Q: Suppose the cost function for producing $$x$$ units of a product is $$C(x) = 2200 + 90x - 0.7x^2$. What is the marginal cost when x = 41$?

$$C'(41) = 90 - 1.4 \times 41 = 32.6$$

Q: A company will sell $$N$$ units of a product after spending $$x$$ thousand dollars in advertising, where $$N = 60x - x^2$ for 5$$ \leq x \leq 30$$. Use differentials to approximate the increase in sales when the advertising budget increases from x = 15$ to x = 16$.

$$dN = (60 - 2x)dx = (60 - 2 \times 15) \times 1 = 30$$

Q: The revenue from selling $$x$$ car seats is $$R(x) = 32x - 0.010x^2$. What is the instantaneous rate of change of revenue at x = 1000$?

$$R'(1000) = 32 - 0.020 \times 1000 = 12$$

Q: A company’s total sales (in millions) $$t$$ months from now is $$S(t) = 0.03t^3$ + 0.3t^2$ + 8t + 7. $$What is the rate of change of sales at $$t = 10$$ months?

$$S'(10) = 0.09 \times 100 + 0.6 \times 10 + 8 = 9 + 6 + 8 = 23$$

Question 1

Suppose the cost function for producing $$x$$ units of a product is $$C(x) = 2200 + 90x - 0.7x^2$. What is the marginal cost when x = 41$?

Accepted Answer

$$C'(41) = 90 - 1.4 	imes 41 = 32.6$$

Question 2

Given $$f(x) = \dfrac{x^2$ - x - 72}{x - 9}$$, find $$\displaystyle \lim_{x 	o 9}.$$

Accepted Answer

$$17$$

Question 3

A company will sell $$N$$ units of a product after spending $$x$$ thousand dollars in advertising, where $$N = 60x - x^2$ for 5$$ \leq x \leq 30$$. Use differentials to approximate the increase in sales when the advertising budget increases from x = 15$ to x = 16$.

Accepted Answer

$$dN = (60 - 2x)dx = (60 - 2 	imes 15) 	imes 1 = 30$$

Question 4

Find the horizontal and vertical asymptotes of $$f(x) = \dfrac{5x^2$ + 19x - 4}{x^2$ + 3x - 28}.$$

Accepted Answer

Horizontal: $$y = 5$$; Vertical: $$x = -7, x = 4$$

Question 5

Given the piecewise function $$f(x) = \begin{cases} -4x & x \u003c 1 \ -4 & x \geq 1 \end{cases}$$, at which value of $$x is$$ $$f(x)$$ nondifferentiable?

Accepted Answer

$$x = 1$$

Question 6

The revenue from selling $$x$$ car seats is $$R(x) = 32x - 0.010x^2$. What is the instantaneous rate of change of revenue at x = 1000$?

Accepted Answer

$$R'(1000) = 32 - 0.020 	imes 1000 = 12$$

Question 7

Find $$\displaystyle \lim_{x 	o -2} \dfrac{x^2$ - 1}{2 - x}.$$

Accepted Answer

$$\dfrac{3}{4}$$

Question 8

If $$y = x^9$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$9x^8$$

Question 9

Suppose $$m(x) = \dfrac{x + 6}{(x - 6)(x - 7)}. On $$which intervals is $$m(x)$$ continuous?

Accepted Answer

$$(-\infty, 6) \cup (6, 7) \cup (7, \infty)$$

Question 10

Given $$f(x) = 2x^2$, compute $$\displaystyle \lim_{x 	o 0} \dfrac{f(1 + x) - f(1)}{x}$$.

Accepted Answer

$$4$$

Question 11

A company’s total sales (in millions) $$t$$ months from now is $$S(t) = 0.03t^3$ + 0.3t^2$ + 8t + 7. $$What is the rate of change of sales at $$t = 10$$ months?

Accepted Answer

$$S'(10) = 0.09 	imes 100 + 0.6 	imes 10 + 8 = 9 + 6 + 8 = 23$$

Question 12

Given $$f(x) = 14 - x^2$ for x$$ \leq 0$$ and f(x) = 14$$ + $$x^2$$ for $$x \u003e 0$$, what is $$\displaystyle \lim_{x 	o 0} f(x)$$?

Accepted Answer

$$14$$

Question 13

If the profit function is $$P(x) = 20x - 0.02x^2$ - 750$$, what is the marginal average profit at $$x = 50$$?

Accepted Answer

$$P_A'(50) = 0.28$$

Question 14

Find the derivative of $$y = 34 + 11x - x^3$.

Accepted Answer

$$dy = 11 - 3x^2$

Question 15

A spherical bird egg has an inner radius of $$4 mm $$and an outer radius of $$4.3 mm. $$Use differentials to approximate the volume of the shell. (Recall $$V(r) = \dfrac{4}{3}\pi r^3$.)

Accepted Answer

$$\Delta V \approx 4\pi r^2$ \Delta r = 4\pi (4)^2$ (0.3) = 60.32 mm$^3$$

Question 16

Given $$C(x) = 176 + 0.9x$$, what is the marginal cost function?

Accepted Answer

$$C'(x) = 0.9$$

Question 17

If $$f(x) = -2t^2$ + 4t + 7$$, what is $$f'(t)$$?

Accepted Answer

$$f'(t) = -4t + 4$$

Question 18

Given $$f(x) = 2 + x$$ for $$x \u003c -2$$ and $$f(x) = 4 - x$$ for $$x \geq -2$$, at which $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

$$x = -2$$

Question 19

Suppose $$R(x) = x(28 - 0.03x). $$What is the marginal revenue function $$R'(x)$$?

Accepted Answer

$$R'(x) = 28 - 0.06x$$

Business Calculus Chapter 2 Review – Step-by-Step Study Guidance

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