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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