91Ó°ÊÓ

Show that \(f\) satisfics the conditions of Rolle's theorem on the indicated interval and find all numbers \(c\) on the interval for which \(f^{\prime}(c)=0\) $$f(x)=x^{4}-2 x^{2}-8 ; \quad|-2,2|$$

An object moves along a coordinate line, its position al each time \(t \geq 0\) given by \(x(t)\). Find the position, velocity, and acceleration at time \(t_{0} .\) What is the speed at time \(t_{0} ?\) $$x(t)=5 t-t^{3} ; \quad t_{0}=3$$.

Sketch the graph of the function using the approach presented in this section. $$f(x)=1-(x-2)^{2}$$

Use a differential to estimate the area of a ring of inner radius \(r\) and width \(h .\) What is the exact area?

Find the dimensions of the rectangle of perimeter 24 that has the largest area.

Find the critical points and the local extreme values. $$f(x)=2 x^{4}-4 x^{2}+6$$.

Show that \(f\) satisfies the conditions of Rolle's theorem on the indicated interval and find all the numbers \(c\) on the interval for which \(f^{\prime}(c)=0\) $$f(x)=\sin x+\cos x-1 ; \quad[0,2 \pi]$$

Use a differential to estimate the value of the indicated expression. Then compare your estimate with the result given by a calculator. $$\sqrt{1002}$$

A particle is moving along the parabola \(y^{2}=4(x+2) .\) As it passes through the point \((7,6),\) its \(y\) -coordinate is increasing at the rate of 3 units per second. How fast is the \(x\) -coordinate changing at this instant?

Find the intervals on which \(f\) increases and the intervals on which \(f\) decreases. $$f(x)=x+\frac{1}{x}$$