Q: A function $f(x)$ has a relative maximum at $x = a$ if:

$f'(a) = 0$ \,\text{and}\, f''(a) < 0$

Q: Suppose $f'(x)$ changes from negative to positive at $x = a$. What does this indicate about $f(x)$ at $x = a$?

$f(x)$ has a relative minimum at $x = a$

Q: A point $x = a$ is an inflection point of $f(x)$ if:

$f''(a) = 0$ \text{ and the sign of } f''(x) \text{ changes at } x = a$

Q: Which of the following best describes a horizontal asymptote of a function $f(x)$?

A straight, horizontal line that the graph approaches as $x$ increases or decreases without bound.

Q: A company wants to maximize the area of a rectangular garden with 300 meters of fencing. What are the dimensions of the garden with the largest area?

$75 \text{ m} \times 75 \text{ m}$

Q: Which of the following is true about the relationship between the first and second derivatives and the shape of the graph?

If $f'(x) 0$, $f(x)$ is decreasing and concave up.

Question 1

Which of the following statements best describes when a function $f(x)$ is increasing on an interval?

Accepted Answer

When $f'(x) > 0$ for all $x$ in the interval.

Question 2

A function $f(x)$ has a relative maximum at $x = a$ if:

Accepted Answer

$f'(a) = 0$ \,	ext{and}\, f''(a) < 0$

Question 3

Which of the following is NOT a necessary step in the general approach to curve sketching?

Accepted Answer

Find the area under the curve.

Question 4

Suppose $f'(x)$ changes from negative to positive at $x = a$. What does this indicate about $f(x)$ at $x = a$?

Accepted Answer

$f(x)$ has a relative minimum at $x = a$

Question 5

A point $x = a$ is an inflection point of $f(x)$ if:

Accepted Answer

$f''(a) = 0$ 	ext{ and the sign of } f''(x) 	ext{ changes at } x = a$

Question 6

Which of the following best describes a horizontal asymptote of a function $f(x)$?

Accepted Answer

A straight, horizontal line that the graph approaches as $x$ increases or decreases without bound.

Question 7

Given $f(x) = x^3 - 3x^2 + 2$, find all critical points.

Accepted Answer

$x = 1$ and $x = 2$

Question 8

A company wants to maximize the area of a rectangular garden with 300 meters of fencing. What are the dimensions of the garden with the largest area?

Accepted Answer

$75 	ext{ m} 	imes 75 	ext{ m}$

Question 9

If $f''(x) > 0$ for all $x$ in an interval, what can you conclude about the graph of $f(x)$ on that interval?

Accepted Answer

The graph is concave up.

Question 10

A profit function is given by $P(x) = -x^2 + 6x + 15$. What is the value of $x$ that maximizes profit?

Accepted Answer

$x = 3$

Question 11

A bookstore sells 8000 books per year. The ordering cost per order is $40$, and the carrying cost per book is $2$. What is the economic order quantity (EOQ) that minimizes total inventory cost?

Accepted Answer

$x = 400$

Question 12

If $f'(x) = 0$ and $f''(x) < 0$ at $x = a$, what does this tell you about $f(x)$ at $x = a$?

Accepted Answer

$f(x)$ has a relative maximum at $x = a$

Question 13

A swimming club offers memberships at $p = 200 - x$, where $x$ is the number of memberships sold above 100. What number of memberships maximizes revenue?

Accepted Answer

$x = 150$

Question 14

Which of the following is true about the relationship between the first and second derivatives and the shape of the graph?

Accepted Answer

If $f'(x) < 0$ and $f''(x) > 0$, $f(x)$ is decreasing and concave up.

Question 15

A function $f(x)$ has a critical point at $x = 2$. The first derivative changes from positive to negative at $x = 2$. What does this mean?

Accepted Answer

$f(x)$ has a relative maximum at $x = 2$

Chapter 2: Applications of the Derivative – Business Calculus Study Notes

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