TEST 6 System Mathematica

Problem 1 : Let   f (x) = (x^4 + 1)/(x^3 - x^2 + x + 1)  (a) Sh ... to the graph (e) Calculate the  local minimums and maximums, if they exist . 

f[x_] := (x^4 + 1)/(x^3 - x^2 + x + 1)

(a) Show as a sum with minimal denominators

Apart[f[x]]

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

(b) Find the roots of the denominator

NSolve[Denominator[f[x]] 0]

RowBox[{{, RowBox[{RowBox[{{, RowBox[{x, , RowBox[{RowBox[{0.771845, }], +, Ro ... 14, , }]}]}], }}], ,, RowBox[{{, RowBox[{x, , RowBox[{-, 0.543689}]}], }}]}], }}]

(c) Draw a graph of the function  f (x)   in a suitable interval

gf = Plot[f[x], {x, -6 , 6}]

[Graphics:HTMLFiles/index_12.gif]

⁃Graphics⁃

(d) Find any eventual inclined asymptotes and add them to the graph

k1 = Limit[f[x]/x, x -∞] b1 = Limit[f[x] - k1 * x, x -&# ... ] k2 = Limit[f[x]/x, x∞] b2 = Limit[f[x] - k2 * x, x∞]

1

1

1

1

asymptote = k1 * x + b1gasym = Plot[asymptote, {x, -6, 6}] Show[gf, gasym]

1 + x

[Graphics:HTMLFiles/index_22.gif]

⁃Graphics⁃

[Graphics:HTMLFiles/index_24.gif]

⁃Graphics⁃

(e) Calculate the  local minimums and maximums, if they exist .

FindMaximum[f[x], {x, -2}] FindMinimum[f[x], {x, 0}]

RowBox[{{, RowBox[{RowBox[{-, 0.912306}], ,, RowBox[{{, RowBox[{x, , RowBox[{-, 1.21447}]}], }}]}], }}]

RowBox[{{, RowBox[{0.771751, ,, RowBox[{{, RowBox[{x, , 0.533452}], }}]}], }}]

Problem 2 : Let    g (x, y) = ^(-(x + y)^(1/2))  S ... w a graph in a suitable interval (c) Find ∫_0^∞∫_0^xg (x, y) yx

g[x_, y_] := ^(-(x + y)^(1/2))

RowBox[{(а), , Calculate, , g, RowBox[{(, RowBox[{1, ,, 1.5}], )}]}]

x = 1 ; RowBox[{RowBox[{y, =, 1.5}], ;}] g[x, y]

0.205741

(b) Draw a graph in a suitable interval

Plot3D[g[x, y], {x, 0, 20}, {y, 0, 20}]

[Graphics:HTMLFiles/index_37.gif]

⁃SurfaceGraphics⁃

(c) Find ∫_0^∞∫_0^xg (x, y) yx

x=. ; y=. ; ∫_0^∞∫_0^xg[x, y] yx

6

Created by Mathematica  (February 11, 2008)