Demonstration Tours - A Survey of MuPAD

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Calculus Consider the rational function >> f:= (x-1)^2/(x-2)+a: for real x and a. To analyze f first without simply plot its graph at a certain interval we look at some characteristics: >> def:= discont( f, x ); {2} See discont if you want to have more details. The zeros of f can be computed with solve: >> ns:= solve( f,x ); { 2 1/2 2 1/2 } { a (4 a + a ) a (4 a + a ) } { - - - ------------- + 1, - - + ------------- + 1 } { 2 2 2 2 } Now, let us ask for extremal points: >> f1:= diff( f, x ); 2 2 x - 2 (x - 1) ------- - -------- x - 2 2 (x - 2) >> ep:= solve( f1=0, x ); {1, 3} >> diff( f, x, x ): eval(subs( %, x=1 )), eval(subs( %, x=3 )); -2, 2 We see from the result that f has a maximum at x=1 and a minimum at x=3. The corresponding values are: >> maximum:= subs( f,x=1 ); minimum:= subs( f,x=3 ); a a + 4 Let us finally ask for the asymptotic behaviour of f. f has a singularity at x=2 where it tends to -infinity from the left and infinity from the right: >> limit( f, x=2, Left ), limit( f, x=2, Right ); -infinity, infinity For large x, the function f is asymptotically equal to the linear term x: >> series( f,x=infinity ); / 1 \ 1 2 4 | -- | x + a + - + -- + -- + O| 4 | x 2 3 \ x / x x You can have a look at a graph of this function for some a with the function plotfunc: >> a:= 0: plotfunc( f,x=-5..5 );