Question 1

A company’s revenue function is $$R(x) = 50x$$, and its cost function is $$C(x) = 30x + 2000$$, where $$x is $$the number of units sold. What is the break-even point?

Accepted Answer

$$x = 100$$

Question 2

Given the function $$f(x) = 3x^2$ - 4x + 7$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x - 4$$

Question 3

A pharmaceutical company models the rate of change of a drug concentration in the bloodstream as $$C'(t) = -0.5C(t)$$, where $$C(t) is $$the concentration at time $$t. If$$ $$C(0) = 100$$, what is $$C(2)$$?

Accepted Answer

$$C(2) = 100 	imes e$$^{-1}$$

Question 4

If the demand function for a product is $$p = 100 - 2x$$, where $$p is $$price and $$x is $$quantity, what is the marginal revenue when $$x = 20$$?

Accepted Answer

$$MR = 60$$

Question 5

A business wants to maximize its profit, given by $$P(x) = -2x^2$ + 40x - 100. $$What is the number of units $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 6

Evaluate the definite integral $$\int_{0}^{3} (2x + 1) \, dx.$$

Accepted Answer

$$10.5$$

Question 7

A company’s cost function is $$C(x) = 500 + 20x. $$What is the average cost per unit when 50 units are produced?

Accepted Answer

$$C_{avg} = 30$$

Question 8

If $$f(x) = e$$^{2x}$$, what is f$$''(x)$$?

Accepted Answer

$$f''(x) = 4e$$^{2x}$$

Question 9

A hospital’s patient intake rate is modeled by $$r(t) = 5t^2$ + 3t$$, where $$t is $$hours since midnight. How many patients have been admitted by 4 PM (16 hours after midnight)?

Accepted Answer

$$\int_{0}^{16} (5t^2$ + 3t) \, dt = 6832$$

Question 10

A company’s revenue function is $$R(x) = 100x - x^2$. What is the elasticity of demand at x = 20$?

Accepted Answer

$$E = 0.8$$

Question 11

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 12

A company’s profit function is $$P(x) = 40x - 0.5x^2$ - 200. $$What is the maximum profit?

Accepted Answer

$$P_{max} = 600$$

Question 13

If $$f(x) = x^3$ - 6x^2$ + 9x$$, find all critical points.

Accepted Answer

$$x = 0, 3$$

Question 14

A company’s cost function is $$C(x) = 1000 + 25x$$, and its revenue function is $$R(x) = 40x. $$What is the profit function?

Accepted Answer

$$P(x) = 15x - 1000$$

Question 15

Evaluate $$\int_{1}^{4} x^2$ \, dx.$$

Accepted Answer

$$21$$

Question 16

A company’s marginal cost function is $$MC(x) = 10 + 0.5x. $$What is the total cost function $$C(x) if$$ $$C(0) = 200$$?

Accepted Answer

$$C(x) = 200 + 10x + 0.25x^2$

Question 17

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 18

A company’s revenue function is $$R(x) = 80x$$, and its cost function is $$C(x) = 60x + 1200. $$What is the minimum number of units to sell to make a profit?

Accepted Answer

$$x = 60$$

Question 19

If $$f(x) = x^2$ e^x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = (2x + x^2$) e^x$$

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