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8. Evaluate the line integral f where C consists of line segments joining successively the points (1,0), (I, 1), (-1, 1) and (-1,0). Answer : __________

其他申論題

4. Evaluate the integral Answer : ____________

5. Find the point on the graph of z = x2 + y2 + 10 nearest the plane x + 2y- z = 0. Answer :__________

6. The derivative of f(x, y) at P0(1, 2) in the direction of i + j is and in the direction -2j is -3. What is the derivative of f in the direction of -i -2j?Answer :__________

7.the area of cap cut from the sphere x2 + y2 + z2 == 2 by the cone z =.Answer :__________

(a) Show that f'(0)=0.(6分)

(b) Is the derivative function f'(x) continuous at x= 0? Explain. (6 分)

(a) Find the limit: (6分)

(b) Let a and b be constants with0 ＜ a ＜ b. Does the sequence {(an +bn)1/n} converges? If it does converge, what is the limit? (6 分)

(a) A number a is called a fixed point of a function f if f(a) = a. Prove that if f'(x)≠1 for all real numbers x, then f has at most one fied point. (6 分)

(b) Show that if converges，thenconverges.(6分)

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