1. The differential equation dy/dx equals x-2/y-2

I .produces a slope field with horizontal tangents at y = 2
II. produces a slope field with vertical tangents at y = 2
III. produces a slope field with columns of parallel segments

A. I only
B. II only
C. I and II only
D. III only

2. Given the table below for selected values of f(x), use 6 right rectangles to estimate the value of the integral from 1 to 10 of f of x dx.

x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
Numerical Answers Expected!

(I got 99 as my answer, but the system marked it wrong. Isn't that the right answer)

3. Which of the following values would be obtained using 10 circumscribed rectangles of equal width (an upper sum) to estimate the integral from 0 to 1 of x^2 dx?

A. 0.275
B. 0.385 <--- My Choice
C. 1.380
D. 2.310

PLEASE Help! I'm really struggling on these problems! If you can, Please provide the explanation of how you'll got to the answer. Thank you in advance and I extremely appreciate this!

Thank you!

#1

I. clearly false, since dy/dx is undefined at y=2
II. therefore, II is true
III. False. For a given value of x, dy/dx depends on y.
See https://www.desmos.com/calculator/p7vd3cdmei
and change g(x,y)

#2
x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
using right-side rectangles, that would clearly be
8*2 + 6*1 + 10*2 + 10*1 + 12*2 + 16*1 = 92
Too bad you didn't show your work ...

#3 Looks good

1. To determine whether the given differential equation produces a slope field with various characteristics, we can examine the equation carefully.

dy/dx = (x-2)/(y-2)

Looking at the denominator, we can see that at y = 2, the differential equation becomes undefined since division by zero is not allowed. Therefore, the slope field will have a vertical tangent at y = 2. Thus, statement II is correct.

Now, if we consider what happens when x = 0 and y = 2, we can see that the numerator is negative (0-2 = -2), and the denominator is also negative (2-2 = 0), resulting in a positive slope. Similarly, for x = 4 and y = 2, the numerator is positive (4-2 = 2), and the denominator is still negative (2-2 = 0), resulting in a negative slope. Therefore, at y = 2, the slope field will have both positive and negative slopes, giving rise to parallel segments. Hence, statement III is also correct.

In summary, both statements II and III are correct. The answer is C. I and II only.

2. To estimate the value of the integral using the given table and six right rectangles, we need to calculate the width of each rectangle first. Since the interval from 1 to 10 is divided into six equal parts, each rectangle will have a width of (10-1)/6 = 1.5.

Now, we can calculate the area of each rectangle by multiplying the width by the corresponding function value:

Rectangle 1: 1.5 * 4 = 6
Rectangle 2: 1.5 * 8 = 12
Rectangle 3: 1.5 * 6 = 9
Rectangle 4: 1.5 * 10 = 15
Rectangle 5: 1.5 * 10 = 15
Rectangle 6: 1.5 * 12 = 18

Finally, we sum up the areas of all six rectangles:

6 + 12 + 9 + 15 + 15 + 18 = 75

Therefore, the estimate for the value of the integral from 1 to 10 of f(x) dx using six right rectangles is 75.

3. To estimate the integral from 0 to 1 of x^2 dx using 10 circumscribed rectangles, we divide the interval into 10 equal parts, resulting in rectangles of equal width.

The width of each rectangle is (1-0)/10 = 0.1.

Now, to estimate the area of each rectangle, we need to find the height. Since we are using circumscribed rectangles (upper sum), we take the maximum value of x^2 within each interval as the height.

For the first rectangle, the maximum value of x^2 in the interval [0, 0.1] is (0.1)^2 = 0.01. Thus, the area of the first rectangle is 0.1 * 0.01 = 0.001.

By repeating this process for all 10 rectangles, we get the areas:

Rectangle 1: 0.001
Rectangle 2: 0.004
Rectangle 3: 0.009
Rectangle 4: 0.016
Rectangle 5: 0.025
Rectangle 6: 0.036
Rectangle 7: 0.049
Rectangle 8: 0.064
Rectangle 9: 0.081
Rectangle 10: 0.1

Finally, we sum up the areas of all 10 rectangles:

0.001 + 0.004 + 0.009 + 0.016 + 0.025 + 0.036 + 0.049 + 0.064 + 0.081 + 0.1 = 0.395

Therefore, the value obtained using 10 circumscribed rectangles to estimate the integral from 0 to 1 of x^2 dx is approximately 0.395. The closest option is B. 0.385.

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.