Chapter 2: Problem 3
Find the derivative of the function. $$ y=8 $$
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In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=2 \tan ^{3} x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4}, 2\right)}\)
Given that \(g(5)=-3, \quad g^{\prime}(5)=6, \quad h(5)=3,\) and \(h^{\prime}(5)=-2,\) find \(f^{\prime}(5)\) (if possible) for each of the following. If it is not possible, state what additional information is required. (a) \(f(x)=g(x) h(x)\) (b) \(f(x)=g(h(x))\) (c) \(f(x)=\frac{g(x)}{h(x)}\) (d) \(f(x)=[g(x)]^{3}\)
The frequency \(F\) of a fire truck siren heard by a stationary observer is \(F=\frac{132,400}{331 \pm v}\) where \(\pm v\) represents the velocity of the accelerating fire truck in meters per second. Find the rate of change of \(F\) with respect to \(v\) when (a) the fire truck is approaching at a velocity of 30 meters per second (use \(-v)\) (b) the fire truck is moving away at a velocity of 30 meters per second (use \(+v\) ).
Find the second derivative of the function. \(g(x)=\sqrt{x}+e^{x} \ln x\)
Prove that \(\arcsin x=\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\)
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