Question 1

Suppose $D(x) = 9 - x^2$ and $S(x) = x^2 + 21$ represent the demand and supply functions for a product, where $x$ is the number of units. What is the equilibrium price and quantity?

Accepted Answer

Equilibrium at $x = 0$, price $P = 9$

Question 2

Big Break Billiards sells 100 pool tables per year. It costs $20 to store one pool table per year. To reduce storage, there is a cost of $40 per shipment plus $16 for each pool table. How many times per year should the store order pool tables, and in what lot size, to minimize inventory costs?

Accepted Answer

Order 5 times per year, 20 tables per order

Question 3

Given the cost function $C(x) = 2500 + 0.1x^2$ and the price function $p = 16 - 0.5x$, what is the profit-maximizing quantity?

Accepted Answer

$x = 12$

Question 4

Evaluate the definite integral $\int_4^8 \frac{5}{x} dx$.

Accepted Answer

$5 \ln 2$

Question 5

Use substitution to evaluate $\int \frac{2x}{x^2 - 6} dx$.

Accepted Answer

$\ln|x^2 - 6| + C$

Question 6

A company estimates that its sales will grow continuously at a rate given by $S'(t) = 14e^t$ where $S'(t)$ is the rate at which sales are increasing in dollars per day. Find the total sales from the 3rd to the 6th day.

Accepted Answer

$14(e^6 - e^3)$

Question 7

Find the area bounded by the graphs of $y = x^2 - 5x + 9$ and $y = x^2 - 7x + 11$.

Accepted Answer

$2$

Question 8

Find the average value of the function $f(x) = x^3 + 2x$ on the interval $[2, 5]$.

Accepted Answer

$60$

Question 9

Evaluate $\int 2e^{-3x} dx$.

Accepted Answer

$-\frac{2}{3}e^{-3x} + C$

Question 10

Suppose $E'(t)$ is the rate of change of employees at a company, and $\int_0^{24} E'(t) dt$ is calculated. What does this integral represent?

Accepted Answer

The net change in the number of employees over 24 years

Question 11

If a company has 2000 employees on Jan 1, 1981, and the net change in employees over 24 years is $656$, how many employees are there on Jan 1, 2005?

Accepted Answer

2656

Question 12

Find $\int \frac{1}{3} (\sqrt{x}) dx$.

Accepted Answer

$\frac{2}{9} x^{3/2} + C$

Question 13

Given the graph of $E'(t)$, the derivative of the number of employees, which year did the company have the least number of employees?

Accepted Answer

Year 11

Question 14

Was the number of employees greater in Jan 1, 1995 or Jan 1, 1996?

Accepted Answer

Jan 1, 1995

Question 15

Find $\int_2^5 \frac{5}{x} dx$.

Accepted Answer

$5 \ln \frac{5}{2}$

Question 16

If the total cost of producing $x$ units is $C(x) = 2500 + 0.1x^2$, what is the marginal cost when $x = 10$?

Accepted Answer

$3$

Question 17

Given $p = 16 - 0.5x$ and $C(x) = 2500 + 0.1x^2$, what is the revenue function $R(x)$?

Accepted Answer

$R(x) = 16x - 0.5x^2$

Question 18

Find the integral $\int 5x^3 dx$.

Accepted Answer

$\frac{5}{4}x^4 + C$

Question 19

Find the integral $\int \sqrt{16 - x^2} dx$ using trigonometric substitution.

Accepted Answer

$\frac{x}{2}\sqrt{16 - x^2} + 8 \arcsin\left(\frac{x}{4}\right) + C$

Business Calculus Study Guide: Applications of Differentiation and Integration

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