Q: Which of the following statements about limits is TRUE?

If $$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$, then $$f(x) is $$continuous at $$x = a.$$

Question 1

Which of the following best describes the concept of the derivative in the context of business applications?

Accepted Answer

The derivative gives the instantaneous rate of change of a function, such as marginal cost or marginal revenue.

Question 2

A company’s profit function is given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. What is the number of units that maximizes profit?

Accepted Answer

$$x = 10$$

Question 3

Given the cost function $$C(x) = 500 + 20x + 0.5x^2$, what is the marginal cost when x = 10$?

Accepted Answer

$$MC = 30$$

Question 4

Which of the following statements about limits is TRUE?

Accepted Answer

If $$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$, then $$f(x) is $$continuous at $$x = a.$$

Question 5

Suppose the revenue function for a product is $$R(x) = 100x - x^2$. What is the marginal revenue when x = 20$?

Accepted Answer

$$MR = 60$$

Question 6

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

Accepted Answer

$$2e^{2x}$$

Question 7

A pharmaceutical company models the spread of a drug in the bloodstream by $$C(t) = 50e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the instantaneous rate of change of concentration at t = 5$ hours?

Accepted Answer

$$C'(5) = -10e$$^{-1}$$

Question 8

Which of the following is the correct application of the chain rule for $$y = \ln(3x^2$ + 2)$$?

Accepted Answer

$$y' = \frac{6x}{3x^2$ + 2}$$

Question 9

A company’s cost and revenue functions are $$C(x) = 100 + 5x$$ and $$R(x) = 20x. $$What is the break-even quantity?

Accepted Answer

$$x = 20$$

Question 10

Evaluate the following limit: $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 11

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It states that the definite integral of a function over $$[a, b] is $$equal to the difference of its antiderivative evaluated at $$b$$ and $$a.$$

Question 12

A business wants to maximize its revenue, given by $$R(x) = 50x - x^2$. What is the maximum revenue?

Accepted Answer

$$625$$

Question 13

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

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 14

A company’s demand function is $$p = 100 - 2x. $$What is the elasticity of demand when $$x = 20$$?

Accepted Answer

$$-1$$

Question 15

Which of the following is the correct derivative of $$f(x) = x^3$ e^x$$?

Accepted Answer

$$3x^2 e^x$$ + $$x^3 e^x$

Question 16

A company’s marginal cost is given by $$MC(x) = 4x + 10. If $$the fixed cost is $$200$$, what is the total cost function $$C(x)$$?

Accepted Answer

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

Question 17

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$ + 2x) dx.$$

Accepted Answer

$$20$$

Question 18

Which of the following is NOT a property of definite integrals?

Accepted Answer

$$\int_a^b f(x) dx = \int_a^b f(y) dy + c$$

Question 19

A company’s revenue and cost functions are $$R(x) = 60x$$ and $$C(x) = 40x + 200. $$What is the profit-maximizing quantity if the market can support up to 10 units?

Accepted Answer

$$x = 10$$

Question 20

If $$f(x) = \frac{1}{x}$$, what is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = -\frac{1}{4}x + \frac{3}{2}$$

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