Question 1

Suppose the demand function for a product is $$D(q) = 100 - 2q$$ and the market price is $$p_0 = 60$. If the equilibrium quantity is q_0$ = 20$$, what is the consumers' surplus? (Use the formula $$\$$int_0$$^{q_0$} [D(q) - p_0$] \, dq.)$$

Accepted Answer

$$\800$

Question 2

Evaluate the improper integral $$\$$int_1$$^{\infty} \frac{1}{x^2$} \, dx. $$Does it converge or diverge?

Accepted Answer

It converges to $$1.$$

Question 3

Find the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx = 4/3$$

Question 4

Which of the following integrals would you solve using integration by parts?

Accepted Answer

$$\int x e^x \, dx$$

Question 5

A company receives a continuous income at a rate of $$f(t) = 1000 e$$^{0.05t}$$ dollars per year for T = 10$ years. If the interest rate is r = 0.05$, what is the present value of this income stream? (Use $$\$$int_0$$^{T} f(t) $$e^{-rt}$$ \, dt$$.)

Accepted Answer

$$1000 \$$int_0$$^{10} e$$^{0.05t}$$ e$$^{-0.05t}$$ \, dt = 1000 	imes 10 = 10000$$

Question 6

Evaluate $$\$$int_0$$^1 3x^2$ \, dx.$$

Accepted Answer

$$1$$

Question 7

Which of the following is the correct formula for the average value of a function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) \, dx$$

Question 8

Given $$\$$int_0$$^4 (8 - 2x) \, dx$$, which integration technique is most appropriate?

Accepted Answer

Basic rules (power rule)

Question 9

A Riemann sum using right endpoints for $$f(x) = x^2$ on $$[0,2]$$ with n=2$ subintervals is:

Accepted Answer

$$(\Delta x)[f(1) + f(2)] = 1[1 + 4] = 5$$

Question 10

Evaluate the indefinite integral $$\int (2x + 3) \, dx.$$

Accepted Answer

$$x^2 + 3x + C$

Question 11

Which of the following is an example of an improper integral?

Accepted Answer

$$\$$int_1$$^{\infty} \frac{1}{x} \, dx$$

Question 12

If $$S(q) = 20 + 3q is $$the supply function and $$p_0 = 35$, q_0$ = 5$$, what is the producers' surplus? (Use $$\$$int_0$$^{q_0$} [p_0$ - S(q)] \, dq.)$$

Accepted Answer

$$\$$int_0$$^5 [35 - (20 + 3q)] \, dq = 37.5$$

Question 13

Which of the following integrals would you solve using substitution?

Accepted Answer

$$\int 2x \cos(x^2$) \, dx$$

Question 14

Evaluate $$\$$int_0$$^1 e$$^{-2x}$$ \, dx.$$

Accepted Answer

$$\frac{1}{2}(1 - e$$^{-2}$$)$$

Question 15

If $$f(x) is $$negative on $$[a, b]$$, what does $$\int_a^b f(x) \, dx$$ represent?

Accepted Answer

The signed area below the x-axis (negative value)

Question 16

Which of the following is the correct formula for the accumulated amount of a money flow $$f(t) at $$interest rate $$r$$ over $$[0, T]$$?

Accepted Answer

$$e^{rT}$$ \$$int_0^T f(t$$) $$e^{-rt}$$ \, dt$$

Question 17

Evaluate $$\int x e^x \, dx$$ using integration by parts.

Accepted Answer

$$x e^x - e^x + C$$

Question 18

Which of the following is the correct expression for $$\Delta x in a $$Riemann sum for $$n$$ subintervals on $$[a, b]$$?

Accepted Answer

$$\Delta x = \frac{b-a}{n}$$

Question 19

Which of the following integrals diverges?

Accepted Answer

$$\$$int_1$$^{\infty} \frac{1}{x} \, dx$$

Integration Techniques and Applications in Business Calculus (Chapters 7.1–8.4)

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