Question 1

Which of the following best describes the Fundamental Theorem of Calculus, Part 2?

Accepted Answer

It states that $$\int_a^b f(x) dx = F(b) - F(a)$$, where $$F(x) is $$any antiderivative of $$f(x).$$

Question 2

A particle moves along the x-axis with velocity $$v(t) = -(t-2)^2$ + 1$$ for $$0 \leq t \leq 5. $$For which intervals is the particle moving to the left?

Accepted Answer

$$t \in (0,1) \cup (3,5)$$

Question 3

Given $$v(t) = -(t-2)^2$ + 1$$, and $$s(0) = 0$$, what is the position function $$s(t)$$?

Accepted Answer

$$s(t) = -\frac{t^3$}{3} + 2t^2$ - 3t$$

Question 4

For the function $$f(x) = x^2$ and g(x) = 1$, what is the area between the curves from x = -1$ to x = 1$?

Accepted Answer

$$\frac{4}{3}$$

Question 5

Evaluate the definite integral $$\$$int_0$$^{60} 0.2 \sin\left(\frac{2\pi}{4} t\right) + 0.2 \, dt to $$model the total air inhaled in 60 seconds. What is the result?

Accepted Answer

$$12 L$$

Question 6

If $$u = x+1$$, what is $$\int (x+1$$)^{1000}$$ dx$$?

Accepted Answer

$$\frac{(x+1$$)^{1001}$$}{1001} + C$$

Question 7

Evaluate $$\$$int_1$$^3 5 \sin(1-2x) dx$$ using substitution.

Accepted Answer

$$\frac{5}{2} [\cos(-5) - \cos(-1)]$$

Question 8

What is the average value of a continuous function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

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

Question 9

Given $$\int x e$$^{1-2x^2}$$ dx$$, what is the correct substitution and resulting integral?

Accepted Answer

Let $$u = 1-2x^2$, du = -4x dx$, so dx$$ = -\frac{du}{4x}$$, resulting in $$-\frac{1}{4} \int e^u du$$

Question 10

If the birth rate and death rate of a population are $$b(t) = 2200e$$^{0.024t}$$ and d(t$$) = $$1460e^{0.018t}$$, what does $$\$$int_0$$^{10} [b(t) - d(t)] dt$$ represent?

Accepted Answer

The net population increase over 10 years

Question 11

Evaluate $$\$$int_2$$^3 \frac{3}{1-x^2$} dx$$ using substitution.

Accepted Answer

$$-\frac{3}{2} \ln(8)$$

Question 12

For the area between $$y = x+1$$ and $$y = e$$^{-x}$$ on $$[-1,1]$$, which of the following is the correct setup for the area?

Accepted Answer

$$\int_{-1}^0 [e$$^{-x}$$ - (x+1)] dx + \$$int_0$$^1 [(x+1) - e$$^{-x}$$] dx$$

Question 13

Given $$\int \frac{x+1}{u} (-du)$$ with $$u = 5-x$$, what is the result?

Accepted Answer

$$5-x-\ln|5-x| + C$$

Question 14

A fish population's growth rate is $$G'(t) = \frac{60000 e$$^{-0.06t}$$}{1+5e$$^{-0.06t}$$}$$ with $$G(0) = 25000. $$What is the predicted biomass after 20 years?

Accepted Answer

$$\approx 41665.87 kg$$

Question 15

Which of the following is the correct formula for the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] dx$$

Question 16

If $$f(x) is $$continuous on $$[a, b]$$, which of the following statements is true about $$\int_a^b f(x) dx$$?

Accepted Answer

It represents the net area between $$f(x)$$ and the $$x-$$axis from $$x=a to$$ $$x=b.$$

Question 17

Which substitution is appropriate for evaluating $$\int \sqrt{x^4$+1} 4x^3$ dx$$?

Accepted Answer

Let $$u = x^4$ + 1$$, $$du = 4x^3$ dx$$

Question 18

If $$\int f(x) dx = F(x) + C$$, what does the constant $$C$$ represent?

Accepted Answer

The family of all antiderivatives of $$f(x)$$

Definite and Indefinite Integrals, Area Between Curves, and Substitution Techniques

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