Calculus

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!

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  1. #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

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    oobleck

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