TEST 31 System Mathematica

Problem 1 : Let   g (x, y) = (x^2 + y^2)/(x^4 + y^4)  (a) Calcu ... ;the  region  x∈[-100, 100],   y∈[-200, 200] . 

In[7]:=

f1[x_, y_] := (x^2 + y^2)/(x^4 + y^4)

(a)   Calculate ∂f/∂x , ∂^2f/(∂x ∂y )

In[8]:=

∂_x f1[x, y] Simplify[%]

Out[8]=

-(4 x^3 (x^2 + y^2))/(x^4 + y^4)^2 + (2 x)/(x^4 + y^4)

Out[9]=

-(2 (x^5 + 2 x^3 y^2 - x y^4))/(x^4 + y^4)^2

In[13]:=

∂_ (x, y) f1[x, y] Simplify[%]

Out[13]=

(32 x^3 y^3 (x^2 + y^2))/(x^4 + y^4)^3 - (8 x^3 y)/(x^4 + y^4)^2 - (8 x y^3)/(x^4 + y^4)^2

Out[14]=

-(8 x y (x^6 - 3 x^4 y^2 - 3 x^2 y^4 + y^6))/(x^4 + y^4)^3

(b) Find   lim f (x, y)   when  x∞, y∞

In[16]:=

Limit[Limit[f1[x, y], x∞], y∞]

Out[16]=

0

(c) Draw a graph of the function  f (x, y)   in  the  region  x∈[-1, 1],   y∈[-1, 1] . 

In[22]:=

Plot3D[f1[x, y], {x, -1, 1}, {y, -1, 1}]

[Graphics:HTMLFiles/index_16.gif]

Out[22]=

⁃SurfaceGraphics⁃

In[23]:=

RowBox[{f1, [, RowBox[{1000, ,, 1000.}], ]}]

Out[23]=

1.*10^-6

Problem 2 : Calculate ∫_1^2∫_1/y^yx^yxy

In[24]:=

∫_1^2∫_1/y^yx^yxy

Out[24]=

∫_1^2 (y^(-1 - y) (-1 + y^(2 + 2 y)))/(1 + y) y

In[25]:=

N[∫_1^2∫_1/y^yx^yxy, {10, 10} ]

Out[25]=

1.071915896

In[26]:=

NIntegrate[∫_1/y^yx^yx, {y, 1, 2}, AccuracyGoal10]

Out[26]=

1.07192

RowBox[{1.07192, }]

Created by Mathematica  (February 11, 2008)