91Ó°ÊÓ

Conjecture Consider the functions \(f(x)=x^{2}\) and \(g(x)=x^{3}\). (a) Graph \(f\) and \(f^{\prime}\) on the same set of axes. (b) Graph \(g\) and \(g^{\prime}\) on the same set of axes. (c) Identify a pattern between \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h^{\prime}(x)\) if \(h(x)=x^{n},\) where \(n\) is an integer and \(n \geq 2\) (d) Find \(f^{\prime}(x)\) if \(f(x)=x^{4}\). Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.

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. $$ g(x)=x^{2}+e^{x}, \quad[0,1] $$

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1,2] . (c) Find the instantaneous velocities when \(t=1\) and \(t=2\). (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact.

Use the position function \(s(t)=-16 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A ball is thrown straight down from the top of a 220 -foot building with an initial velocity of -22 feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{y=\frac{1}{x}+\sqrt{\cos x}} \quad \frac{\text { Point }}{\left(\frac{\pi}{2}, \frac{2}{\pi}\right)}\)

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. \(\frac{\text { Function }}{y=\cos 3 x} \quad \frac{\text { Point }}{\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right)}\)

Find equations of both tangent lines to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) that passes through the point (4,0).

Find the tangent line(s) to the curve \(y=x^{3}-9 x\) through the point (1,-9).

(a) Find an equation of the normal line to the ellipse \(\frac{x^{2}}{32}+\frac{y^{2}}{8}=1\) at the point (4,2) . (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line intersect the ellipse?

Find the derivative of the function. \(y=\log _{5} \sqrt{x^{2}-1}\)