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Chapter 2: Problem 56
Sketch the graph of a function \(f\) such that \(f^{\prime}>0\) for all \(x\) and the rate of change of the function is decreasing.
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Where are the functions \(f_{1}(x)=|\sin x|\) and \(f_{2}(x)=\sin |x|\) differentiable?
\( \text { Radway Design } \) Cars on a certain roadway travel on a circular arc of radius \(r\). In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude \(\theta\) from the horizontal (see figure). The banking angle must satisfy the equation \(r g \tan \theta=v^{2},\) where \(v\) is the velocity of the cars and \(g=32\) feet per second per second is the acceleration due to gravity. Find the relationship between the related rates \(d v / d t\) and \(d \theta / d t\)
In Exercises 103 and \(104,\) the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). \(g(x)=f\left(x^{2}\right)\)
In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{f(x)=\frac{3}{x^{3}-4}} \quad \frac{\text { Point }}{\left(-1,-\frac{3}{5}\right)}\)
Slope Find all points on the circle \(x^{2}+y^{2}=25\) where the slope is \(\frac{3}{4}\).
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