Q: Suppose the cost function for producing $$x$$ units of a product is given by $$C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when x = 10$?

$$C'(10) = 20 + 2 \times 0.5 \times 10 = 30$$

Question 1

Which of the following best describes the primary focus of QMS 702: Calculus for Business at Toronto Metropolitan University?

Accepted Answer

To introduce basic calculus concepts with applications in business and management.

Question 2

Suppose the cost function for producing $$x$$ units of a product is given by $$C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when x = 10$?

Accepted Answer

$$C'(10) = 20 + 2 	imes 0.5 	imes 10 = 30$$

Question 3

If the demand function for a product is $$p = 100 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold, what is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 4

A company’s profit function is $$P(x) = -3x^2$ + 60x - 100. $$What is the production level $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 5

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It relates differentiation and integration, stating that integration is the reverse process of differentiation.

Question 6

Given the function $$f(x) = e$$^{2x}$$, what is the derivative f$$'(x)$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 7

A business wants to minimize the cost of packaging a cylindrical can with a fixed volume. Which calculus concept is most appropriate to solve this problem?

Accepted Answer

Optimization using derivatives

Question 8

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

Accepted Answer

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

Question 9

A company’s revenue is increasing at a rate given by $$R'(t) = 500e$$^{0.05t}$$ dollars per month, where t$ is in months. What is the total increase in revenue over the first 12 months?

Accepted Answer

$$\$$int_0$$^{12} 500e$$^{0.05t}$$ dt$$

Question 10

Which of the following is the correct application of the chain rule for $$y = (3x^2$ + 2x)^5$?

Accepted Answer

$$y' = 5(3x^2$ + 2x)^4$ \cdot (6x + 2)$$

Question 11

If the supply function is $$S(x) = 10x + 100$$ and the demand function is $$D(x) = 200 - 5x$$, what is the equilibrium quantity?

Accepted Answer

$$x = 20$$

Question 12

Which of the following best describes a continuous function?

Accepted Answer

A function with no breaks, jumps, or holes in its graph.

Question 13

Evaluate the definite integral $$\$$int_0$$^2 (4x + 1) dx.$$

Accepted Answer

$$12$$

Question 14

A company’s cost function is $$C(x) = 1000 + 25x. $$What does the term $$25x$$ represent?

Accepted Answer

The variable cost, which depends on the number of units produced.

Question 15

If $$f(x) = x^3$ - 6x^2$ + 9x + 15$$, at what value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 16

Which of the following is an example of a real-world application of improper integrals in business?

Accepted Answer

Calculating the present value of a perpetuity.

Question 17

If $$f(x) = \frac{1}{x}$$, what is $$\$$int_1$$^e f(x) dx$$?

Accepted Answer

$$1$$

Question 18

A business uses the function $$S(t) = 1000e$$^{-0.1t}$$ to model the amount of a chemical in storage over time t$ (in years). What does the derivative S$$'(t)$$ represent in this context?

Accepted Answer

The rate at which the chemical is decreasing in storage at time $$t.$$

Question 19

If $$f(x) = x^2$ and g(x) = 3x + 1$, what is the derivative of the product f(x)g(x$$)$$?

Accepted Answer

$$f'(x)g(x) + f(x)g'(x) = 2x(3x + 1) + x^2$(3)$$

QMS 702: Calculus for Business – Course Syllabus and Study Guide

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