91Ó°ÊÓ

Writing In Exercises \(87-90\) , explain why the function has a zero in the given interval. $$ f(x)=x^{2}-2-\cos x \quad[0, \pi] $$

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$

Removable and Nonremovable Discontinuities Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following descriptions. (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)

Free-Falling Object In Exercises 103 and 104 , use the position function \(s(t)=-4.9 t^{2}+200\) , which gives the height (in meters) of an object that has fallen for \(t\) seconds from a height of 200 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\) At what velocity will the object impact the ground?

Dirichlet Function Show that the Dirichlet function $$ f(x)=\left\\{\begin{array}{l}{0, \text { if } x \text { is rational }} \\ {1, \text { if } x \text { is irrational }}\end{array}\right. $$ is not continuous at any real number.

Approximation (a) Find \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}\) (b) Use your answer to part (a) to derive the approximation \(\cos x \approx 1-\frac{1}{2} x^{2}\) for \(x\) near 0 . (c) Use your answer to part (b) to approximate \(\cos (0.1)\) . (d) Use a calculator to approximate \(\cos (0.1)\) to four decimal places. Compare the result with part (c).