91Ó°ÊÓ

In Exercises 83-86, explain why the function has at least one zero in the given interval. $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {f(x)=x^{3}+5 x-3} & {[0,1]}\end{array}$$

Inscribe a rectangle of base \(b\) and height \(h\) in a circle of radius one, and inscribe an isosceles triangle in a region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of \(h\) do the rectangle and triangle have the same area?

In Exercises 83-86, explain why the function has at least one zero in the given interval. $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {f(x)=x^{2}-2-\cos x \quad[0, \pi]}\end{array}$$

Using the Intermediate Value Theorem In Exercises 89-94, use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$h(\theta)=\tan \theta+3 \theta-4$$

Using the Intermediate Value Theorem In Exercises \(95-100,\) verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$f(x)=\sqrt[3]{x}+8, \quad[-9,-6], \quad f(c)=6$$

In Exercises 101 and \(102,\) use the position function\(s(t)=-16 t^{2}+500,\) which gives the height (in feet) of an object that has fallen for \(t\) seconds from a height of 500 feet. The velocity at time \(t=a\) seconds is given by $$\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}$$ A construction worker drops a full paint can from a height of 500 feet. How fast will the paint can be falling after 2 seconds?

In Exercises 101 and \(102,\) use the position function\(s(t)=-16 t^{2}+500,\) which gives the height (in feet) of an object that has fallen for \(t\) seconds from a height of 500 feet. The velocity at time \(t=a\) seconds is given by $$\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}$$ A construction worker drops a full paint can from a height of 500 feet. When will the paint can hit the ground? At what velocity will the paint can impact the ground?

Removable and Nonremovable Discontinuities Describe the difference between a discontinuity that is removable and a discontinuity that is nonremovable. Then give an example of a function that satisfies each description. (a) A function with a nonremovable discontinuity at x = 4 (b) A function with a removable discontinuity at x = -4 (c) A function that has both of the characteristics described in parts (a) and (b)

If \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c),\) then either \(f\) or \(g\) is not continuous at \(c .\)

The Intermediate Value Theorem guarantees that \(f(a)\) and \(f(b)\) differ in sign when a continuous function \(f\) has at least one zero on \([a, b] .\)