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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