91Ó°ÊÓ

A wire of length \(L\) can be shaped into a circle or a square, or it can be cut into two pieces, one of which is formed into a circle and the other a square. How is the minimum total area obtained?

In each of Exercises \(41-48\), use the given information to find \(F(c)\). $$ F^{\prime}(x)=4 / x, \quad F\left(e^{2}\right)=7, \quad c=e^{3} $$

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is twice differentiable and \(f^{\prime \prime}(x) \geq 1\) for all \(x\). Prove that the graph of \(f\) cannot have any horizontal asvmptotes.

Find and test the critical points of the functions in Exercises 43-47. $$ f(x)=2 x^{2}-2 x^{2} $$

If \(R_{1}\) and \(R_{2}\) are parallel variable resistances, then the resulting resistance \(R\) satisfies \(1 / R=1 / R_{1}+1 / R_{2} .\) If \(R_{1}\) increases at the rate of \(0.6 \Omega / \mathrm{s}\) when \(R_{1}=40 \Omega,\) and \(R_{2}=20 \Omega,\) then at what rate must \(R_{2}\) decrease if \(R\) remains constant? (The unit by which resistance in an electric circuit is usually measured, the \(o h m,\) is denoted by \(\Omega .)\)

In Exercises \(43-48,\) the second derivative \(f^{\prime \prime}\) of a function \(f\) is given. Determine every \(x\) at which \(f\) has a point of inflection. $$ f^{\prime \prime}(x)=x(x+1) $$

Use trigonometric identities to compute the indefinite integrals. Evaluate \(\int 2^{x \ln (2)} d x\)

Find and test the critical points of the functions in Exercises 43-47. $$ f(x)=|x-7|+x^{2} $$

At \(1 \mathrm{PM}\), a car traveling east at a constant speed of \(30 \mathrm{mph}\) passes through an intersection. At \(2 \mathrm{PM},\) a car traveling south at a constant speed of 40 mph passes through the same intersection. How fast is the distance between the two cars changing at 3 PM?

A wire of length \(L\) can be shaped into a (closed) semicircle or a square, or it can be divided into two pieces. With one, a (closed) semicircle is formed and with the other a square. Find the largest and smallest areas that are possible.