Question 1

Which of the following is the correct formula for the present value $$PV of a $$continuous income stream $$R(t)$$ over $$[0, T]$$ with a constant interest rate $$r$$?

Accepted Answer

$$PV = \$$int_0$$^T R(t) e$$^{-rt}$$ \, dt$$

Question 2

Evaluate the following definite integral using substitution: $$\$$int_0$$^1 2x e$$^{x^2}$$ \, dx.$$

Accepted Answer

$$e - 1$$

Question 3

A company invests $$\10$$,000$$ at an annual interest rate of 5$$\%$$ compounded continuously. What will the investment be worth after 8$ years?

Accepted Answer

$$10,000 e$$^{0.05 	imes 8}$$

Question 4

Which of the following integrals represents the area between $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$?

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx$$

Question 5

Suppose $$F'(x) = 3x^2$ - 4x + 2. $$What is $$F(x) if$$ $$F(0) = 5$$?

Accepted Answer

$$x^3$$ - $$2x^2 + 2x + 5$

Question 6

A business receives a continuous income stream at a rate of $$R(t) = 5000 e$$^{0.03t}$$ dollars per year for 10$ years. If the interest rate is 4$$\%$$ compounded continuously, what is the present value of this income stream?

Accepted Answer

$$\$$int_0$$^{10} 5000 e$$^{0.03t}$$ e$$^{-0.04t}$$ \, dt$$

Question 7

Which substitution is most appropriate for evaluating $$\int x \sqrt{x^2$ + 1} \, dx$$?

Accepted Answer

$$u = x^2$ + 1$$

Question 8

If $$A(t) is $$the amount in an account after $$t$$ years with continuous compounding at rate $$r$$ and initial deposit $$P$$, which formula is correct?

Accepted Answer

$$A(t) = P e$$^{rt}$$

Question 9

Find the value of $$k so $$that $$\$$int_0$$^1 kx^2$ \, dx = 1.$$

Accepted Answer

$$3$$

Question 10

A loan of $$\20$$,000$$ is to be repaid in 5$ years with equal annual payments at 6$$\%$$ interest compounded annually. Which formula gives the annual payment R$?

Accepted Answer

$$R = 20000 \frac{0.06}{1 - (1 + 0.06$$)^{-5}$$}$$

Question 11

Evaluate $$\$$int_1$$^e \frac{1}{x} \, dx.$$

Accepted Answer

$$1$$

Question 12

Which of the following is the correct indefinite integral for $$\int (4x^3$ - 2x + 7) \, dx$$?

Accepted Answer

$$x^4$$ - $$x^2 + 7x + C$

Question 13

A company wants to accumulate $$\50$$,000$$ in 12$ years by making equal annual payments into an account earning 5$$\%$$ interest compounded annually. Which formula gives the required payment R$?

Accepted Answer

$$R = 50000 \frac{0.05}{(1 + 0.05$$)^{12}$$ - 1}$$

Question 14

Which of the following is the correct application of the Fundamental Theorem of Calculus for $$\int_a^b f(x) \, dx$$?

Accepted Answer

$$F(b) - F(a)$$ where $$F'(x) = f(x)$$

Question 15

If $$\$$int_0$$^2 f(x) \, dx = 5$$ and $$\$$int_2$$^5 f(x) \, dx = 7$$, what is $$\$$int_0$$^5 f(x) \, dx$$?

Accepted Answer

$$12$$

Question 16

A business receives a continuous income stream of $$R(t) = 10000$$ per year for $$15$$ years. If the interest rate is $$3\%$$ compounded continuously, what is the present value?

Accepted Answer

$$\$$int_0$$^{15} 10000 e$$^{-0.03t}$$ \, dt$$

Question 17

Which of the following is the correct formula for the future value $$FV of an $$ordinary annuity with payment $$R$$, interest rate $$i$$, and $$n$$ periods?

Accepted Answer

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

Business Calculus Exam Review: Integration, Applications, and Financial Mathematics

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