91Ó°ÊÓ

Find all points on the curve \(y=\cot x, 0

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\tan ^{-1} 2\) b. \(\cos ^{-1} 2\)

At what rate is the angle between a clock's minute and hour hands changing at 4 o'clock in the afternoon?

Find the derivatives of the functions in Exercises \(23-50\). $$r=\sin \left(\theta^{2}\right) \cos (2 \theta)$$

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=(4 / 3) \pi r^{3}\) of a sphere when the radius changes from \(r_{0}\) to \(r_{0}+d r\)

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin \theta) \sqrt{\theta+3}$$

Find the first and second derivatives of the functions. $$y=\frac{x^{3}+7}{x}$$

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=x^{3}\) of a cube when the edge lengths change from \(x_{0}\) to \(x_{0}+d x\)

Find the first and second derivatives of the functions. $$s=\frac{t^{2}+5 t-1}{t^{2}}$$

An explosion at an oil rig located in gulf waters causes an elliptical oil slick to spread on the surface from the rig. The slick is a constant 9 in. thick. After several days, when the major axis of the slick is 2 mi long and the minor axis is \(3 / 4\) mi wide, it is determined that its length is increasing at the rate of \(30 \mathrm{ft} / \mathrm{hr},\) and its width is increasing at the rate of \(10 \mathrm{ft}\) hr. At what rate (in cubic feet per hour) is oil flowing from the site of the rig at that time?