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Hands on Mathematics for CS
differentiable functions
Stefan Nesbigall
Sebastian Germesin
2030115
2029918
Definition: Differentiability
A function f is differentiable at the point x in its domain if
f x f
lim
x
x
exists.
The whole function is differentiable, if it is differentiable in each point of
its domain.
exercise: f(x) = x²
f x
x2
f x
f
f x
lim
x
x
lim
lim x
2
f
x
x
x
x
x
x
2
2
x
x
exercise: f(x) = 1/x
f x
1
x
f x
f
1
x
x
x
1
x
x
x
x
x
1
x
x
x
x
x
1 x
x
x
1
x
x
x
1
x
x
exercise: f(x) = 1/x
1
x
f x
f x
lim
x
x
f
1
x
lim
x
x
lim
x
x
1
x
1
2
exercise: f(x) = |x–1|
f x
x 1
case 1 : x 1
1 x 1
f x
x 1
x
1 1
lim
x
f
f x
x
lim
x
1 x
x
lim 1
x
1
f
x 1
1
x
1 x
exercise: f(x) = |x–1|
f x
x 1
case 2 : x 1
x 1
f x
x 1
1
x
1 1 x
lim
x
f
f x
x
lim
x
x
x
lim 1
x
1
f
exercise: f(x) = |x–1|
f x
x 1
case 3 : examination in point 1
let
1
f x
lim
x
1
lim
x
1
f 1
x 1
x 1 1 1
x 1
x 1
lim
x 1 x 1
0
0
not defined
not differentiable
exercise: f(x) = {...
f x
{
x
4
x
,x 4
1 ,x 4
case 1 : x 4
f x
x
1
4
x 4
f
f x
lim
x
x
1
4
4
4
x 4
4
4
4
x
4
f
x
lim
x
4
x
x
lim
4 x
x
1
lim
4
x
1
4
exercise: f(x) = {...
f x
{
x
4
x
,x 4
1 ,x 4
case 2 : 0 x 4
f x
f
x
x
x
x
x
2
x
2
x
x
x
f x
lim
x
x
lim
x
x
x
x
x
f
lim
x
x
x
x
lim
x
1
x
1
2
exercise: f(x) = {...
f x
{
x
4
x
,x 4
1 ,x 4
case 2 : x 0
a : x is not defined for x 0
b :x 0
not differentiable
1
limit is not defined because
is not defined in point 0
2 x
not differentiable
How to solve exercises
in general?
Is f x differentiable ?
1 Solve f x
f
2 Solve lim Y
x
Y
in the domain of f.
x
a if limit not exists
f x is not differentiable
b if limit exists
f x is differentiable
Conclusion: Problems
●
Correct arithmetical convertions
●
Problems while solving the limit
●
Critial Points
●
Case differentation