Question 1

Which of the following statements best describes the relationship between the graphs shown in the images above, which represent antiderivatives of $$f(x) = x$$?

Accepted Answer

They are vertical translations of each other.

Question 2

Given that $$F(x) is an $$antiderivative of $$f(x)$$, which of the following is always true?

Accepted Answer

$$F'(x) = f(x)$$ for all $$x in $$the domain.

Question 3

If $$F(x)$$ and $$G(x)$$ are both antiderivatives of $$f(x) on an $$interval $$(a, b)$$, what is the relationship between $$F(x)$$ and $$G(x)$$?

Accepted Answer

$$F(x) = G(x) + k$$ for some constant $$k.$$

Question 4

What is the general form of the antiderivative of $$f(x) = x$$?

Accepted Answer

$$F(x) = \frac{1}{2}x^2$ + C$$

Question 5

Which property of indefinite integrals allows you to write $$\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$$?

Accepted Answer

Linearity of integration

Question 6

If the marginal cost of producing $$x$$ units is $$C'(x) = 0.3x^2$ + 2x$$ and the fixed cost is $$3,000$$, what is the cost function $$C(x)$$?

Accepted Answer

$$C(x) = 0.1x^3$ + x^2$ + 3,000$$

Question 7

A curve passes through the point $$(3, 5)$$ and its slope at any point $$x is $$given by $$\frac{dy}{dx} = 2x. $$What is the equation of the curve?

Accepted Answer

$$y = x^2$ + C$$, where $$C = -4$$

Question 8

Which of the following best describes the constant $$C in $$the indefinite integral $$\int f(x) dx = F(x) + C$$?

Accepted Answer

It is the constant of integration.

Question 9

If $$\int 3x^2$ dx = F(x) + C$$, what is $$F(x)$$?

Accepted Answer

$$F(x) = x^3$

Question 10

A radio station wants to increase its daily listeners from 27,000 to 41,000. If the growth rate is modeled by $$S'(t)$$ and the campaign started 16 days ago, what is the best approach to determine how many more days the campaign should last?

Accepted Answer

Integrate $$S'(t) to $$find $$S(t)$$ and solve $$S(t) = 41,000$$ for $$t.$$

Question 11

Which of the following is NOT a property of indefinite integrals?

Accepted Answer

$$\int f(x)g(x) dx = \int f(x) dx \cdot \int g(x) dx$$

Question 12

Which of the following best explains why indefinite integration and differentiation are considered reverse operations?

Accepted Answer

Integration undoes differentiation except for the constant of integration.

Question 13

If $$\int f(x) dx = F(x) + C$$, what happens if you differentiate $$F(x) + C$$?

Accepted Answer

$$f(x)$$

Question 14

Which of the following is the correct indefinite integral for $$\int 4 dx$$?

Accepted Answer

$$4x + C$$

Question 15

If $$F(x) is an $$antiderivative of $$f(x)$$, which of the following is true about the family of antiderivatives?

Accepted Answer

All antiderivatives differ by a constant.

Question 16

Which of the following is the correct indefinite integral for $$\int x^3$ dx$$?

Accepted Answer

$$\frac{1}{4}x^4$ + C$$

Question 17

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

Accepted Answer

The variable of integration is $$x.$$

Question 18

Which of the following is a real-world application of antiderivatives discussed in the study materials?

Accepted Answer

Finding the cost function from marginal cost.

Question 19

If the cost of producing 30 units is required and the cost function is $$C(x) = 0.1x^3$ + x^2$ + 3,000$$, what is $$C(30)$$?

Accepted Answer

$$C(30) = 0.1 	imes 30^3$ + 30^2$ + 3,000$$

Integration: Antiderivatives and Indefinite Integrals

Guida di studio - Domande di pratica

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