Exam 4: Using the Derivative

A rectangle is inscribed between the function y= -x² + 49 and the x-axis.What is the maximum area of the square if the base if two vertices of the square lie on the x-axis?

Graph the function $f(x)=x^{2} e^{-2 x}$ , labeling all important points.

The equations $x=\cos \left(t^{5}\right)$ , $y=\sin \left(t^{5}\right)$ parameterize a circle.

If

What does the Extreme Value Theorem allow us to conclude about f if f is continuous on [-10, 80]? Mark all that apply.

Starting at time t = 0, water is poured at a constant rate into an empty vase (pictured below).It takes ten seconds for the vase to be filled completely to the top.Let h = f(t)be the depth of the water in the vase at time t.Is h = f(t)concave up or down on the region 3 < t < 9?

Consider the function $f(x)=a x(1-x)$ for $a>0$ .For what value of x on the region $0 \leq x \leq 1$ does f have a local maximum? If there is more than one value, give the smallest one.

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .What is the largest value of f? Round to 2 decimal places.

One fine day you take a hike up a mountain path.Using your trusty map you have determined that the path is approximately in the shape of the curve $y=3\left(x^{3}-12 x^{2}+48 x+36\right)$ .Here y is the elevation in feet above sea level and x is the horizontal distance in miles you have traveled, but your map only shows the path for 7 miles, horizontal distance.Which of the following is a good estimate of the elevation after 7.5 horizontal miles.(You do not know the shape of the path explicitly after 7 miles!)

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Which of the following values of x are local minima of f?

Let f(x)be a function with positive values and let $g=\sqrt{f}$ .If f has a local maximum at $x=x_{0}$ , what about g?

If f is always decreasing and concave down, then f must have at least one root.

Draw the graph of a polynomial of degree four that has a local minimum at (-5, -14)and inflection points at (-2, -6)and the origin.

If you want to maximize profit, you should minimize average cost.

Determine the $\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{x-a}$ , given a=5

The function $y=0.2(x+2)^{2}-2 x+10$ gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

The revenue for selling q items is $R(q)=600 q-2 q^{2}$ and the total cost is C(q)= 120 + 60q.What quantity maximizes profit?

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Suppose that you are told that $f(0)$ = 3.Which of the following is an exact expression for $f(2)$ ?

Let f be a function.Is it true or false that the inflection points of f are the local extrema of f '.

The equations $x=\frac{3}{\pi} \cos (\pi / 90), y=\frac{3}{\pi} \sin (\pi / 90)$ describe the motion of a particle moving on a circle.Assume x and y are in miles and t is in days.What is the period of the circular motion (in days)?