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Christoph J. Scherr 2024-09-16 17:05:08 +02:00
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build
src/exercise

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== Exercise Sheet 5 - Partial Derivatives
=== Exercise 1 @Exercise[5, 1]
#block(
fill: luma(230),
inset: 8pt,
radius: 4pt,
[
Where is the function
$
f: RR^2 -> RR
(x,y) |-> x dot sqrt(y^2 + x^2)
$
partially differentiable? Calculate the first-order partial derivatives.
])
$
g(x,y) &= x ; (delta g)/(delta x) = 1 ; (delta g)/(delta y) = 0 \
h(x,y) &= sqrt(y^2+x^2) ; (delta h)/(delta x) = sqrt(y^2+x^2) dot 2x ; (delta h)/(delta y) = sqrt(y^2+x^2) dot 2y \
\
(delta f)/(delta x) &= g dot h' + g' dot h = x dot sqrt(y^2+x^2) dot 2x + 1 dot sqrt(x^2+y^2) \
&= sqrt(x^2+y^2) (2x^2 + 1) \
(delta f)/(delta y) &= g dot h' + g' dot h = x dot sqrt(x^2+y^2) dot 2y + 0 dot dots \
&= x dot sqrt(x^2+y^2) dot 2y
\ "dödö falsch"
$

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#include "3.typ" #include "3.typ"
#pagebreak() #pagebreak()
#include "4.typ" #include "4.typ"
#pagebreak()
#include "5.typ"