Chapter 2: Problem 104
In Exercises 103 and \(104,\) the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). \(g(x)=f\left(x^{2}\right)\)
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Angle of Elevation A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
In Exercises 35 and 36, find an equation of the tangent line to the graph of the equation at the given point. $$ \arcsin x+\arcsin y=\frac{\pi}{2}, \quad\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$
Let \(u\) be a differentiable function of \(x\). Use the fact that \(|u|=\sqrt{u^{2}}\) to prove that \(\frac{d}{d x}[|u|]=u^{\prime} \frac{u}{|u|}, \quad u \neq 0\).
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.
Consider the function \(f(x)=\sin \beta x\), where \(\beta\) is a constant. (a) Find the first-, second-, third-, and fourth-order derivatives of the function. (b) Verify that the function and its second derivative satisfy the equation \(f^{\prime \prime}(x)+\beta^{2} f(x)=0\) (c) Use the results in part (a) to write general rules for the even- and odd- order derivatives \(f^{(2 k)}(x)\) and \(f^{(2 k-1)}(x)\) [Hint: \((-1)^{k}\) is positive if \(k\) is even and negative if \(k\) is odd.]
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