Exam 4: Using the Derivative

A lady bug moves on the xy-plane according the the equations $x=2 t(t-6)$ , $y=5-t$ . When does the lady bug stop moving?

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Graph the function and use your graph to find how many roots there are to the equation $f(x)=1$ .

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, sketch a graph of f.

A window with a rectangular base is topped by a semicircle creating a Norman window.A plate of glass 24 ft² in area.What are the dimensions that would minimize the metal frame around the window? [round to 3 decimal places] Base: _____________, Height: ________________, Radius: ___________________

Given the table of data about the second derivative of a function f, which of the following types of a function could f be? Assume b > 0.The other constants can be positive or negative. x 0 1 2 3 1 -2 -5 -8 (x)

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 $y=5\left(x^{3}-12 x^{2}+48 x+36\right)$ .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.If your friend, who is not in "good shape" followed this path for 15 miles total, in horizontal distance, the day before, it makes sense that the equation for the elevation continues to hold much beyond the 7 mile mark.

Consider the function $f(x)=x+2 \cos x$ , for $0 \leq x \leq 2 \pi$ .Is f increasing or decreasing at x = 3.82?

When light strikes a shiny surface, much of it is reflected in the direction shown.However some of it may be scattered on either side of the reflected light.If the intensity (brightness)of the scattered light at the angle $\theta$ (shown in the picture)is I, the Phong model says that $I=k \cos ^{n}(\theta)$ where k and n are positive constants depending on the surface.Thus this function gives an idea of how "spread-out" the scattered light is.What effect does decreasing the parameter n have of the graph of I?

The number of plants in a terrarium is given by the function $P(c)=-1.4 c^{2}+3 c+15$ , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Which of the following diagrams represents the parametric curve $x=-\sin t, y=\cos t, 0 \leq t \leq \pi$ ?

Which function dominates as $x \rightarrow \infty$ ?

The regular air fare between Boston and San Francisco is $600.An airline flying 747s with a capacity of 480 on this route observes that they fly with an average of 400 passengers.Market research tells the airlines' managers that each $20 fare reduction would attract, on average, 20 more passengers for each flight.How should they set the fare to maximize their revenue?

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 $f(0)$ = 5.Estimate f(2).

What is the shortest distance from the point (0,1)to the curve $y=e^{x}$ ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

Given below are the graphs of two functions f(x)and g(x).Let $h(x)=f(g(x))$ .Is $h(x)$ increasing or decreasing on the interval 3 < x < 4?

The cost C(q)(in dollars)of producing a quantity q of a certain product is shown in the graph below.The average cost is given by $a(q)=\frac{C(q)}{q}$ .Find on the graph the quantity $90$ where a(q)is minimal.Now suppose that the fixed costs (i.e., the costs of setting up before production starts)are doubled.Sketch the new cost function $C_{1}(g)$ on the same set of axes as the original one and let $q_{1}$ be the quantity where the new $a_{1}(g)$ is minimal.Which of the following is true?

A particle is travelling along the x-axis according to the function $s(t)=$ ( t-3 )( t-2 )².Assuming t $\geq$ 0, when is the acceleration of the particle equal to 0?

If you throw a stone into the air at an angle of $\theta$ to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{160}\left(1+\tan ^{2} \theta\right)$ , where y is the height of the stone above the ground, x is the horizontal distance.Suppose the stone is to be thrown over a wall at a fixed horizontal distance l away from you.If you can vary $\theta$ , what is the highest wall that the stone can go over? (Your answer will contain l.)

Suppose f is a cubic function with critical points at x = 8 and x = 6.What is the x-coordinate of the inflection point of f?

Given below are the graphs of two functions f(x)and g(x).Let $h(x)=f(g(x))$ .Is the point x = 1 a local minimum, a local maximum, or neither for the function $h(x)$ ?