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Chapter 3: Problem 38
Find the derivative of the following functions. $$y=\frac{\tan t}{1+\sec t}$$
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A store manager estimates that the demand for an energy drink decreases with increasing price according to the function \(d(p)=\frac{100}{p^{2}+1},\) which means that at price \(p\) (in dollars), \(d(p)\) units can be sold. The revenue generated at price \(p\) is \(R(p)=p \cdot d(p)\) (price multiplied by number of units). a. Find and graph the revenue function. b. Find and graph the marginal revenue \(R^{\prime}(p)\). c. From the graphs of \(R\) and \(R^{\prime}\), estimate the price that should be charged to maximize the revenue.
A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance \(x\) (in inches) of the mass from its equilibrium position after \(t\) seconds is given by the function \(x(t)=10 \sin t-10 \cos t,\) where \(x\) is positive when the mass is above the equilibrium position. a. Graph and interpret this function. b. Find \(\frac{d x}{d t}\) and interpret the meaning of this derivative. c. At what times is the velocity of the mass zero? d. The function given here is a model for the motion of an object on a spring. In what ways is this model unrealistic?
Identifying functions from an equation. The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}(x+2)=x^{2}(6-x) \text { (trisectrix) }$$
a. Identify the inner function \(g\) and the outer function \(f\) for the composition \(f(g(x))=e^{k x},\) where \(k\) is a real number. b. Use the Chain Rule to show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\).
Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.
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