Q: If $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix}$$, what is $$A + B$$?

$$\begin{bmatrix} 3 & 2 \\ 4 & 6 \end{bmatrix}$$

Q: Given the matrix $$A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$, what is $$\det(A)$$?

$$2 \times 4 - 1 \times 3 = 5$$

Question 1

A company’s total cost function is given by $$C(x) = 2x^2$ + 75x + 100. $$What is the marginal cost when 10 units are produced?

Accepted Answer

$$C'(10) = 4(10) + 75 = 95$$

Question 2

Suppose the demand and supply equations for a market are $$Q_d = 12 - p$$ and $$Q_s = -4 + 0.5p. $$What is the equilibrium price?

Accepted Answer

$$p = 10$$

Question 3

Which of the following best describes the derivative of a product of two functions $$f(x)$$ and $$g(x)$$?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)$$

Question 4

A business invests $$5000 at an $$annual interest rate of $$6\%$$ compounded monthly for $$3$$ years. What is the maturity value?

Accepted Answer

$$A = 5000(1 + \frac{0.06}{12}$$)^{36}$$

Question 5

Given the revenue function $$R(x) = 15x$$ and the cost function $$C(x) = 5x + 400$$, what is the break-even point?

Accepted Answer

$$x = 40$$

Question 6

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 7

A loan of $$10,000 is to be $$repaid in $$5$$ years with monthly payments at an annual interest rate of $$4\%$$ compounded monthly. Which formula gives the monthly payment $$R$$?

Accepted Answer

$$R = \frac{P \cdot i}{1 - (1 + i$$)^{-n}$$}$$

Question 8

Which of the following is the correct formula for the present value of an ordinary annuity?

Accepted Answer

$$PV = R \left[ \frac{1 - (1 + i$$)^{-n}$$}{i} \right]$$

Question 9

If $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \ 1 & 2 \end{bmatrix}$$, what is $$A + B$$?

Accepted Answer

$$\begin{bmatrix} 3 & 2 \ 4 & 6 \end{bmatrix}$$

Question 10

A company’s profit function is $$P(x) = R(x) - C(x)$$, where $$R(x) is $$the revenue and $$C(x) is $$the cost. If $$R(x) = 20x$$ and $$C(x) = 5x + 100$$, what is the profit when $$x = 10$$?

Accepted Answer

$$P(10) = 20(10) - (5(10) + 100) = 200 - 150 = 50$$

Question 11

Which of the following is the correct formula for the effective annual rate (EAR) if the nominal rate is $$r$$ compounded $$m$$ times per year?

Accepted Answer

$$EAR = \left(1 + \frac{r}{m}\right)^m - 1$$

Question 12

Suppose $$f(x) = \ln(x^2$ + 1). $$What is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 13

A business wants to accumulate $$20,000 in 5 $$years by making equal quarterly deposits into an account paying $$6\%$$ compounded quarterly. Which formula should be used to find the required deposit $$R$$?

Accepted Answer

$$R = \frac{A \cdot i}{(1 + i)^n - 1}$$

Question 14

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

Accepted Answer

$$x = 3$$

Question 15

Which of the following is the correct formula for the sum of the first $$n$$ terms of a geometric series with first term $$a$$ and ratio $$r$$?

Accepted Answer

$$S_n = a \frac{1 - r^n}{1 - r}$$

Question 16

Given the matrix $$A = \begin{bmatrix} 2 & 1 \ 3 & 4 \end{bmatrix}$$, what is $$\det(A)$$?

Accepted Answer

$$2 	imes 4 - 1 	imes 3 = 5$$

Question 17

If $$f(x) = x^2$ e^x$$, what is $$f'(x)$$?

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 18

A company’s cost function is $$C(x) = 1000 + 5x$$, and its revenue function is $$R(x) = 20x. $$What is the minimum number of units $$x$$ that must be sold to break even?

Accepted Answer

$$x = 50$$

Business Calculus: Comprehensive Study Notes

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