Question 1

Given $$P = \dfrac{4}{\sqrt{(4t^5$ - Q)^3$}}$$, compute $$\dfrac{dP}{dt}.$$

Accepted Answer

$$\dfrac{-60t^4$}{\sqrt{(4t^5$ - Q)^5$}}$$

Question 2

Compute $$\dfrac{d}{dr}(7r^4$ - \dfrac{8}{r}).$$

Accepted Answer

$$28r^3$$ + \dfrac{8}{$$r^2$$}$$

Question 3

Compute $$\dfrac{d}{dr}(r^7$).$$

Accepted Answer

$$7r^6$$

Question 4

For $$y = 7t^6$ - 7.8t^4$ + t - 499.2$$, compute $$\dfrac{d^2$y}{dt^2$}\Big|_{t=3}.$$

Accepted Answer

$$882 - 93.6 = 788.4$$

Question 5

Which notation represents the second derivative of a function $$f(x)$$?

Accepted Answer

$$\dfrac{d^2$}{dx^2$}f(x)$$

Question 6

If $$C(x) is a $$cost function, what does the marginal cost $$C'(a)$$ estimate?

Accepted Answer

The approximate additional cost incurred when production increases from $$a to$$ $$a+1$$ units.

Question 7

Given $$R(45) = 17.5$$, $$C(45) = 15.4$$, what is the profit at a production level of 45 items?

Accepted Answer

$$2.1$$

Question 8

Given $$R'(45) = -2.75$$ and $$C'(45) = 3.12$$, what is the marginal profit at a production level of 45 items?

Accepted Answer

$$-5.87$$

Question 9

If the marginal revenue at a production level of 100 items is $$R'(100) = 5$$, how should this result be interpreted?

Accepted Answer

Producing one more item (the 101st) will increase revenue by approximately $5.

Question 10

Given $$C(a) = 100$$ and $$C(a+1) = 105$$, estimate the marginal cost at $$a$$ units.

Accepted Answer

$$5$$

Question 11

If $$y = x^5$ - 3x^2$ + 7$$, what is $$\dfrac{dy}{dx}$$ evaluated at $$x = 2$$?

Accepted Answer

$$68$$

Question 12

Which of the following best describes the economic meaning of the derivative of a profit function $$P(x) at$$ $$x = a$$?

Accepted Answer

It estimates the change in profit if production increases from $$a to$$ $$a+1$$ units.

Question 13

If $$C(x) = 2x^2$ + 5x + 10$$, what is the marginal cost function $$C'(x)$$?

Accepted Answer

$$4x + 5$$

Question 14

Suppose $$R(x) = 10x - x^2$ and C(x) = 2x + 5$. What is the marginal profit function P$$'(x)$$?

Accepted Answer

$$10 - 2x - 2$$

Question 15

If $$f(x) = x^3$ - 2x + 1$$, what is $$f''(x)$$?

Accepted Answer

$$6x$$

Question 16

Given $$y = 5x^4$ - 3x^2$ + 2x$$, compute $$\dfrac{d^2$y}{dx^2$} at$$ $$x = 1.$$

Accepted Answer

$$80 - 6 = 74$$

Question 17

If $$C(x) is a $$cost function, which formula best estimates the marginal cost at $$x = a$$?

Accepted Answer

$$C(a+1) - C(a)$$

Question 18

If $$R(x) = 20x$$ and $$C(x) = 15x + 100$$, what is the profit at $$x = 10$$?

Accepted Answer

$$50$$

Question 19

Suppose $$C(x) = x^2$ + 2x + 1. $$What is the marginal cost at $$x = 3$$?

Accepted Answer

$$8$$

Question 20

If $$R(x) = 5x^2$ and C(x$$) = $$3x^2$$, what is the marginal profit at $$x = 4$$?

Accepted Answer

$$16$$

1.7

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