91Ó°ÊÓ

a. How close does the semicircle \(y=\sqrt{16-x^{2}}\) come to the point \((1, \sqrt{3}) ?\) b. Graph the distance function and \(y=\sqrt{16-x^{2}}\) together and reconcile what you see with your answer in part (a).

Sketch the graph of a differentiable function \(y=f(x)\) that has a. a local minimum at (1,1) and a local maximum at (3,3) b. a local maximum at (1,1) and a local minimum at (3,3) c. local maxima at (1,1) and (3,3) d. local minima at (1,1) and (3,3)

Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(b)

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \cos \theta(\tan \theta+\sec \theta) d \theta$$

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2 / 3}(x+2)$$

Sketch the graph of a continuous function \(y=g(x)\) such that a. \(g(2)=2,02,\) and \(g^{\prime}(x) \rightarrow-1^{+}\) as \(x \rightarrow 2^{+}\) b. \(g(2)=2, g^{\prime}<0\) for \(x<2, g^{\prime}(x) \rightarrow-\infty\) as \(x \rightarrow 2^{-}\) \(g^{\prime}>0\) for \(x>2,\) and \(g^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 2^{+}\)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow(\pi / 2)} \frac{\sec x}{\tan x}$$

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{\csc \theta}{\csc \theta-\sin \theta} d \theta$$

Sketch the graph of a continuous function \(y=h(x)\) such that a. \(h(0)=0,-2 \leq h(x) \leq 2\) for all \(x, h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{-}\) and \(h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{+}\) b. \(h(0)=0,-2 \leq h(x) \leq 0\) for all \(x, h^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 0^{-}\) and \(h^{\prime}(x) \rightarrow-\infty\) as \(x \rightarrow 0^{+}\)

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow 0^{+}} \frac{\cot x}{\csc x}$$