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Mathematics-Online course: Prepcourse Mathematics - Analysis - Exercises and Test

Exercises

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Shortly after beginning his studies, freshman Felix is convinced he needs a high-end PC in order to solve his maths problems.

costs: 5000 DM - Savings account balance: 0 DM.

a)
His bank offers him a credit that is to be repayed in $ 24$ equal monthly rates. For the remainders of his debt Felix has to pay $ 1\%$ interest per month; the first rate is to be payed after one month. How much are the monthly rates $ r$?

b)
Felix decides to repay the credit not in monthly rates, but the entire sum as a whole after two years. How much is that sum when interest calculation is carried out once a year, every three months, monthly, daily or at steady interest with $ 12\%$ , $ 3\%$ or $ 1\%$ respectively? Determine each time the yearly effective yield.

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)
Differentiate:

$\displaystyle (\ln x)^{\ln x}, \quad \tan \frac{1+x}{1-x}, \quad \sqrt{x^2+\sqrt{x^2+1}} \; . $

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)
For $ m,n\in\mathbb{N}$ let $ f$ be

$\displaystyle f(x)=x^m(1-x)^n, \qquad x\in\mathbb{R}\,. $

a)
Examine $ f$ with regard to zero points and local extrema. How does $ f(x)$ behave for $ x\longrightarrow\pm\infty$ ?
b)
Sketch the graph of $ f$ for $ m=2$ and $ n=3$ .
c)
Together with the $ x$ -axis, the graph of $ f$ encloses a finite area. Determine its size.
(Authors: Höllig/Apprich/Abele)
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)
Given a cuboid with side lengths $ a, b, c$ . Where does the point $ P$ have to be situated so that the polygon $ OPQ$ is the shortest link from $ O$ to $ Q$ along the cuboid's surface?
\includegraphics[width=8.4cm]{A775_1_bild.eps}
(Authors: Höllig/Abele)
Which way is the fastest for a man standing at point $ A$ to get to the island $ I$ , if he runs five times as fast as he can swim?

\includegraphics[width=.7\linewidth]{A775_2_bild.eps}
(Authors: Höllig/Abele)
Determine a primitive function for

$\displaystyle {\bf {a)}} \ \frac{(\ln x)^3}{x}\, \qquad {\bf {b)}} \ \cos^3 x\, \sin^5 x\, \qquad {\bf {c)}} \ \sqrt{4-x^2} $

(Authors: Höllig/Abele)
Use partial Integration to determine

$\displaystyle {\bf {a)}}\quad \int (\ln x)^2\, dx \,,\qquad {\bf {b)}}\quad \in... ..._0^{\pi}{\rm {e}}^x \sin x\, dx\,,\qquad {\bf {c)}}\quad \int \cos^4 x\, dx\,. $

(Authors: Höllig/Abele)
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 9/18/2007