Q: Given the function $$f(x) = \frac{1}{x}$$, which of the following statements about its domain and monotonicity is correct?

The domain is $$x \in \mathbb{R}, x \neq 0$$ and $$f(x) is $$strictly decreasing on both $$(-\infty, 0)$$ and $$(0, \infty).$$

Q: If $$A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$, what is the product $$AB$$?

$$\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$$

Question 1

Which of the following is the correct derivative of the function $$f(x) = e$$^{2x}$$ \cdot \ln(x)$$ with respect to $$x$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$ \cdot \ln(x) + \frac{e$$^{2x}$$}{x}$$

Question 2

A pharmaceutical company invests $$10,000 at an annual interest rate of 5\% compounded quarterly. What will be the value of the investment after 3 years? (Use the compound interest formula A = P$$ \left(1 + \frac{r}{n}\$$right)^{nt}$)

Accepted Answer

$$A = 10,000 \left(1 + \frac{0.05}{4}\$$right)^{12}$$ \approx 11,614$$

Question 3

Which of the following statements about the monotonicity of a function $$f(x) is $$correct?

Accepted Answer

If $$f'(x) \u003e 0$$ for all $$x in an $$interval, then $$f(x) is $$strictly increasing on that interval.

Question 4

Given the function $$f(x) = \frac{1}{x}$$, which of the following statements about its domain and monotonicity is correct?

Accepted Answer

The domain is $$x \in \mathbb{R}, x 
eq 0$$ and $$f(x) is $$strictly decreasing on both $$(-\infty, 0)$$ and $$(0, \infty).$$

Question 5

A company wants to minimize the cost function $$C(q) = 100 + 20q + q^2$. What is the output q$ that minimizes the cost?

Accepted Answer

$$q = -10$$

Question 6

If $$A = \begin{pmatrix} 2 & 1 \ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$, what is the product $$AB$$?

Accepted Answer

$$\begin{pmatrix} 2 & 1 \ 3 & 4 \end{pmatrix}$$

Question 7

Suppose the demand function for a product is $$Q = 100 - 2P. $$What is the price elasticity of demand when $$P = 20$$?

Accepted Answer

$$\varepsilon = -0.5$$

Question 8

Which of the following is the correct formula for the present value $$PV of an $$annuity that pays $$A$$ per period for $$n$$ periods at an interest rate $$r$$ per period?

Accepted Answer

$$PV = A \cdot \frac{1 - (1 + r$$)^{-n}$$}{r}$$

Question 9

Given the function $$f(x) = x^3$ - 3x^2$ + 4$$, at which $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 10

A company’s revenue function is $$R(q) = 50q - q^2$. What is the marginal revenue when q = 10$?

Accepted Answer

$$MR = 30$$

Question 11

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

Accepted Answer

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

Question 12

Which of the following statements about the geometric interpretation of the derivative is correct?

Accepted Answer

The derivative at a point gives the slope of the tangent line to the function at that point.

Question 13

A bank offers a loan with an annual interest rate of 6\% compounded monthly. What is the effective annual rate (EAR)?

Accepted Answer

$$EAR = (1 + \frac{0.06}{12}$$)^{12}$$ - 1 \approx 0.0617$$

Question 14

If $$f(x) = x^2$ + 2x + 1$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$f''(x) = 2$$

Question 15

Suppose a company’s profit function is $$\pi(q) = -2q^2$ + 40q - 100. $$What is the output $$q$$ that maximizes profit?

Accepted Answer

$$q = 10$$

Question 16

Which of the following is the correct formula for the derivative of $$f(x) = \sqrt{x}$$?

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

A company’s cost function is $$C(q) = 500 + 10q. $$What is the marginal cost?

Accepted Answer

$$MC = 10$$

Question 19

Which of the following is the correct formula for the future value $$FV of a $$principal $$P$$ invested at an annual interest rate $$r$$ compounded $$n$$ times per year for $$t$$ years?

Accepted Answer

$$FV = P \left(1 + \frac{r}{n}\$$right)^{nt}$$

Mathematics and Financial Calculus: Study Notes for Economics Students

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