91Ó°ÊÓ

A sprinter accelerates during the first \(20 \mathrm{~m}\) of a race at the rate of \(4 \mathrm{~m} / \mathrm{s}^{2}\). Of course, she begins at rest. How fast is she running at the moment she hits the \(20 \mathrm{~m}\) mark?

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=x-\ln (x) $$

An object is dropped from a window \(100 \mathrm{ft}\) above the ground. At what speed is the object traveling at the moment of impact with the ground?

An automobile is cruising at a constant speed of \(55 \mathrm{mi} / \mathrm{hr}\). To pass another vehicle, the car accelerates at a constant rate. In the course of one minute the car covers 1.3 miles. What is the rate at which the car is accelerating? What is the speed of the car at the end of this minute?

If \((x-c)^{2}\) is a factor of a polynomial \(p(x)\) but \((x-c)^{3}\) is not, then \(c\) is a root of \(p(x)\) of multiplicity \(2 .\) The graph of \(y=p(x)\) touches the \(x\) -axis at a root of multiplicity 2 but does not cross the \(x\) -axis there. Plot the given polynomial \(p(x)\) in the specified viewing rectangle. Identify a rational number \(c\) that is a root of \(p\) with multiplicity \(2 .\) Use the Newton-Raphson Method with initial estimate \(x_{1}=c+1 / 2\) to obtain iterates \(x_{2}, x_{3}, \ldots, x_{n} .\) Terminate the process at the smallest value of \(n\) for which \(\left|x_{N}-c\right|>5 \times 10^{-4}\). What is \(N ?\) You will notice that the convergence is slow. Record the value of \(N\) so that it can be used for comparison in Exercise \(37 .\) $$ p(x)=x^{4}-2 x^{3}+3 x^{2}-4 x+2,[-2,3] \times[-5,50] $$

A right circular cone of height \(h\) and base radius \(r\) has volume \(\pi r^{2} h / 3\) and lateral surface area \(\pi r \sqrt{r^{2}+h^{2}}\). What is the greatest volume that such a cone can have if its surface area, including the base, is \(2 \pi\) ?

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

A spherical balloon is being inflated. At a certain instant its radius is \(12 \mathrm{~cm}\) and its area is increasing at the rate of \(24 \mathrm{~cm}^{2} / \mathrm{min}\). At that moment, how fast is its volume increasing?

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

The second derivative \(f^{\prime \prime}\) of a function \(f\) is given. Determine every \(x\) at which \(f\) has a point of inflection. $$ f^{\prime \prime}(x)=x^{2}(x+2) $$