### [試題] 107-2 李瑩英 微積分3 第一次小考

1. Assume that f(x,y) = xy^2/(x+y^4) , (x,y) =/= (0,0)
0 , (x,y) = (0,0)
(a) (3pts) Determine and explain whether f(x,y) is continuous.
(b) (10pts) Compute f_x(x,y) and f_y(x,y) if it exists.
(c) (3pts) Is f(x,y) differentiable at (0,0)? Give your reasons.
2. Assume that f(x,y,z) = x^3*(y^2+z^2)^(1/2),
(a) (8pts) Determine the level surface of f that pass through (2,3,4) and
its tangent plane at that point.
(b) (6pts) Use the linear approximation of f at (2,3,4) to estimate
(1.98)^3*(3.01^2+3.97^2)^(1/2)
3. (10pts) Find z_x and z_y (z對x和對y的偏導數) for yz+zlny = z^2
4. (12pts) Use the chain rule to compute z_s and z_t ,
where z = arctan(x^2+y^2) , s=lnt , y=te^s.
5. Assume that f(x,y) = ln(1+x^2+y^2)-xy^2
(a) (8pts) Find the directional derivative of f at point (1,1) in the
direction of (1,-1).
(b) (7pts) In which direction will f increase the most rapidly at(1,1)?
What is the rate?
6. f(x,y) = 4xy^2-x^2y^2-xy^3 , D is the closed triangular region in the xy
plane with vertices (0,0) , (0,6) and (6,0).
(a) (10pts) Determine the critical points of f in the interior of D and
their types.
(b) (8pts) Find the absolute maximum and minimum values of f.
7. (15pts) A pentagon is formed by placing an isosceles triangle on a
rectangle, as shown in the figure. If the pentagon has fixed
perimeter P, find the lengths of the sides of the pentagon that
maximize the area of the pentagon.
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p.s. 示意圖我盡力了QQ 就是一個等腰三角形放在一個長方形上面，給定一個ψ角