Q: Given the function $f(x) = x^2 - 6$, which of the following is the correct graph of $f(x)$?

A parabola opening upwards with vertex at $(0, -6)$

Question 1

Given the function $f(x) = x^2 - 6$, what is the simplified form of $f(x + h) - f(x)$?

Accepted Answer

$2xh + h^2$

Question 2

For the linear function $f(x) = 5x - 5$, what is the value of $\frac{f(x + h) - f(x)}{h}$?

Accepted Answer

$5$

Question 3

A box is constructed by cutting squares of side $x$ from each corner of a $10$ inch by $12$ inch sheet and folding up the sides. What is the volume $V(x)$ of the box in terms of $x$?

Accepted Answer

$V(x) = x(12 - 2x)(10 - 2x)$

Question 4

What is the domain of the volume function $V(x) = x(12 - 2x)(10 - 2x)$, given the physical restrictions of the box?

Accepted Answer

$0 < x < 5$

Question 5

Given the linear equation $y = 3x - 9$, what is the $x$-intercept?

Accepted Answer

$x = 3$

Question 6

For the function $f(x) = x^2 - 6$, what is the average rate of change from $x = 2$ to $x = 5$?

Accepted Answer

$7$

Question 7

If $f(x) = 5x - 5$, what is the $y$-intercept of the function?

Accepted Answer

$-5$

Question 8

A company models its profit $P(x)$ by $P(x) = 2x^2 - 8x + 10$, where $x$ is the number of units sold. What is the profit when $x = 3$?

Accepted Answer

$P(3) = 4$

Question 9

Given $f(x) = x^2 - 6$, which of the following represents the difference quotient $\frac{f(x + h) - f(x)}{h}$?

Accepted Answer

$\frac{2xh + h^2}{h}$

Question 10

A business finds that its revenue $R(x)$ is given by $R(x) = 100x - 2x^2$, where $x$ is the number of items sold. What is the maximum possible revenue?

Accepted Answer

$2500$

Question 11

For the function $f(x) = 5x - 5$, what is the slope?

Accepted Answer

$5$

Question 12

Suppose the cost $C(x)$ to produce $x$ units is $C(x) = 4x + 20$. What is the marginal cost?

Accepted Answer

$4$

Question 13

If $f(x) = x^2 - 6$, what is the value of $f(4)$?

Accepted Answer

$10$

Question 14

A company wants to maximize the volume of a box made from a $10$ inch by $12$ inch sheet by cutting squares from each corner. What value of $x$ (side of the square) gives the maximum volume?

Accepted Answer

$x = 2$

Question 15

Given the function $f(x) = x^2 - 6$, which of the following is the correct graph of $f(x)$?

Accepted Answer

A parabola opening upwards with vertex at $(0, -6)$

Question 16

If the revenue function is $R(x) = 100x - 2x^2$, at what value of $x$ is the revenue maximized?

Accepted Answer

$x = 25$

Question 17

For the function $f(x) = 5x - 5$, what is the equation in slope-intercept form?

Accepted Answer

$y = 5x - 5$

Question 18

A company’s cost function is $C(x) = 4x + 20$. What is the cost to produce $10$ units?

Accepted Answer

$60$

Question 19

Given $f(x) = x^2 - 6$, what is the $y$-intercept?

Accepted Answer

$-6$

Question 20

If $f(x) = x^2 - 6$, what is the slope of the tangent line at $x = 2$?

Accepted Answer

$4$

Fundamental Concepts in Business Calculus: Functions, Difference Quotients, and Applications

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