2 \sin \theta\) \(x=2 \cos \theta, \quad y=2 \sin \theta\) (b) Express \(\theta\) as a function of time \(t\) . (c) Find the rate of horizontal movement and the rate of vertical movement of a point on the edge of the wheel when it is at the position given by \(\theta=\pi / 4, \pi / 2,\) and \(\pi .\)
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 \(b\).
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