M&M's
Given f'(x) = x^2 + 5x + 4, at what x values will f(x) have a maxima or minima?
x = -4, x = -1
Determine when f(x) is concave up and down where f(x) = 2x^2 + 25x + 100
f(x) is always concave down
Consider two nonnegative number x and y such that x+y = 10. Maximize and minimize the quantities? Identify the constraint function.
x + y = 10
Consider a pizzeria that sells pizzas for a revenue of R(x) = ax and costs C(x) = b + cx + dx^2, where x represents the number of pizzas; What equation do we want to maximize?
R(x) = ax
Revenue
Write a concise (clear) definition for a critical point
C is a critical point of f(x) if f'(x) = 0 or is undefined
Given f'(x) = (5x^2 + 2x - 7) / (3 + x)^(1/2) what are the critical points?
x= - 3, x= 1, x= 7/5
Find when f(x) is concave up or down given that f(x) = 1/3x^3 + 5x^2 + 7.
f(x) is concave up over the interval (-oo, -5) and concave down from (-5, oo)
You have 400ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?
2w + 2l = 400
Find the volume of the largest right circular cylinder that fits in a sphere of radius 1
V = \pi (r^2) h
Suppose you are looking to find the maximum of f(x) over an interval [0,2]. Suppose that f'(x) = 0 at x=1/2 and x=3/2. List all values of x which f(x) could have a maximum at.
x = 1/2, 3/2, 0, 2
Find the critical points and the corresponding max and min of f(x) = cos(x) over the interval [0,2pi).
Max = 1, Min = -1
Critical points: x = 0, pi
Find the intervals of x where f(x) is concave up and concave down.
Let f(x) = (x^2−1)/x
f''(x)>0 (-1,oo)
f''(x)< 0 (-oo,1)
You have a garden row of 20 watermelon plants that produce an average of 30 watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?
yield = -x + 30 for x > 20
You are building five identical pens adjacent to each other with a total area of 1000 m^2, as shown in the following figure. What dimensions should you use to minimize the amount of fencing?
P = 5x + 5y, where x and y represent the lengths and widths of each sub-pen.
What is an inflection point? Write a clear definition.
An inflection point is a point on the function were f(x) changes concavity. Or when the second derivative changes signs(ie. positive -> negative or negative -> positive) If f(x) changes concavity at a the inflection point would be (a,f(a)).
Find the critical points of f(x) where f(x) = tan(x) over the domain [0,2pi).
x= pi/2, 3pi/2
Find the intervals of x where f(x) is concave up and concave down.
Let f(x) = e^x(sin(x))
f(x) is concave up when x=(-pi/2, pi/2) and concave down
when x=(-pi, -pi/2) U (pi/2,pi).
You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $5/ft^2 and the material for the sides costs $2/ft^2. You need a box with volume 4ft^3. Find the dimensions of the box that minimize cost. Use x to represent the length of the side of the box.
4 = x^2h
You are the manager of an apartment complex with 50 units. When you set rent at $800/month, all apartments are rented. As you increase rent by $25/month, one fewer apartment is rented. Maintenance costs run $50/month for each occupied unit. What is the rent that maximizes the total amount of profit?
p = rx - 50x where x represents the total amount of apartments rented and r represents the rent charged.
True/False and explain. Can a polynomial of degree 2 have an inflection point? (Note: the degree of a polynomial is the highest power on x)
No, it cannot because the second derivative of a degree 2 polynomial is always a constant value and thus it never changes.
Find the critical points of f(x) where f(x) = (3x^2+4x -4)^(1/2)
x = [-2, -2/3]
Find the intervals where f(x) is concave up and concave down given
f(x) = (x-2)^2 * (x-4)^2
f(x) is concave up from (-oo,2.577) U (3.577,oo)
f(x) is concave down from (2.577,3.577)
An island is 2 mi due north of its closest point along a straigt shoreline. A visitor is staying at a cabin on the shore that is 6 mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8 mph and swims at a rate of 3 mph. How far should the visitor run before swimming to minimize the time it takes to reach the island?
D = x + sqrt{ (6-x)^2 + 2^2}
A window is composed of a semicircle placed on top of a rectangle. If you have 20ft of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use r to represent the radius of the semicircle.
A = wh + (pi r^2) / h, where w and h represent the widths and lengths of the base, and r represents the radius of the semi-circle.
In precalculus, you learned a formula for the position of the maximum or minimum of a quadratic equation y=ax^2 +bx + c, which was h = -b\(2a). Prove this formula using calculus.
Since the maximum or minimum occurs at a critical point we can simply solve for the critical points.