TEST 21 System Mathematica

Problem 1 : Let   f (x) = f (x, y) = xy/((xy + 1)^(1/2) - 1)  ( ... ; the  region  x∈[0, 100],   y∈[0, 200] . 

f[x_, y_] := (x * y)/((x * y + 1)^(1/2) - 1)

(a) Calculate ∂f/∂x , ∂f/∂y , ∂^2f/∂x^2

In[49]:=

∂_x f[x] Simplify[%]

Out[49]=

-(x y^2)/(2 (1 + x y)^(1/2) (-1 + (1 + x y)^(1/2))^2) + y/(-1 + (1 + x y)^(1/2))

Out[50]=

y/(2 (1 + x y)^(1/2))

In[55]:=

∂_x f[x] ; Simplify[%]

Out[55]=

-y^2/(4 (1 + x y)^(3/2))

In[53]:=

∂_ (x, x) f[x] Simplify[%]

Out[53]=

(x y^3)/(2 (1 + x y) (-1 + (1 + x y)^(1/2))^3) + (x y^3)/(4 (1 + x y)^(3/2) (-1 + (1 + x y)^(1/2))^2) - y^2/((1 + x y)^(1/2) (-1 + (1 + x y)^(1/2))^2)

Out[54]=

-y^2/(4 (1 + x y)^(3/2))

(b)   Find    lim f (x, y)   when  x0, y0

r1 = Limit[Limit[f[x, y], x0], y0]

2

r1 = Limit[f[x, y], x0]

2

(c) Draw a graph of the function  f (x, y)   in  the  region  x∈[0, 100],   y∈[0, 200] .

In[33]:=

RowBox[{Plot3D, [, RowBox[{f[x, y], ,, , RowBox[{{, RowBox[{x, ,, 0.0001, ,, , 100}], }}], ,, RowBox[{{, RowBox[{y, ,, 0.0001, ,, , 200}], }}]}], ]}]

[Graphics:HTMLFiles/index_20.gif]

Out[33]=

⁃SurfaceGraphics⁃

Problem 2 :  (a) Solve the system 3 2  ...       x - y = 2xy 2 x + 3y = 4 (b) Calculate х_1 + у_1

In[34]:=

f1 = x^3 - y^2 - 2 * x * y RowBox[{f2, , =, RowBox[{x^2, +, 3 * y, -, 4.}]}]

Out[34]=

x^3 - 2 x y - y^2

Out[35]=

RowBox[{RowBox[{-, 4.}], +, x^2, +, 3 y}]

In[37]:=

sol = Solve[{f10, f20}, {x, y}]

Out[37]=

RowBox[{{, RowBox[{RowBox[{{, RowBox[{RowBox[{y, , RowBox[{-, 77.86}]}], ,, RowBox[{x, ... Box[{RowBox[{y, , 1.13723}], ,, RowBox[{x, , RowBox[{-, 0.767005}]}]}], }}]}], }}]

In[39]:=

Print[x + y =, sol_〚1〛_〚1〛_〚2〛 + sol_〚1〛_〚2〛_〚2〛]  1 1

RowBox[{x + y =, , RowBox[{-, 62.4464}]}] 1 1

Created by Mathematica  (February 11, 2008)