1. To determine the characteristics of the slope field, we need to analyze the given differential equation, dy/dx = (x-2)/(y-2).
First, let's examine the denominator (y-2). When the denominator is equal to zero, division by zero occurs, resulting in an undefined value. Therefore, we must determine the values of y that make y-2 equal to zero. Solving the equation, we find that y = 2.
I. The slope field has horizontal tangents at y = 2: To determine if horizontal tangents exist, we need to analyze the numerator (x-2). When x-2 equals zero, the numerator equals zero, indicating the possible presence of horizontal tangents. Solving the equation x-2 = 0, we find that x = 2. Therefore, the slope field has horizontal tangents where x = 2. Hence, statement I is correct.
II. The slope field has vertical tangents at y = 2: To determine if vertical tangents exist, we need to examine the denominator (y-2). Since the denominator does not depend on x, there are no restrictions on x in the presence of vertical tangents. Therefore, the slope field does not have vertical tangents. Hence, statement II is incorrect.
III. The slope field has columns of parallel segments: To establish if the slope field contains columns of parallel segments, we need to investigate the given differential equation. When a slope field exhibits segments parallel to the x-axis or y-axis, it suggests that the differential equation is not explicit in one of the variables. In this case, the given equation is explicit in both x and y variables, and there are no columns of parallel segments. Hence, statement III is incorrect.
Considering the above analysis, the correct answer is A. I only.
2. To estimate the value of the integral from 1 to 10 of f(x) dx using 6 right rectangles, we'll use the Right Rectangle Rule.
The Right Rectangle Rule estimates the area under the curve by multiplying the width of the rectangles by the height of the function at the right endpoint of each rectangle.
Given the table of selected values of f(x), we can use the formula:
Estimated Integral = width * (f(2) + f(4) + f(6) + f(8) + f(10) + f(12))
width = (10 - 1) / 6 = 1.5
Estimated Integral = 1.5 * (8 + 6 + 10 + 10 + 12 + 16)
= 1.5 * 62
= 93
Therefore, the estimated value of the integral is 93. Since you obtained 99 as your answer, it seems there might have been a calculation error. Double-check your calculations to verify the correct answer.
3. To estimate the integral from 0 to 1 of x^2 dx using 10 circumscribed rectangles, we'll use the Upper Sum method.
The Upper Sum method estimates the area under the curve by drawing rectangles above the curve in such a way that the top-right corner of each rectangle aligns with the curve.
To obtain the estimate, divide the interval [0, 1] into 10 equal sub-intervals, each with a width of 1/10. Then, evaluate the function x^2 at the right endpoint of each interval and multiply it by the width of the interval.
Width of each rectangle = 1/10
Estimate = (1/10) * (f(1/10) + f(2/10) + f(3/10) + ... + f(10/10))
Since f(x) = x^2, the estimate becomes: Estimate = (1/10) * ((1/10)^2 + (2/10)^2 + (3/10)^2 + ... + (10/10)^2)
Evaluating this expression, we find:
Estimate = (1/10) * (1/100 + 4/100 + 9/100 + ... + 100/100)
Now, we need to calculate this sum. Simplifying, we get:
Estimate = (1/10) * (385/100)
= 38.5/100
= 0.385
Therefore, the approximate value obtained using 10 circumscribed rectangles is 0.385. Hence, the correct answer is B. 0.385.