Q: A company’s revenue $$R(x)$$ from selling $$x$$ units of a product is given by $$R(x) = 100x - 0.5x^2$. What is the marginal revenue when x = 20$ units?

$$R'(20) = 100 - 1 \times 20 = 80$$

Q: A bank offers continuous compounding on savings at an annual rate of $$6\%. If $$you deposit $$\2000$, how much will you have after 5 years? (Use A$$ = $$Pe^{rt}$)

$$A = 2000e$$^{0.06 \times 5}$$ \approx 2699.72$$

Q: A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the production level $$x$$ that maximizes profit?

$$x = -\frac{b}{2a} = -\frac{40}{2 \times -2} = 10$$

Q: A business’s cost function is $$C(x) = 1000 + 15x + 0.05x^2$. What is the marginal cost when x = 40$?

$$C'(40) = 15 + 0.1 \times 40 = 19$$

Question 1

A company’s revenue $$R(x)$$ from selling $$x$$ units of a product is given by $$R(x) = 100x - 0.5x^2$. What is the marginal revenue when x = 20$ units?

Accepted Answer

$$R'(20) = 100 - 1 	imes 20 = 80$$

Question 2

If the cost function for producing $$x$$ items is $$C(x) = 500 + 20x + 0.1x^2$, what is the average cost per item when x = 50$?

Accepted Answer

$$\frac{C(50)}{50} = \frac{500 + 1000 + 250}{50} = 35$$

Question 3

A bank offers continuous compounding on savings at an annual rate of $$6\%. If $$you deposit $$\2000$, how much will you have after 5 years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$A = 2000e$$^{0.06 	imes 5}$$ \approx 2699.72$$

Question 4

Given the demand function $$p = 50 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold, what is the elasticity of demand when $$x = 10$$?

Accepted Answer

$$E = \left| \frac{p}{x} \cdot \frac{dx}{dp} \right| = \left| \frac{30}{10} \cdot \frac{1}{-2} \right| = 1.5$$

Question 5

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the production level $$x$$ that maximizes profit?

Accepted Answer

$$x = -\frac{b}{2a} = -\frac{40}{2 	imes -2} = 10$$

Question 6

If $$f(x) = e$$^{3x}$$, what is f$$''(x)$$?

Accepted Answer

$$f''(x) = 9e$$^{3x}$$

Question 7

A pharmaceutical company models the spread of a drug in the bloodstream by $$C(t) = 10e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -2e$$^{-0.6}$$ \approx -1.098$$

Question 8

Suppose the supply function for a product is $$S(p) = 5p^2$ + 10$$, where $$p is $$the price. What is the rate of change of supply with respect to price when $$p = 2$$?

Accepted Answer

$$S'(2) = 10p = 20$$

Question 9

A business’s cost function is $$C(x) = 1000 + 15x + 0.05x^2$. What is the marginal cost when x = 40$?

Accepted Answer

$$C'(40) = 15 + 0.1 	imes 40 = 19$$

Question 10

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

Accepted Answer

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

Question 11

A company’s revenue is $$R(x) = 200x - x^2$ and cost is C(x) = 50x + 1000$. What is the break-even quantity?

Accepted Answer

$$x^2 - 150x + 1000 = 0$$ \implies x = 142.96, 7.04$$

Question 12

If $$f(x) = 3x^2$ - 4x + 7$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = 8x - 9$$

Question 13

A company’s profit is maximized when marginal revenue equals marginal cost. If $$R(x) = 120x - x^2$ and C(x) = 40x + 400$, what is the profit-maximizing output?

Accepted Answer

$$MR = MC \implies 120 - 2x = 40 \implies x = 40$$

Question 14

If $$f(x) = \sqrt{4x + 1}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2}{\sqrt{4x + 1}}$$

Question 15

A company’s cost function is $$C(x) = 5000e$$^{0.02x}$$. What is the marginal cost when x = 10$?

Accepted Answer

$$C'(10) = 5000 	imes 0.02 	imes e$$^{0.2}$$ \approx 122.10$$

Question 16

If $$f(x) = \ln(x)$$ and $$g(x) = x^2$, what is the derivative of h(x) = f(g(x$$))$$?

Accepted Answer

$$h'(x) = \frac{2x}{x^2$}$$

Question 17

A company’s revenue is $$R(x) = 50x \ln(x)$$ for $$x \u003e 0. $$What is the marginal revenue when $$x = 2$$?

Accepted Answer

$$R'(2) = 50 \ln(2) + 25 \approx 84.66$$

Question 18

If $$f(x) = x^3$ - 6x^2$ + 9x + 15$$, find all critical points.

Accepted Answer

$$f'(x) = 3x^2$ - 12x + 9 = 0 \implies x = 1, 3$$

Question 19

A company’s cost function is $$C(x) = 1000 + 20x. If $$the company wants to minimize average cost, what is the optimal production level?

Accepted Answer

There is no minimum; average cost decreases as $$x$$ increases.

Question 20

If $$f(x) = \frac{1}{x}$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = -\frac{1}{4}x + \frac{3}{2}$$

Syllabus and Course Structure: Calculus for Business and Social Sciences (MATH-1325)

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