91Ó°ÊÓ

In your own words, state the guidelines for implicit differentiation.

Show that the graphs of the two equations \(y=x\) and \(y=1 / x\) have tangent lines that are perpendicular to each other at their point of intersection.

Find equations of the tangent lines to the graph of \(f(x)=\frac{x+1}{x-1}\) that are parallel to the line \(2 y+x=6\). Then graph the function and the tangent lines.

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.

The radius of a right circular cylinder is given by \(\sqrt{t+2}\) and its height is \(\frac{1}{2} \sqrt{t}\), where \(t\) is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.

Newton's Law of Universal Gravitation states that the force \(F\) between two masses, \(m_{1}\) and \(m_{2}\), is $$F=\frac{G m_{1} m_{2}}{d^{2}}$$ where \(G\) is a constant and \(d\) is the distance between the masses. Find an equation that gives an instantaneous rate of change of \(F\) with respect to \(d\). (Assume \(m_{1}\) and \(m_{2}\) represent moving points.)

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. \(f(t)=2 t+7, \quad[1,2]\)

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. $$ g(t)=\tan 2 t, \quad\left(\frac{\pi}{6}, \sqrt{3}\right) $$

When satellites observe Earth, they can scan only part of Earth's surface. Some satellites have sensors that can measure the angle \(\theta\) shown in the figure. Let \(h\) represent the satellite's distance from Earth's surface and let \(r\) represent Earth's radius. (a) Show that \(h=r(\csc \theta-1)\). (b) Find the rate at which \(h\) is changing with respect to \(\theta\) when \(\theta=30^{\circ} .\) (Assume \(r=3960\) miles. \()\)

Use the position function \(s(t)=-4.9 t^{2}+v_{0} t+s_{0}\) for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds?