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

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

Question 1

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

The average rate of change is the slope of the secant line between $$x = a$$ and $$x = a + h$$, while the instantaneous rate of change is the slope of the tangent line at $$x = a.$$

Question 2

Given $$f(x) = x^2$ - 6x$$, what is the equation of the tangent line to the graph at $$x = 2$$?

Accepted Answer

$$y = -2x - 4$$

Question 3

Suppose $$C(x) is $$the cost of producing $$x$$ units of a product. If $$C'(100) = 300$$, what does this value represent in practical business terms?

Accepted Answer

The cost to produce the 101st unit is approximately $300.

Question 4

Using the limit definition, what is the derivative of $$g(x) = \frac{-2}{x+5}$$?

Accepted Answer

$$g'(x) = \frac{2}{(x+5)^2$}$$

Question 5

If $$s(x) = \sqrt{x}$$, what is $$s'(x)$$ using the limit definition?

Accepted Answer

$$s'(x) = \frac{1}{2\sqrt{x}}$$

Question 6

The revenue from selling $$x$$ containers of matcha tea is $$R(x) = 20x - \frac{1}{500}x^2$. What is the marginal revenue function R$$'(x)$$?

Accepted Answer

$$R'(x) = 20 - 0.004x$$

Question 7

If $$R'(2000) = 12$$ for the revenue function $$R(x) = 20x - \frac{1}{500}x^2$, what does this value mean in a business context?

Accepted Answer

Selling the 2001st container will generate approximately $12 in additional revenue.

Question 8

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

Accepted Answer

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

Question 9

A company finds that $$C(100) = 80,000. $$What does this tell you?

Accepted Answer

The total cost to produce 100 units is $80,000.

Question 10

Suppose $$f(x) is $$differentiable. Which of the following statements is true?

Accepted Answer

The derivative $$f'(x)$$ gives the instantaneous rate of change of $$f(x) at $$any $$x.$$

Question 11

A business uses the function $$C(x) to $$model the cost of producing $$x$$ items. If $$C'(x) is $$large and positive, what does this suggest about the cost structure?

Accepted Answer

The cost of producing additional items is increasing rapidly.

Question 12

If $$f(x) = x^2$ - 6x$$, what is $$f'(x)$$ using the limit definition?

Accepted Answer

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

Question 13

A company’s cost function is $$C(x). $$Which of the following best describes the marginal cost at $$x = a$$?

Accepted Answer

It is the approximate cost to produce one more unit after $$a$$ units have been produced.

Question 14

Suppose $$s(x) = \sqrt{x}. $$What is the value of $$s'(9)$$?

Accepted Answer

$$\frac{1}{6}$$

Question 15

If the marginal revenue $$R'(x)$$ for a product is negative at a certain production level, what does this indicate?

Accepted Answer

Selling additional units will decrease total revenue.

Question 16

A business calculates $$R(100) = 1500$$ and $$R'(100) = 10. $$Which of the following best estimates the revenue from selling 101 units?

Accepted Answer

$$1510$$

Question 17

Which algebraic technique is used to simplify the difference quotient for $$s(x) = \sqrt{x} in $$the limit definition of the derivative?

Accepted Answer

Multiplying numerator and denominator by the conjugate.

Question 18

If $$g(x) = \frac{-2}{x+5}$$, what is the value of $$g'(0)$$?

Accepted Answer

$$\frac{2}{25}$$

Question 19

Which of the following best describes the role of the derivative in short-term business predictions?

Accepted Answer

It estimates how a cost, revenue, or profit will change with the production or sale of one additional unit.

Derivatives and Their Applications in Business Calculus

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