Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 16
- Warm-up-exercises
Exercise
Find a zero for the function
- Failed to parse (syntax error): {\displaystyle f \colon \R \longrightarrow \R , x \longmapsto f(x) <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 16]] __NOINDEX__ x^2+x-1 , }
in the interval using the interval bisection method with a maximum error of .
Exercise
Let
be a continuous function. Show that is not surjective.
Exercise
Give an example of a bounded interval and a continuous function
such that the image of is bounded, but the function admits no maximum.
Exercise
Let
be a continuous function. Show that there exists a continuous extension
of .
Exercise
Let
be a continuous function defined over a real interval. The function has at points , , local maxima. Prove that the function has between and has at least one local minimum.
Exercise
Determine directly, for which the power function
has an extremum at the point zero.
Exercise *
Show that the Intermediate value theorem for continuous functions from to does not hold.
Exercise
Determine the limit of the sequence
- Hand-in-exercises
Exercise (2 marks)
Determine the minimum of the function
Exercise (5 marks)
Find for the function
- Failed to parse (syntax error): {\displaystyle f \colon \R \longrightarrow \R , x \longmapsto f(x) <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 16]] __NOINDEX__ x^3 -3x+1 , }
a zero in the interval using the interval bisection method, with a maximum error of .
Exercise (2 marks)
Determine the limit of the sequence
The next task uses the notion of an even and an odd function.
Exercise (4 marks)
Let
be a continuous function. Show that one can write
with a continuous even function and a continuous odd function .
The following task uses the notion of fixed point.
Exercise (4 marks)
Let
be a continuous function from the interval into itself. Prove that has a fixed point.