Question 1

Given the function $$f(x) = 4x^5$ - x^3$ + 5x - 9$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 20x^4$ - 3x^2$ + 5$$

Question 2

Find the derivative of $$f(x) = 5x^4$ - 3x^2$ + x - 9.$$

Accepted Answer

$$f'(x) = 20x^3$ - 6x + 1$$

Question 3

If $$f(x) = \frac{3}{\sqrt[3]{x}}$$, what is $$f'(x)$$?

Accepted Answer

Question 4

Given $$f(x) = \frac{3}{\sqrt[3]{x}}$$, which of the following is the correct derivative?

Accepted Answer

$$f'(x) = -x$$^{-4/3}$$

Question 5

Find the derivative of $$f(x) = \frac{x^2$ - 4}{x^2$ + 4}$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{8x}{(x^2$ + 4)^2$}$$

Question 6

Suppose $$f(x) = \frac{2x}{3x - 5}. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{-10}{(3x - 5)^2$}$$

Question 7

A company’s total cost function for producing air conditioners is $$C(x) = 15,000 + 30x + x^3$. What is the marginal cost function?

Accepted Answer

$$C'(x) = 30 + 3x^2$

Question 8

If air conditioners sell for $$450 per unit, what is the revenue function R(x$$)$$ in terms of x$?

Accepted Answer

$$R(x) = 450x$$

Question 9

Given $$C(x) = 15,000 + 30x + x^3$ and R(x) = 450x$, what is the profit function P(x$$)$$?

Accepted Answer

$$P(x) = 450x - (15,000 + 30x + x^3$)$$

Question 10

If $$f(x) = x^2$ - 4$$, what is the average rate of change from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$5$$

Question 11

A business finds that its cost function is $$C(x) = 15,000 + 30x + x^3$. What is the marginal cost when x = 10$?

Accepted Answer

$$C'(10) = 330$$

Question 12

If $$f(x) = x^2$ - 4$$, what is $$f'(x)$$?

Accepted Answer

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

Question 13

Given $$f(x) = \frac{2x}{3x - 5}$$, what is the value of $$f'(2)$$?

Accepted Answer

$$-0.4$$

Question 14

If the revenue function is $$R(x) = 450x$$ and the cost function is $$C(x) = 15,000 + 30x + x^3$, what is the marginal profit function?

Accepted Answer

$$P'(x) = 450 - (30 + 3x^2$)$$

Question 15

A company’s cost function is $$C(x) = 15,000 + 30x + x^3$. If the company produces 20 units, what is the marginal cost at that level?

Accepted Answer

$$1,230$$

Business Calculus Test 2 Review – Derivatives and Applications

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