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Chapter 2: Problem 10
Find the slope of the tangent line to the graph of the function at the given point. \(h(t)=t^{2}+3, \quad(-2,7)\)
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The displacement from equilibrium of an object in harmonic motion on the end of a spring is \(y=\frac{1}{3} \cos 12 t-\frac{1}{4} \sin 12 t\) where \(y\) is measured in feet and \(t\) is the time in seconds. Determine the position and velocity of the object when \(t=\pi / 8\).
\( \text { Radway Design } \) 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 \(d \theta / d t\)
Determine the point(s) in the interval \((0,2 \pi)\) at which the graph of \(f(x)=2 \cos x+\sin 2 x\) has a horizontal tangent line.
Find the second derivative of the function. \(f(x)=(3+2 x) e^{-3 x}\)
Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ g(x)=x^{2}+e^{x}, \quad[0,1] $$
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