Question 1

A company sells scientific calculators for $75.99 each. They cost only $19.56 to make. Fixed setup costs are $120,000. Let $x$ be the number of calculators produced and sold. What is the revenue function, $R(x)$?

Accepted Answer

$R(x) = 75.99x$

Question 2

Given the cost function $C(x) = 120,000 + 19.56x$, what is the profit function $P(x)$ if the calculators are sold for $75.99$ each?

Accepted Answer

$P(x) = 75.99x - (120,000 + 19.56x)$

Question 3

How many calculators must be sold to break even? (Round to the nearest calculator.)

Accepted Answer

$x = 1,579$

Question 4

Given the following data showing average time spent on a computer per month:

Age (years): 8.5, 14.5, 21, 29.5
Average time (minutes): 643, 945, 1377, 1727

What is the $R^2$ value for the quadratic regression found in part B? (Round to 3 decimal places)

Accepted Answer

$R^2 = 0.999$

Question 5

Suppose the supply and demand for a product are given by $S(p) = 2p + 5$ and $D(p) = 43 - 3p$. What is the equilibrium price $p$?

Accepted Answer

$p = 6.4$

Question 6

Find the limit: $\lim\limits_{x 	o 2} \frac{x^2 - 1}{x^2 - 4}$

Accepted Answer

$\frac{3}{0}$ (does not exist)

Question 7

Given $C(x) = -0.052x^2 + 60x$, where $C(x)$ is in dollars and $x$ is the number of units, find the average rate of change of $C(x)$ as $x$ increases from 350 to 550.

Accepted Answer

$21$

Question 8

Using the limit of the difference quotient, find $f'(x)$ for $f(x) = \frac{1}{x}$.

Accepted Answer

$f'(x) = -\frac{1}{x^2}$

Question 9

Find the derivative of $f(x) = \sqrt{x} - \frac{1}{2}x^2 + 7x^3 - 1$ using appropriate differentiation techniques.

Accepted Answer

$f'(x) = \frac{1}{2\sqrt{x}} - x + 21x^2$

Question 10

Given the cost function $C(x) = 770 + 39x - 0.063x^2$, what is the average cost function $\overline{C}(x)$?

Accepted Answer

$\overline{C}(x) = \frac{770 + 39x - 0.063x^2}{x}$

Question 11

Given the cost function $C(x) = 770 + 39x - 0.063x^2$, find the rate at which the average cost is changing when 175 units have been produced. (Round to 2 decimal places.)

Accepted Answer

$-0.12$

Question 12

A company’s revenue function is $R(x) = 50x$ and its cost function is $C(x) = 20x + 5000$. What is the minimum number of units $x$ that must be sold to make a profit?

Accepted Answer

$x = 250$

Question 13

If the demand function for a product is $p = 100 - 2q$ and the supply function is $p = 20 + 3q$, what is the equilibrium quantity $q$?

Accepted Answer

$q = 16$

Question 14

Evaluate the following limit: $\lim\limits_{x 	o 3} \frac{x^2 - 9}{x - 3}$

Accepted Answer

$6$

Question 15

A business finds that the marginal cost of producing $x$ units is $MC(x) = 5x^2 - 2x + 10$. What is the total cost of producing the first 3 units if the fixed cost is $100$?

Accepted Answer

$C(3) = 100 + \int_0^3 (5x^2 - 2x + 10) dx$

Question 16

If $f(x) = x^3 - 2x^2 + 5x - 7$, what is $f'(2)$?

Accepted Answer

$f'(2) = 9$

Question 17

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. For what value of $x$ is the profit maximized?

Accepted Answer

$x = 10$

Question 18

Given $y = \frac{2x^3 - 5}{x^2}$, find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx} = 6x - \frac{10}{x^3}$

Question 19

A company’s cost function is $C(x) = 500 + 25x$. If the company produces 100 units, what is the marginal cost at $x = 100$?

Accepted Answer

$25$

Business Calculus: Exam 1 Study Guide and Practice Problems

Study Guide - Practice Questions

Study Guide - Flashcards