Question 1

A company manufactures calculators. The cost function is $$C(x) = 60{,}000 + 30x$$, the revenue function is $$R(x) = 20x - \frac{2}{x}$$, and the profit function is $$P(x) = R(x) - C(x). If $$production is increasing at a rate of 400 calculators per week when production output is 5,000 calculators, what is the rate of increase in profit at this production level?

Accepted Answer

$$7{,}960$$ per week

Question 2

Find the instantaneous velocity function $$v(t)$$ for an object moving along a line with position $$s(t) = 7t^2$ - 8t (in $$feet, $$t in $$seconds), and determine $$v(1).$$

Accepted Answer

$$v(1) = 6 ft/sec$$

Question 3

Given $$y = x^7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$7x^6$$

Question 4

If $$y(x) = x^2$ + 11$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$2x$$

Question 5

Find the derivative of $$f(x) = (2x^4$ - x)^6$.

Accepted Answer

$$6(2x^4$$ - $$x)^5$$ \cdot $$(8x^3 - 1$$)$$

Question 6

The demand function is $$x = p^3$ - 6p^2$ + 500. $$Find the rate of change of $$p$$ with respect to $$x by $$differentiating implicitly.

Accepted Answer

$$\frac{dp}{dx} = \frac{1}{3p^2$ - 12p}$$

Question 7

Find the marginal cost function for $$C(x) = 176 + 1.2x.$$

Accepted Answer

$$C'(x) = 1.2$$

Question 8

Given $$f(x) = 7 - 14 \ln x$$, find $$f'(x).$$

Accepted Answer

$$f'(x) = -\frac{14}{x}$$

Question 9

Find the derivative of $$f(x) = x^7$ + 6x + 7e^x.$$

Accepted Answer

$$f'(x) = 7x^6$ + 6 + 7e^x$$

Question 10

If $$f(x) = x^2$ + 5x - 3$$, use the four-step process to find $$f'(2).$$

Accepted Answer

$$f'(2) = 9$$

Question 11

Which of the following is the correct implicit differentiation of $$9x^4$$ + $$y^4 = 0$?

Accepted Answer

$$36x^3$$ + $$4y^3$$ \frac{dy}{dx} = 0$$

Question 12

Given $$f(x) = (x^2$ + 7)^2$, find f$$'(x)$$.

Accepted Answer

$$f'(x) = 4x(x^2$ + 7)$$

Question 13

Find the equation of the tangent line(s) to the graph of $$xy - 12x = 0 at$$ $$x = 3.$$

Accepted Answer

$$y = 4x - 12$$

Question 14

If $$f(x) = x^2$ + 5x - 3$$, find $$f'(x)$$ and $$f'(3).$$

Accepted Answer

$$f'(x) = 2x + 5$$, $$f'(3) = 11$$

Question 15

Find the marginal revenue function for $$R(x) = x(16 - 0.2x).$$

Accepted Answer

$$R'(x) = 16 - 0.4x$$

Question 16

Given $$f(x) = x^2$ + 7$$, find $$\lim_{x 	o 3} f(x).$$

Accepted Answer

$$16$$

Question 17

Is the function $$f(x) = \frac{x^2$ - 3}{x - 3}$$ continuous at $$x = 3$$?

Accepted Answer

No, because the denominator is zero at $$x = 3.$$

Question 18

If $$y = e$$^{7x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$7e^{7x}$$

Question 19

Suppose the cost function for a product is $$C(x) = 500 + 0.5x^2$. What is the marginal cost when x = 10$?

Accepted Answer

$$10$$

Business Calculus: Derivatives, Marginal Analysis, and Applications Study Guide

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