Exam 4: Using the Derivative

The equations $x=\frac{2}{\pi} \cos (\pi / 180), y=\frac{2}{\pi} \sin (\pi / 180)$ 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 radius of the circle (in miles)? Round to 2 decimal places.

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Which of the following values are inflection points of f?

A particle is travelling along the x-axis according to the function $s(t)=$ ( t-3 )( t-1 )^2.When is the velocity of the particle equal to 0?

Consider $f(x)=x^{2} e^{-x}$ for $-1 \leq x \leq 3$ .For which value(s)of x is $f(x)$ least?

Sketch a graph of a function with two local minima, no global maximum, but a global minimum.

In the function y=-4 sin (x)+4.96, in the interval from 0 $\leq x \leq$ $\pi$ , what is the global maximum value?

Write a formula for total cost as a function of quantity r when fixed costs are $30,000 and variable costs are $1,600 per item.

Consider $f(x)=x^{2} e^{-x}$ for $-1 \leq x \leq 3$ .Is f increasing or decreasing on the interval 0 < x < 2?

Total cost and revenue are approximated by the functions C = 1900 + 4q and R = 6q, both in dollars.Give a formula for the profit function.

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.Where on the path is a nice flat place to stop for a picnic?

A Brian's candy sugar wand is made from flavored sugar inside a straw.The straw is 210 mm long and 5 mm in diameter.The child accidentally poked a hole in the bottom, making the height of the sugar fall at a rate of 1 mm per second.The child realizes that there is a hole after 1 seconds.What was the rate of change of the volume of the sugar at this time? [ $V=\pi r^{2} h$ ]

The function $C(r)=10 r^{2}-30$ gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

For which interval(s)is the function $f(x)=x^{5}-$ 16x³ decreasing?

Daily production levels in a plant can be modeled by the function $G(t)=-3 t^{2}$$+18t-9$ , which gives units produced t hours after the factory opened at 8am.At what time during the day is factory productivity a maximum? Answer in the form "_:_ _" (without an "am" or "pm").

What is $f^{\prime}(c) ?$

Find the limit: $\lim _{x \rightarrow 0} \frac{2 \sin x \cos x}{x}$

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Where is f increasing most rapidly?

Given below are the graphs of two functions f(x)and g(x).Graph $h(x)=f(g(x))$ on a similar set of axes.

Consider the function $f(x)=\frac{1}{1+a e^{-x}}$ , for $a>0$ .As a increases, what happens to the horizontal asmyptotes of f?

The curve represented by the parametric equations $x=5-t^{3}$ and $y=6+t^{3}$ is