Q: Which of the following is the correct definition of the derivative of a function $f(x)$ at a point $x=a$?

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Q: Which of the following statements about continuity is correct?

A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value a function approaches as the input approaches a certain point.

Question 2

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

Accepted Answer

$f'(x) = 6x + 2$

Question 3

A company’s profit $P(x)$ in thousands of dollars is given by $P(x) = 100 - 5x^2$, where $x$ is the number of units produced (in thousands). What is the marginal profit when $x = 4$?

Accepted Answer

$P'(x) = -10x$, so P'(4) = -40$

Question 4

Which of the following is the correct definition of the derivative of a function $f(x)$ at a point $x=a$?

Accepted Answer

$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 5

If the cost function for producing $x$ units is $C(x) = 50 + 10x + x^2$, what is the marginal cost when $x = 5$?

Accepted Answer

$C'(x) = 10 + 2x$, so C'(5) = 20$

Question 6

A business finds that its revenue $R(x)$ in dollars from selling $x$ units is $R(x) = 200x - 4x^2$. What is the maximum revenue, and at what value of $x$ does it occur?

Accepted Answer

Maximum at $x = 25$, $R(25) = 2500$

Question 7

Which rule is used to find the derivative of the product of two functions $f(x)$ and $g(x)$?

Accepted Answer

Product Rule: $[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)$

Question 8

Suppose the demand function for a product is $p(x) = 100 - 2x$, where $p$ is the price and $x$ is the quantity sold. What is the elasticity of demand when $x = 20$?

Accepted Answer

Elasticity $E = \left| \frac{p}{x} \cdot \frac{dp}{dx} \right| = \left| \frac{60}{20} \cdot (-2) \right| = 6$

Question 9

Which of the following statements about continuity is correct?

Accepted Answer

A function is continuous at $x=a$ if $\lim_{x 	o a} f(x) = f(a)$

Question 10

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

Accepted Answer

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

Question 11

A manufacturer’s cost function is $C(x) = 500 + 20x - x^2$. At what production level $x$ is the marginal cost zero?

Accepted Answer

$C'(x) = 20 - 2x$, set to zero: $20 - 2x = 0 \Rightarrow x = 10$

Question 12

Which of the following is the correct application of the chain rule for $f(x) = \sin(3x)$?

Accepted Answer

$f'(x) = 3\cos(3x)$

Question 13

A company’s revenue function is $R(x) = 120x - x^3$. Find the production level $x$ that maximizes revenue.

Accepted Answer

Set $R'(x) = 120 - 3x^2 = 0 \Rightarrow x = 20$

Question 14

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

Accepted Answer

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

Question 15

A business’s cost function is $C(x) = 1000 + 15x + 0.5x^2$. What is the average cost per unit when $x = 10$?

Accepted Answer

Average cost $= \frac{C(10)}{10} = \frac{1000 + 150 + 50}{10} = 120$

Question 16

Which of the following is the correct formula for the elasticity of demand?

Accepted Answer

$E = \left| \frac{p}{x} \cdot \frac{dp}{dx} \right|$

Question 17

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

Accepted Answer

Set $f'(x) = 3x^2 - 12x + 9 = 0 \Rightarrow x = 1, x = 3$

Question 18

A company’s profit function is $P(x) = -2x^2 + 40x - 100$. What is the maximum profit and at what value of $x$ does it occur?

Accepted Answer

Maximum at $x = 10$, $P(10) = 300$

MATH 209: Business Calculus Syllabus and Course Structure

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