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7. (12 points) Evaluate the double integral dA where R is the square with vertices (0,1), (1,0), (2,1), (1,2).

其他申論題

4. (8 points) Let . Find (f-1)'(0).

5. (12 points) Consider the function given by f(x) = Find the interval of convergence for f(x).

6. (12 points) Find all local maxima, local minima, and saddle points of the function f(x, y) = 2x3 + 3xy + 2y3.

(a) (6 points) Compute fx(0,0) and fy(0,0).

(b) (10 points) Show that f is not continuous at (0,0).

1. (10 分) Use Newton's method to find the x-coordinate of the point where the curve y = x² - 2x crosses the horizontal line y = 1. Specifically apply Newton's method with the starting value x0 = 1 to get the first approximation x₁ = ? and then the second approximation x₂ = ? and omit all the other approximations xₙ, n ＞ 2.

2. (10 分) Find the length of the parameterized curve x = , 0 ≤ t ≤ 3.

(a) (10 分) Finddx.

(b) (10 分) Solve the differential equation .

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