Exam 4: Using the Derivative

If you throw a stone into the air at an angle of $\theta$ to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{100}\left(1+\tan ^{2} \theta\right)$ , where y is the height of the stone above the ground, x is the horizontal distance.If the angle $\theta$ is fixed, what value of x gives the maximum height? (Your answer will $\theta$ .)

A water tank is constructed in the shape of a sphere seated atop a circular cylinder.If water is being pumped into the tank at a constant rate, let $h(t)$ be the height of the water as a function of time.Which of the following is true at the point where $h(t)=$ b?

Frank decided to ride in a hot air balloon.His brother Damien was going to videotape the lift off from a distance of 30 feet away.The hot air balloon rises to a height of 2000 feet in 19 minutes.What is the rate at which the camera's angle should be raised in order to follow the balloon? (specify units)

Find the quantity q which maximizes profit if the total revenue, R(q), and the total cost, C(q), are given in dollars by $R(q)=8 q-0.02 q^{2}$ $C(q)=300+1.9 q$ , where $0 \leq q \leq 600$ units. Round to the nearest whole number.

Find the marginal cost for q = 100 when the fixed costs in dollars are 1000, the variable costs are $190 per item, and each sells for $310.

A lady bug moves on the xy-plane according the the equations $x=2 t(t-8)$ , $y=2-t$ . When is the lady bug ever moving straight up or down?

Consider a continuous function with the following properties: $f(0)=4$ $\left|f^{\prime}(x)\right|<0.5$ $f^{\prime \prime}(x)<0$ for $x<0$ $f^{\prime}(1)=0$ . Which of the following is true?

A fan is watching a 100-meter footrace from a seat in the bleachers 15 meters back from the midway point.The winning runner is moving approximately 8 meters per second.How fast is the distance from the fan to the winning runner changing when he is x meters into the race?

Below is the graph of the derivative of a function f, i.e., it is a graph of y = f '(x).Where in the interval 0 $\le$ x $\le$ 4 does f achieve its global maximum?

For $f(x)=4 \cos ^{2} x-\sin x$ and $0 \leq x \leq \pi$ , what is the global maximum value of $f(x)$ ?

Below is the graph of the rate r at which people arrive for lunch at Cafeteria Charlotte.Checkers start at 12:00 noon and can pass people through at a constant rate of 5 people/minute.Let f(t)be the length of the line (i.e.the number of people)at time t.Suppose that at 11:50 there are already 150 people lined up.Using the graph together with this information, when is the line the longest?

Find the limit: $\lim _{x \rightarrow \infty} \frac{2 e^{-x}+2 x}{x}$

The revenue for selling q items is $R(q)=600 q-2 q^{2}$ and the total cost is C(q)= 110 + 60q.Which function gives the total profit earned?

Which of the following parametric equations pass through the points (8, 1)and (-1, -1)?

The cost C(q)(in dollars)of producing a quantity q of a certain product is shown in the graph below.Suppose that the manufacturer can sell the product for $2.50 each (regardless of how many are sold), so that the total revenue from selling a quantity q is R(q)= 2.5q.The difference $\pi(q)=R(q)-C(q)$ is the total profit.Let $90$ be the quantity that will produce the maximum profit.What is $C^{1}\left(q_{0}\right)$ ?

The equations $x=\frac{9}{\pi} \cos (\pi / 180), y=\frac{9}{\pi} \sin (\pi / 180)$ 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 speed of the particle (in miles per day)when it passes through the point (-9/ $\pi$ , 0)? Round to 3 decimal places.

Total cost and revenue are approximated by the functions C = 1200 + 3.5q and R = 6q, both in dollars.Identify the marginal cost per item.

A spherical lollipop has a circumference of 7.9 centimeters.A student decides to measure the rate of change of the volume of the lollipop, in $\mathrm{cm}^{3}$ per minute.The student licks the lollipop and measures the circumference every minute.The radius is decreasing at a rate of 0.18 cm/min.Determine the rate at which the volume is changing when the circumference is half of it's original size.[ $V=\frac{4}{3} \pi r^{3}$ ]

A lady bug moves on the xy-plane according the the equations $x=2 t(t-8)$ , $y=3-t$ . Suppose that the temperature at a point (x, y)in the plane depends only on the y coordinate of the point and is equal to $4 y^{2}$ .Find the rate of change of the temperature at the location of the lady bug at time t.

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm³ per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.