Exam 4: Integrals

Determine where the graph of the function $f(x)=\sin 9 x$ is concave upward and where it is concave downward on the interval $0 \leq x \leq \frac{2 \pi}{9}$ . Also, find all inflection points of the function.

A woman at a point A on the shore of a circular lake with radius $4 \mathrm{mi}$ wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of $6 \mathrm{mi} / \mathrm{h}$ and row a boat at $2 \mathrm{mi} / \mathrm{h}$ . How should she proceed? (Find $\theta$ ). Round the result, if necessary, to the nearest hundredth.

Find the local and absolute extreme values of the function on the given interval. $f(x)=x^{3}-6 x^{2}+9 x+4$ , $[2,4]$

Sketch the curve. $y=x \sin 2 x$ , $-\frac{\pi}{2}<x<\frac{\pi}{2}$

Sketch the graph of the function $y=(x+1)^{3 / 2}-2$ using the curve-sketching guidelines.

Find the slant asymptote of the function $f(x)=\frac{x^{2}+7}{x}$ .

At 4:00 P.M. a car's speedometer reads 25 $\mathrm{mi} / \mathrm{h}^{2}$ . At 4:15 it reads 72 $\mathrm{mi} / \mathrm{h}^{2}$ . At some time between 4:00 and 4:15 the acceleration is exactly x $\mathrm{mi} / \mathrm{h}^{2}$ . Find x.

Given $f(x)=\frac{10 x}{x^{2}+36}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.

Determine where the graph of the function $f(x)=9 x-\sqrt{4-x^{2}}$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Estimate the extreme values of the function. Round the answers to the nearest hundredth. $y(x)=\frac{1}{3} x^{3}-7 x^{2}+40 x+15$

Find the absolute maximum value of $y=6 \sin \left(\frac{\pi}{10}\right)$ .

Find all the critical numbers of the function. $g(x)=7 x+\sin (7 x)$

Approximate the zero of $f(x)=x^{3}-x-3$ in $[1,2]$ to within 0.00001.

Find all the critical numbers of the function. $g(\theta)=2 \cos \theta+\sin ^{2} \theta$

Sketch the graph of the function $g(x)=\frac{x-2}{x-1}$ using the curve-sketching guidelines.

The function $f(x)=e^{-\frac{x}{7}}$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0,14] .$ Find all values of c that satisfy the conclusion of the theorem.

Sketch the graph of the function $f(t)=4 t^{4}+3 t^{3}$ using the curve-sketching guidelines.

Let P (x) and Q (x) be polynomials. Find $\lim _{x \rightarrow \infty} \frac{P(x)}{Q(x)}$ if the degree of P (x) is 3 and the degree of Q (x) is 7.

Sketch the graph of the function $f(x)=e^{\frac{x}{2}}$ on $(-\infty, 1]$ and find its absolute maximum and absolute minimum values, if any.

Given $f(x)=\sin ^{2} 5 x$ $\quad$ $\quad$ $0<$$x<\frac{\pi}{5}$ . (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f.