From 89a4f3b8ffe13060e83e3a71ad21db0dc010a0e9 Mon Sep 17 00:00:00 2001 From: "Christoph J. 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z=Bc*|lOQWp(S2XzEmT(VV`goBVtO@kmq6$LyJjl`hHTN&|4 zNFh~^@zA@gDV1)sVW?OUDc5IPXwa25l3JEh9jIkdB+`Zr@^AFx(O4peD#v&T;go$F$E0NVTHi@ zFAZ8ILmdSi5jxo-)=wWB!wwot(AkU12%bP$O0?B50fIHPgLo2#a@oJdK|Bcyt>mDl zK~VogBmX*M;_)N`fjW1166v36NmA2gu;p%FKw infinity) b_m &= 1/(m+2-8/m) = 1/(m+2) = 0 \ + + c_m &= (m^2+7)/((m+1)(m-1)) \ + &= (m^2+7)/(m^2-1) \ + lim_(m -> infinity) c_m &= (m^2+7)/(m^2-1) = 1/1 = 1 \ + + => a_m &= (b_m , c_m) => lim_(m -> infinity) a_m = (0, 1) checkmark +$ + +#pagebreak() +=== Exercise 2 @Exercise[4, 2] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Is the sequence +$ + a_m = (sqrt(m+10) - sqrt(m) , sqrt(m+ sqrt(m)) - sqrt(m)) +$ + convergent or divergent? If it converges, determine its limit. +]) + +$ + b_m &= sqrt(m + 10) - sqrt(m) \ + &= ((sqrt(m + 10) - sqrt(m))(sqrt(m + 10) + sqrt(m)))/(sqrt(m + 10) + sqrt(m)) \ + &= (m+10 - m)/(sqrt(m + 10) + sqrt(m)) = 10/(sqrt(m + 10) + sqrt(m)) \ + lim_(m -> infinity) + b_m &= 10/(sqrt(m + 10) + sqrt(m)) = 0 \ \ + + c_m &= sqrt(m+ sqrt(m)) - sqrt(m) \ + &= ((sqrt(m+ sqrt(m)) - sqrt(m))(sqrt(m+ sqrt(m)) + sqrt(m)))/(sqrt(m+ sqrt(m)) + sqrt(m)) \ + &= (m+ sqrt(m) -m)/(sqrt(m+ sqrt(m)) + sqrt(m)) = sqrt(m)/(sqrt(m+ sqrt(m)) + sqrt(m)) \ + &= 1/(sqrt(m +sqrt(m))/sqrt(m) + 1) = 1/(sqrt((m +sqrt(m))/m) + 1) \ + &= 1/(sqrt(1+1/sqrt(m)) + 1) \ + lim_(m -> infinity) + c_m &= 1/(sqrt(1+1/sqrt(m)) + 1) = 1/2 \ + + => a_m &= (b_m , c_m) => lim_(m -> infinity) a_m = (0, 1/2) checkmark +$ + +#pagebreak() +=== Exercise 3 @Exercise[4, 3] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Investigate whether the real function +$ + f: RR^2 -> RR \ + + (x_1, x_2) |-> cases( + (x_1^4 - x_2^4)/(x_1^2 + x_2^2) + quad &"for" (x_1, x_2) &!= (0,0), + 0 quad &"for" (x_1, x_2) &= (0,0), + ) +$ + is continuous in $(0, 0)$. +]) + +$ + "let" x &= x_1; y = x_2 \ + + f(x,y) &= (x^4 - y^4)/(x^2 + y^2) \ + &= ((x^2-y^2)(x^2+y^2))/(x^2 + y^2) \ + &= x^2-y^2 -> 0 "für" (x,y) -> (0,0) \ + &"Überall stetig" checkmark +$ + +#pagebreak() +=== Exercise 4 @Exercise[4, 4] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Investigate whether the real function +$ + f: RR^2 -> RR \ + + (x_1, x_2) |-> cases( + (x_1^3 + x_2^3)/(x_1^2 + x_2^2) + quad &"for" (x_1, x_2) &!= (0,0), + 0 quad &"for" (x_1, x_2) &= (0,0), + ) +$ + is continuous in $(0, 0)$. Hint: Use polar coordinates. +]) + +$ + "let" x &= x_1; y = x_2 \ + x &= r cos(phi); y = r sin(phi) \ + + f(x,y) &= (r^3 cos(phi)^3 + r^3 sin(phi)^3)/(x^2 + y^2) \ + &= r ( cos(phi)^3 + sin(phi)^3 ) \ + lim_(r -> 0) + f(x,y) &= r ( cos(phi)^3 + sin(phi)^3 ) = 0 checkmark +$ + +#pagebreak() +=== Exercise 5 @Exercise[4, 5] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Investigate whether the real function +$ + f: RR^2 -> RR \ + + (x,y) |-> e^(x-y^4) dot sin((x-x^2y^3)/(x^2+5)) \ +$ + is continuous on $RR$. +]) + +$x^2 + 5 = 0$ hat keine Lösung in $RR$. Die Funktion $f$ ist stetig, weil sie +stetige Funktionen auf stetige Weise kombiniert. (müsste die Funtkionen zerlegen, +aber kein Bock.) $checkmark$ diff --git a/src/exercise/index.typ b/src/exercise/index.typ index f24a479..5cd43bd 100644 --- a/src/exercise/index.typ +++ b/src/exercise/index.typ @@ -3,3 +3,5 @@ #include "2.typ" #pagebreak() #include "3.typ" +#pagebreak() +#include "4.typ"