Q: A company’s revenue is given by $$R(x) = 120 + 90\sqrt{x}$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 100$$?

$$R'(100) = 45 \times 100$$^{-1/2}$$ = 4.5$$

Question 1

Which of the following correctly states the Product Rule for differentiation?

Accepted Answer

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

Question 2

Given $$f(x) = x^4$ - 2x^3$ - 7$$ and $$g(x) = 3x^2$ - 5x$$, what is $$\frac{d}{dx}[(x^4$ - 2x^3$ - 7)(3x^2$ - 5x)] (do $$not simplify)?

Accepted Answer

$$(x^4$$ - $$2x^3 - 7$$) \cdot \frac{d}{dx}$$(3x^2 - 5x$$) + $$(3x^2 - 5x$$) \cdot \frac{d}{dx}$$(x^4$$ - $$2x^3 - 7$$)$$

Question 3

Which of the following is the correct application of the Quotient Rule for $$Q(x) = \frac{N(x)}{D(x)}$$?

Accepted Answer

$$Q'(x) = \frac{D(x)N'(x) - N(x)D'(x)}{[D(x)]^2}$$

Question 4

Differentiate $$f(x) = \frac{1 + x^2$}{x^3$}$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{x^3$ \cdot 2x - (1 + x^2$) \cdot 3x^2$}{(x^3$)^2$$}$

Question 5

Suppose $$R(x) = 120 + 90\sqrt{x}. $$What is $$R'(x)$$?

Accepted Answer

$$R'(x) = 45x$$^{-1/2}$$

Question 6

If $$A_P(x) = \frac{P(x)}{x}$$, which rule must be used to find $$A_P'(x)$$?

Accepted Answer

Quotient Rule

Question 7

Which of the following is the correct statement of the Chain Rule?

Accepted Answer

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Question 8

Differentiate $$y = (1 + x^2$)^3$ using the chain rule.

Accepted Answer

$$y' = 3(1 + x^2$)^2$$ \cdot 2x$

Question 9

A video game manufacturer wants to find the rate of change of profit in dollars per minute. If the profit per item is changing at $$5$$ dollars per item and the production rate is $$20$$ items per minute, what is the rate of change of profit per minute?

Accepted Answer

$$100$$ dollars per minute

Question 10

Given $$f(x) = \sqrt[4]{\frac{x+3}{x-2}}$$, which rules are needed to differentiate $$f(x)$$?

Accepted Answer

Chain Rule and Quotient Rule

Question 11

If $$s(t) is $$the position of an object at time $$t$$, what is the acceleration $$a(t) in $$terms of $$s(t)$$?

Accepted Answer

$$a(t) = s''(t)$$

Question 12

Which of the following best describes the composition of functions $$f \circ g$$?

Accepted Answer

$$(f \circ g)(x) = f(g(x))$$

Question 13

Suppose $$C(x) is $$the cost to produce $$x$$ units. The average cost is $$A_C(x) = \frac{C(x)}{x}. $$What is the correct formula for the rate of change of average cost with respect to $$x$$?

Accepted Answer

$$A_C'(x) = \frac{xC'(x) - C(x)}{x^2$}$$

Question 14

If $$y = (3x^2$ - 5x)^7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$7(3x^2$$ - $$5x)^6$$ \cdot (6x - 5)$$

Question 15

A company’s revenue is given by $$R(x) = 120 + 90\sqrt{x}$$, where $$x is $$the number of units sold. What is the marginal revenue when $$x = 100$$?

Accepted Answer

$$R'(100) = 45 	imes 100$$^{-1/2}$$ = 4.5$$

Question 16

If $$f(x) = x^2$ and g(x) = 2x + 1$, what is $$\frac{d}{dx}[f(g(x))]$$?

Accepted Answer

$$2(2x + 1) \cdot 2$$

Business Calculus Study Notes: Product Rule, Quotient Rule, Chain Rule, and Applications

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