Question 1

Given the first image, which represents a rectangle with sides $$x$$ and $$y$$, if the perimeter is fixed at 100 ft, what is the area of the largest possible rectangle that can be built?

Accepted Answer

$$625\ 	ext{ft}^2$$

Question 2

For the rectangle in the first image, if the perimeter is $$2x + 2y = 100$$, what are the dimensions $$x$$ and $$y$$ that maximize the area?

Accepted Answer

$$x = 25,\ y = 25$$

Question 3

In the second image, a rancher wants to maximize the area of a rectangular pen with one side against a barn (no fence needed there), and the pen is subdivided into four smaller pens using three interior fences parallel to the shorter side. If the total fencing available is 1500 ft, what is the equation relating $$x$$ and $$y$$?

Accepted Answer

$$x + 5y = 1500$$

Question 4

For the subdivided pen in the second image, what are the dimensions $$x$$ and $$y$$ that maximize the area, given $$x + 5y = 1500$$?

Accepted Answer

$$x = 750,\ y = 150$$

Question 5

Given the function $$f(x) = x^2$ + 8x on $$the interval $$[-5, 3]$$, what is the absolute minimum value?

Accepted Answer

$$-16$$

Question 6

For $$f(x) = x^3$ - 3x^2$ on $$[-2, 4]$$, at which x$ does the absolute maximum occur?

Accepted Answer

$$x = 4$$

Question 7

For $$f(x) = 2x^3$ + 3x^2$ - 120x + 3$$, at which $$x$$ does the local minimum occur?

Accepted Answer

$$x = 4$$

Question 8

Given $$f(x) = x^3$ - 27x$$, what is the interval where the function is decreasing?

Accepted Answer

$$(-3, 3)$$

Question 9

For $$f(x) = x^2$ e^x$$, at which $$x$$ does the local maximum occur?

Accepted Answer

$$x = -2$$

Question 10

For $$f(x) = x^3$ - 18x^2$ + 10$$, at which $$x$$ does the inflection point occur?

Accepted Answer

$$x = 6$$

Question 11

Given $$f(x) = x^5$ + 5x^4$ - 5x + 6$$, which interval is the function concave down?

Accepted Answer

$$(-\infty, -3)$$

Question 12

For $$f(x) = x^3$ - 3x^2$, what are the x$-intercepts?

Accepted Answer

$$x = 0,\ 3$$

Question 13

Given $$y = \frac{2x - 1}{x + 3}$$, what is the vertical asymptote?

Accepted Answer

$$x = -3$$

Question 14

For $$y = \frac{2x - 1}{x + 3}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 15

Of all pairs of numbers whose sum is 80, which pair has the maximum product?

Accepted Answer

$$40$$ and $$40$$

Question 16

What is the maximum product of two numbers whose sum is 80?

Accepted Answer

$$1600$$

Calculus Study Guide: Maxima, Minima, Derivatives, and Optimization

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