Q: If $$\lim_{x \to a^-} f(x) = 2$$ and $$\lim_{x \to a^+} f(x) = 3$$, what can be said about $$\lim_{x \to a} f(x)$$?

$$\lim_{x \to a} f(x)$$ does not exist.

Q: Which of the following is a correct property of limits?

$$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$

Q: If $$f(x) = |x|/x$$, what is $$\lim_{x \to 0^-} f(x)$$ and $$\lim_{x \to 0^+} f(x)$$?

$$\lim_{x \to 0^-} f(x) = -1$$, $$\lim_{x \to 0^+} f(x) = 1$$

Question 1

In the Leontief Input-Output Model, what does the entry $$a_{ij} in $$the Leontief matrix $$A$$ represent?

Accepted Answer

The proportion of the output of sector $$i$$ that is used by sector $$j to $$produce one unit of output in sector $$j.$$

Question 2

Given the Leontief matrix $$A = \begin{bmatrix} 0.2 & 0.3 & 0.2 \ 0.4 & 0.1 & 0.2 \ 0.1 & 0.3 & 0.2 \end{bmatrix}$$ and final demand vector $$D = \begin{bmatrix} 10 \ 5 \ 6 \end{bmatrix}$$, which equation gives the production vector $$X$$?

Accepted Answer

$$X = (I - A$$)^{-1}$$D$$

Question 3

Suppose in a three-sector economy, the Leontief matrix is $$A = \begin{bmatrix} 0.1 & 0.2 & 0.3 \ 0.3 & 0.1 & 0.1 \ 0.2 & 0.3 & 0.2 \end{bmatrix}. If $$sector 2 saves 25% of its inputs from sector 1, what is the new value of $$a_{12}$$?

Accepted Answer

$$0.15$$

Question 4

In the Leontief model, if the final demand vector $$D$$ increases by $$\Delta D$$, how does the production vector $$X$$ change?

Accepted Answer

$$\Delta X = (I - A$$)^{-1}$$\Delta D$$

Question 5

A two-sector economy has the following technical coefficients: to produce one unit of sector 1 requires $$\frac{1}{3}$$ unit of sector 1 and $$\frac{1}{4}$$ unit of sector 2; to produce one unit of sector 2 requires $$\frac{3}{4}$$ unit of sector 1 and no sector 2. If the final demand is 300 units for sector 1 and 500 units for sector 2, which system of equations represents the required production levels $$X_1$$ and $$X_2$$?

Accepted Answer

Question 6

If the Leontief matrix for a three-sector economy is $$A = \begin{bmatrix} 0.4 & 0.5 & 0.3 \ 0.2 & 0.1 & 0.1 \ 0.2 & 0.2 & 0.1 \end{bmatrix}$$ and sector 3 requires 800 units from itself, what is the total output of sector 3?

Accepted Answer

$$8000$$

Question 7

Which of the following is NOT a necessary condition for a function $$f to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 8

Consider the function $$f(x) = \begin{cases} \frac{x^3$ - 1}{x - 1}, & x \u003e 1 \ x + 2, & x \leq 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because both the left and right limits at $$x=1$$ exist and are equal to $$f(1).$$

Question 9

Suppose the Leontief matrix $$A$$ for a three-sector economy is $$A = \begin{bmatrix} 0.1 & 0.2 & 0.3 \ 0.3 & 0.1 & 0.1 \ 0.2 & 0.3 & 0.2 \end{bmatrix}$$ and the final demand vector is $$D = \begin{bmatrix} 39 \ 49 \ 16 \end{bmatrix}. If $$sector 2 saves 25% of its inputs from sector 1, what is the new output of sector 1 (rounded to two decimal places)?

Accepted Answer

$$83.35$$

Question 10

If the sum $$\sum_{i=1}^n a_{ij} \u003c 1$$ for each $$j in $$the Leontief model, what does this imply about the economic system?

Accepted Answer

The total value of inputs required to produce one unit of output in any sector is less than the value of the output itself.

Question 11

Given a function $$f(x) = x^2$ and g(x) = 3x$, what is $$\lim_{x 	o 2} [f(x) + g(x)]$$?

Accepted Answer

$$4 + 6 = 10$$

Question 12

Which of the following is the correct definition of the range of a function $$f: D 	o E$$?

Accepted Answer

The set of all $$y$$ such that $$y = f(x)$$ for some $$x \in D.$$

Question 13

If $$\lim_{x 	o a^-} f(x) = 2$$ and $$\lim_{x 	o a^+} f(x) = 3$$, what can be said about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist.

Question 14

In the Leontief model, if the final demand for sector 1 increases by 3 units, sector 2 by 1 unit, and sector 3 decreases by 2 units, and $$(I - A$$)^{-1}$$ = \begin{bmatrix} 2.56 & 1.66 & 1.04 \ 0.65 & 1.56 & 0.39 \ 0.71 & 0.71 & 1.43 \end{bmatrix}$$, what is the change in output for sector 1?

Accepted Answer

$$7.27$$

Question 15

Which of the following is a correct property of limits?

Accepted Answer

$$\lim_{x 	o a} [f(x)g(x)] = \lim_{x 	o a} f(x) \cdot \lim_{x 	o a} g(x)$$

Question 16

If $$f(x) = |x|/x$$, what is $$\lim_{x 	o 0^-} f(x)$$ and $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\lim_{x 	o 0^-} f(x) = -1$$, $$\lim_{x 	o 0^+} f(x) = 1$$

Question 17

Suppose in a three-sector Leontief model, the production level of sector 2 is 150, and it requires 30 units from sector 1. If $$a_{12} is $$the technical coefficient from sector 1 to sector 2, what is its value?

Accepted Answer

$$0.2$$

Question 18

Which of the following best describes the graph of a function $$f$$?

Accepted Answer

The set of all ordered pairs $$(x, f(x))$$ where $$x \in D.$$

Leontief Input-Output Model and Functions of One Variable: Study Notes for College Algebra

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