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匿名使用者 發問時間: 科學數學 · 1 0 年前

微積分觀念題懂得人幫忙一下!

以下題目請說明對錯或舉例說明

1. the level curves of z=2x^2+3y^2 are ellipses.

2.if fx(0,0) exists,then g(x)=f(x,0) is continuous at x=0.

3. if lim f(x,y) =L, then lim y→0 f(y,y)=L.

(x,y)→(0,0)

4.if f(x,y)=g(x)h(y),where g and h are continuous for all x and y,respectively,then f is continuous on the whole xy-plane.

5. if f(x,y) and g(x,y) have the same gradient , then they are identical functions.

6.if f is differentiable and ▽f(a,b)=0 ,then the graph of z=f(x,y) has a horizontal tangent plane at (a,b).

7. if ▽f(p0)=0,then f has an extreme value at p0.

以上...!

1 個解答

評分
  • 1 0 年前
    最佳解答

    1. YES

    the level curves are

    2x^2+3y^2=k

    i.e.

    x^2/(k/2) + y^2/(k/3) =1

    which are ellipses.

    2. YES

    g'(0)=f_x(0,0) 存在,故 g 在 0 連續

    3. YES

    路徑法則

    4. YES

    Let G(x,y)=g(x), H(x,y)=h(y)

    Then G, H are contiuous, so is their product GH.

    thus f=gh=GH is continuous.

    5. NO

    f and g differ by a constant

    6. YES

    the tangent plane of the graph z=f(x,y) at (a,b) has normal vector equal to

    ▽f(a,b) + k = k, where k is the upward unit vector in the z axis.

    Hence it is horizontal.

    7. NO. For example, f(x,y)=xy. ▽f(0,0)=(0,0), but f has a saddle point at the origin.

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