91Ó°ÊÓ

Use a graphing utility to estimate the absolute maximum and minimum values of \(f\), if any, on the stated interval, and then use calculus methods to find the exact values. $$f(x)=\sin (\cos x) ;[0,2 \pi |$$

Use the Mean-Value Theorem to prove that $$1.71<\sqrt{3}<1.75$$ IHint: Let \(f(x)=\sqrt{x}, a=3,\) and \(b=4\) in the Mean-Value Theorem.]

Use a graphing utility to estimate the absolute maximum and minimum values of \(f\), if any, on the stated interval, and then use calculus methods to find the exact values. $$f(x)=\cos (\sin x) ; \quad[0, \pi]$$

Find the absolute maximum and minimum values of $$ f(x)=\left\\{\begin{array}{ll} 4 x-2, & x < 1 \\ (x-2)(x-3), & x \geq 1 \end{array}\right. $$ on \(\left[\frac{1}{2}, \frac{7}{2}\right]\)

(a) Show that if \(f\) and \(g\) are functions for which $$f^{\prime}(x)=g(x) \quad \text { and } \quad g^{\prime}(x)=-f(x)$$ for all \(x,\) then \(f^{2}(x)+g^{2}(x)\) is a constant. (b) Give an example of functions \(f\) and \(g\) with this property.

(a) Use the chain rule to show that for a particle in rectilinear motion \(a=v(d v / d s)\) (b) Let \(s=\sqrt{3 t+7}, t \geq 0 .\) Find a formula for \(v\) in terms of \(s\) and use the equation in part (a) to find the acceleration when \(s=5\)

(a) Show that if \(f\) and \(g\) are functions for which $$f^{\prime}(x)=g(x) \quad \text { and } \quad g^{\prime}(x)=f(x)$$ for all \(x\), then \(f^{2}(x)-g^{2}(x)\) is a constant. (b) Show that the function \(f(x)=\frac{1}{2}\left(e^{x}+e^{-x}\right)\) and the function \(g(x)=\frac{1}{2}\left(e^{x}-e^{-x}\right)\) have this property.

Find the dimensions of the isosceles triangle of least area that can be circumscribed about a circle of radius \(R\)

Let \(g(x)=x^{3}-4 x+6 .\) Find \(f(x)\) so that \(f^{\prime}(x)=g^{\prime}(x)\) and \(f(1)=2\).

Find the height and radius of the right circular cone with least volume that can be circumscribed about a sphere of radius \(R\)