Question 1

Suppose the cost function for a company is $$C(x) = 180x + 200$$ for $$0 \u003c x \u003c 20. $$What does the definite integral $$\$$int_5$$^{10} 180\,dx$$ represent in this context?

Accepted Answer

The increase in cost when production increases from 5 to 10 units.

Question 2

Given the marginal cost function $$C'(x) = 180$$, what is the area under the curve from $$x = 5 to$$ $$x = 10$$?

Accepted Answer

$$180 	imes (10 - 5) = 900$$

Question 3

If $$C(x) = -7.5x^2$ + 305x + 625$$, what is the marginal cost function $$C'(x)$$?

Accepted Answer

$$-15x + 305$$

Question 4

The Fundamental Theorem of Calculus states that if $$F is an $$antiderivative of $$f on$$ $$[a, b]$$, then $$\int_a^b f(x)\,dx$$ equals:

Accepted Answer

$$F(b) - F(a)$$

Question 5

A management service found that maintenance costs increase at a rate of $$M'(x) = 90x^2$ + 5,000. $$What definite integral represents the total maintenance cost from the end of year 2 to the end of year 7?

Accepted Answer

$$\$$int_2$$^7 (90x^2$ + 5,000)\,dx$$

Question 6

Evaluate $$\$$int_0$$^1 (2x + 4)\,dx$$ using the Fundamental Theorem of Calculus.

Accepted Answer

$$[x^2$ + 4x]_0^1$ = (1 + 4) - (0 + 0) = 5$$

Question 7

Which of the following best describes the relationship between the area under the marginal cost curve and the change in total cost?

Accepted Answer

The area under the marginal cost curve from $$x = a to$$ $$x = b$$ equals the change in total cost from $$x = a to$$ $$x = b.$$

Question 8

Find the average value of $$f(x) = 6x^2$ - 2x$$ over the interval $$[-3, 2].$$

Accepted Answer

$$\frac{1}{2 - (-3)} \int_{-3}^2 (6x^2$ - 2x)\,dx$$

Question 9

Given the supply equation $$p = S(x) = 10e$$^{0.05x}$$, what is the average price over the supply interval $$[25, 35]$$?

Accepted Answer

$$\frac{1}{35 - 25} \int_{25}^{35} 10e$$^{0.05x}$$\,dx$$

Question 10

Which substitution is appropriate for evaluating $$\$$int_0$$^1 \frac{1}{2x + 4}\,dx$$?

Accepted Answer

Let $$u = 2x + 4$$, then $$du = 2dx$$

Question 11

A definite integral $$\int_a^b f(x)\,dx is $$interpreted as:

Accepted Answer

The net area between the graph of $$f$$ and the $$x-$$axis from $$x = a to$$ $$x = b.$$

Question 12

If $$\int_{1.5}^{4.3} \frac{x}{\ln(x)}\,dx is $$evaluated numerically to be approximately $$8.017$$, what does this value represent?

Accepted Answer

The net area under $$f(x) = \frac{x}{\ln(x)}$$ from $$x = 1.5 to$$ $$x = 4.3.$$

Question 13

Which of the following is NOT a correct application of the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f(x)\,dx = F(a) + F(b)$$ where $$F'(x) = f(x)$$

Question 14

If $$C(x) is a $$cost function and $$C'(x) is $$its marginal cost, what does $$\$$int_5$$^{10} C'(x)\,dx$$ compute?

Accepted Answer

The change in total cost as production increases from 5 to 10 units.

Question 15

Which of the following best describes an antiderivative?

Accepted Answer

A function whose derivative is the given function.

Chapter 5: Integration – The Fundamental Theorem of Calculus and Applications

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