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研究所、轉學考(插大)-微積分
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95年 - 95 國立暨南大學轉學生招生考試試題_微積分_(資工系)#125248
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6. (a) Show that
diverges (Hint: by ratio test)
其他申論題
2. Show that the function g(x,y)= has limiting value 0 as (x,y)→(0,0) along any line through the origin, but lim still does not exist.
#532536
3. Definition: If z is irrational, then by eᶻ we mean the unique number which has logarithm z, i.e. ln = z. Prove ˣ for all real x.
#532537
4. A rod of length L is placed on the x-axis from x=0 to x=L. Find the mass of the rod and the center of mass if the density of the rod varies directly as the distance from the x=0 endpoint of the rod.
#532538
5. Find the volume of the solid generated by revolving the region between y = √x, 0≤x≤1 and y= x², 0 ≤ x ≤ 1, around the line x = -2.
#532539
(b) Let r be a positive number. For what values of r (if any) does converge? (Hint: by root test) (Hint: a. ∀ real x, →1 as n→∞.)
#532541
(a) Find the gradient ∇f(x,y), where f(x,y) = x2+y2.
#532542
(b) At the point (1, 2, 5), in what direction does f increase most rapidly? What is the magnitude of its speed?
#532543
(c) Find the directional derivative of the function f at the point (1, 2) in the direction of the vector 2i-3j.
#532544
(d) Determine the path of steepest descent along the surface z=x2+y2 from the point (1, 2, 5).
#532545
(e) Determine the level curve of f that passes through the point (1, 2).
#532546
diverges (Hint: by ratio test)
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diverges (Hint: by ratio test)