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

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