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Mathematics-Online course: Preparatory Course Mathematics - Analysis - Exercises and Test

Exercises

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Let the function $ f$ be differentiable at $ x_0\in\mathbb{R}$ and let $ f$ further suffice the equation $ f(x+y)=f(x)+f(y)$ for all $ x,y\in\mathbb{R}$ .
a)
Use the difference quotient to show that $ f$ is differentiable for all $ x \in \mathbb{R}$.
b)
Prove the existence of a constant $ a\in \mathbb{R}$ with $ f(x)=ax$ for all $ x \in \mathbb{R}$.
(Authors: Wipper/Abele)
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The front of a greenhouse shall have the form of a axes-symmetric pentagon with three right angles (cp.figure). The amount of the glass wall is limited by 20 m; the area within shall be maximised.

What's the height and the width of the greenhouse?

\includegraphics[width=3.5cm]{g25_bild1}

(Authors: Apprich/Höfert)
Given the function $ f(x)=(x+2)\sqrt{4-x^2}$.
a)
For which $ x\in\mathbb{R}$ $ f$ is defined? Check where $ f$ is differentiable and determine $ f'(x)$.
b)
What are the roots and lokal extremums of $ f$?
c)
What's the behaviour of $ f'$ for the boundary of the domain?
d)
Sketch the graph of $ f$.

(Authors: /Höfert)
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Let $ f$ and $ g$ be the real functions given by $ f(x)=\sqrt{3+2x}-x$ and $ g(x)=\ln\,(f(x))$ .
a)
Determine $ f$ 's domain and sketch the graph. What is $ g$ 's domain?
b)
Examine $ g$ with regard to zero points, asymptotes and local extrema.
c)
How do $ g$ and $ g'$ behave at the domain's boundary points?
d)
Draw the graph of the function $ g$ . note: $ \ln 2\approx 0,69; \ \ln 3\approx 1,10$ .

(Authors: Kimmerle/Roggenkamp/Rump/Abele)
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Use appropriate substitutions to determine the following integrals:

$\displaystyle {\bf {a)}} \quad \int \frac{dx}{x\ln x} \qquad\quad {\bf {b)}} \q... ...c{x}{(x^2+1)^3}\,dx\qquad\quad {\bf {c)}} \quad \int \frac{dx}{\sin x\,\cos x} $

(Authors: Kimmerle/Roggenkamp/Rump/Abele)
Use partial integration to determine the following integrals:

$\displaystyle \begin{array}{lll} \hspace*{1.5cm} {\bf {a)}} \quad {\displaystyl... ...space*{2cm} & {\bf {c)}} \quad {\displaystyle{\int \arctan x\,dx}} \end{array} $

(Authors: Kimmerle/Roggenkamp/Rump/Abele)
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  automatically generated 1/9/2017