Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^4$ - 9\sqrt{x} + 8\sqrt[3]{x} + 10$$?

Accepted Answer

$$f'(x) = 28x^3$ - \dfrac{9}{2\sqrt{x}} + \dfrac{8}{3}x$$^{-2/3}$$

Question 2

A company’s cost function is $$C(x) = \sqrt{x}$$ hundred dollars. What is the marginal cost when producing 100 items?

Accepted Answer

$$MC(100) = \dfrac{1}{2\sqrt{100}} = 0.05$$ hundred dollars

Question 3

If $$f(x) = 5x^3$ + 4x^2$ + 0.5x^2$ - 10x + 157$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 15x^2$ + 8x + x - 10$$

Question 4

Given the revenue function $$R(x) = 6x^2$, what is the marginal revenue when x = 5$?

Accepted Answer

$$MR(5) = 12x = 60$$

Question 5

Which rule allows you to find the derivative of $$f(x) = ax^n as$$ $$f'(x) = a n x$$^{n-1}$$?

Accepted Answer

Power Rule

Question 6

A book publisher has a cost function $$C(x) = 1.5x^2$ - 4x + 7. $$What is the instantaneous rate of change of cost when 2 books have been produced?

Accepted Answer

$$C'(2) = 2 \cdot 1.5 \cdot 2 - 4 = 6 - 4 = 2$$

Question 7

If $$f(x) = 10 - x^2$, what is the equation of the tangent line at x = 2$?

Accepted Answer

$$y = -4x + 14$$

Question 8

Which of the following best describes the marginal profit $$MP(x) in $$business calculus?

Accepted Answer

$$MP(x) is $$the derivative of the profit function $$P(x)$$ with respect to $$x$$

Question 9

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

Accepted Answer

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

Question 10

If $$C(x) = x^2$ + 2x + 1$$, what is the average rate of change of cost from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$\dfrac{C(3) - C(1)}{3 - 1} = \dfrac{16 - 4}{2} = 6$$

Question 11

Which of the following is the correct derivative of $$f(x) = \dfrac{2x^2$ + 11x}{2x}$$?

Accepted Answer

$$f'(x) = 1 + \dfrac{11}{2x}$$

Question 12

A company’s profit function is $$P(x) = R(x) - C(x). If$$ $$R(x) = 6x^2$ and C(x$$) = $$x^2 + 2x + 1$, what is P$$'(x)$$?

Accepted Answer

$$P'(x) = 12x - 2x - 2$$

Question 13

If $$f(x) = 3x^2$ + 2x + 1$$, what is the value of $$f'(4)$$?

Accepted Answer

$$f'(4) = 3 \cdot 2 \cdot 4 + 2 = 24 + 2 = 26$$

Question 14

Which of the following statements is true about the derivative of a constant function?

Accepted Answer

The derivative of a constant function is zero.

Question 15

If $$f(x) = 4x^3$, what is the slope of the tangent line at x = 2$?

Accepted Answer

$$f'(2) = 12 \cdot 2^2$ = 48$$

Question 16

A company’s revenue function is $$R(x) = 10x - x^2$. What is the marginal revenue when x = 3$?

Accepted Answer

$$R'(3) = 10 - 2 \cdot 3 = 4$$

Question 17

If $$f(x) = x$$^{-1}$$, what is f$$'(x)$$?

Accepted Answer

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

Question 18

Which of the following is the correct application of the sum rule for derivatives?

Accepted Answer

$$\dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 19

A company’s cost function is $$C(x) = 0.5x^2$ + 2x + 100. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(10) = 0.5 \cdot 2 \cdot 10 + 2 = 10 + 2 = 12$$

Formulas for Derivatives: Business Calculus Section 1.3 Study Notes

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