Q: A company invests $$40,000 at an $$annual rate of $$6\%$$ compounded monthly for $$15$$ years. What is the formula for the future value $$A$$?

$$A = 40,000 \left(1 + \frac{0.06}{12}\$$right)^{12 \times 15}$$

Q: Which of the following is the correct formula for the future value of $$40,000$$ invested at $$6\%$$ compounded continuously for $$15$$ years?

$$A = 40,000 e$$^{0.06 \times 15}$$

Q: For the function $$f(x) = \frac{x^2$ + 2x}{x^2$ - 2x}$$, what is the domain?

$$D = \{x \mid x \neq 0, x \neq 2\}$$

Question 1

A company models its revenue with the function $$R(x) = 3x^2$ + 120x - 1200$$, where $$x is $$the number of units sold. What is the domain of $$R(x) if$$ $$x$$ must be between 0 and 40?

Accepted Answer

$$D = \{x \mid 0 \leq x \leq 40\}$$

Question 2

Given the cost function $$C(x) = 300 + 30x$$, what is the profit function $$P(x) if $$the revenue function is $$R(x) = 3x^2$ + 120x - 1200$$?

Accepted Answer

$$P(x) = 3x^2$ + 90x - 1500$$

Question 3

For the function $$f(x) = \frac{x^2$ + 2x}{x^2$ - 2x}$$, which value of $$x is a $$vertical asymptote?

Accepted Answer

$$x = 2$$

Question 4

A company invests $$40,000 at an $$annual rate of $$6\%$$ compounded monthly for $$15$$ years. What is the formula for the future value $$A$$?

Accepted Answer

$$A = 40,000 \left(1 + \frac{0.06}{12}\$$right)^{12 	imes 15}$$

Question 5

If $$A = P e$$^{rt}$$, what is the expression for the time t$ required to triple an investment at a continuous rate r$?

Accepted Answer

$$t = \frac{\ln 3}{r}$$

Question 6

Given $$\log_8$ x + \log_8$ (x-2) = 2$$, what are the solutions for $$x$$?

Accepted Answer

$$x = 5$$ and $$x = 2$$

Question 7

For the polynomial $$f(x) = -3(x+1)^2$ + 5$$, what are the vertex and maximum value?

Accepted Answer

Vertex at $$(-1, 5)$$, maximum value $$5$$

Question 8

Which of the following is the correct formula for the future value of $$40,000$$ invested at $$6\%$$ compounded continuously for $$15$$ years?

Accepted Answer

$$A = 40,000 e$$^{0.06 	imes 15}$$

Question 9

If $$f(x) = \frac{x^2$ + 2x}{x^2$ - 2x}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 1$$

Question 10

A company wants to know how long it will take for an investment to double at a continuous rate $$r. $$What is the formula for $$r in $$terms of $$t$$?

Accepted Answer

$$r = \frac{\ln 2}{t}$$

Question 11

Given the polynomial $$f(x) = x^5$ - x^3$ - x$$, what is its degree and leading coefficient?

Accepted Answer

Degree $$5$$, leading coefficient $$1$$

Question 12

For the function $$f(x) = \frac{x^2$ + 2x}{x^2$ - 2x}$$, what is the domain?

Accepted Answer

$$D = \{x \mid x 
eq 0, x 
eq 2\}$$

Question 13

If $$A = P(1 + r/n$$)^{nt}$$, what is the value of A$ after 15$ years if P = 40$$,000$$, r = 0.06$, n = 12$?

Accepted Answer

$$A = 40,000 (1 + 0.005$$)^{180}$$

Question 14

Given $$e^{x^2 + 2x + 5} = 20$, what is the solution for x$?

Accepted Answer

$$x = -1 \pm \sqrt{\ln 20 - 5}$$

Question 15

A company’s price-demand function is $$p(x) = 60 - 1.5x$$, and the cost function is $$C(x) = 300 + 30x. $$What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = x(60 - 1.5x)$$

Question 16

For the function $$f(x) = \frac{x^2$ + 2x}{x^2$ - 2x}$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

Does not exist

Question 17

If $$f(x) = -3(x+1)^2$ + 5$$, what is the range of $$f(x)$$?

Accepted Answer

$$(-\infty, 5]$$

Question 18

A company wants to know how long it will take for an investment to triple at $$6\%$$ compounded quarterly. What is the formula for $$t$$?

Accepted Answer

$$t = \frac{\ln 3}{4 \ln(1 + 0.06/4)}$$

Question 19

Given $$f(x) = x^5$ - x^3$ - x$$, which of the following is a zero of $$f(x)$$?

Accepted Answer

$$x = 0$$

Business Calculus Exam Study Guide: Limits, Functions, Polynomials, and Exponential Applications

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