Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 19/en/latex

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\setcounter{section}{19}






\zwischenueberschrift{Warm-up-exercises}

The following task should be solved both directly and through the derivation rules.


\inputexercise
{}
{

Determine the derivative of the function \maabbeledisp {f} {\R} { \R } {x} {f(x)=x^n } {,}


for all $n \in \N$.

}
{} {}




\inputexercise
{}
{

Determine the derivative of the function \maabbeledisp {f} {\R \setminus \{0\}} { \R } {x} {f(x)=x^n } {}


for all $n \in \Z$.

}
{} {}




\inputexercise
{}
{

Determine the derivative of the function \maabbeledisp {f} {\R_+} { \R } {x} {f(x)=x^{\frac{1}{n} } } {,}


for all $n \in \N_+$.

}
{} {}




\inputexercise
{}
{

Determine directly \zusatzklammer {without the use of derivation rules} {} {} the derivative of the function \maabbeledisp {f} {\R} {\R } {x} {f(x) = x^3+2x^2-5x+3 } {,} at any point $a \in \R$.

}
{} {}




\inputexercise
{}
{

Prove that the real absolute value \maabbeledisp {} {\R} {\R } {x} { \betrag { x } } {,} is not differentiable at the point zero.

}
{} {}




\inputexercise
{}
{

Determine the derivative of the function \maabbeledisp {f} {\R \setminus \{0\}} { \R } {x} {f(x)= \frac{x^2+ 1 }{ x^3} } {.}

}
{} {}




\inputexercise
{}
{

Prove that the derivative of a rational function is also a rational function.

}
{} {}




\inputexercise
{}
{

Consider $f(x)=x^3+4x^2-1$ and $g(y) =y^2-y+2$. Determine the derivative of the composite function $h(x)=g(f(x))$ directly and by the chain rule (Theorem 19.8).

}
{} {}




\inputexercise
{}
{

Prove that a polynomial $P \in \R[X]$ has degree $\leq d$ \zusatzklammer {or it is $P=0$} {} {,} if and only if the $(d+1)$-th derivative of $P$ is the zero poynomial.

}
{} {}




\inputexercise
{}
{

Let \maabbdisp {f,g} {\R} {\R } {} be two differentiable functions and consider
\mathdisp {h(x)=(g(f(x)))^2 f(g(x))} { . }

a) Determine the derivative $h'$ from the derivatives of \mathkor {} {f} {and} {g} {.}

b) Let now
\mathdisp {f(x)=x^2-1 \text{ and } g(x) =x+2} { . }
Compute $h'(x)$ in two ways, one directly from $h(x)$ and the other by the formula of part $a)$.

}
{} {} For the \anfuehrungenglisch{linear approximation}{} of differentiable maps we need the definition of affine-linear maps.


Let $K$ be a field and let \mathkor {} {V} {and} {W} {} be vector spaces over $K$. A map \maabbeledisp {\alpha} {V} {W } {v} {\alpha(v) = \varphi(v) +w } {,} where $\varphi$ is a linear map and
\mathl{w \in W}{} is a vector, is called \definitionswort {affine-linear}{.}





\inputexercise
{}
{

Let $K$ be a field and let $V$ be a $K$-vector space. Prove that given two vectors $u,v \in W$ there exists exactly one affine-linear map \maabbdisp {\alpha} {K} {W } {} sucht that \mathkor {} {\alpha(0) = u} {and} {\alpha(1) = v} {.}

}
{} {}




\inputexercise
{}
{

Determine the affine-linear map \maabbdisp {\alpha} {\R} {\R^3 } {} such that \mathkor {} {\alpha(0) = (2,3,4)} {and} {\alpha(1)=(5,-2,-1)} {.}

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{3}
{

Determine the derivative of the function \maabbeledisp {f} {D} { \R } {x} {f(x)= \frac{x^2+x-1 }{ x^3-x+2} } {,}


where $D$ is the set where the denominator does not vanish.

}
{} {}




\inputexercise
{4}
{

Determine the tangents to the graph of the function
\mathl{f(x)=x^3-x^2-x+1}{,} which are parallel to
\mathl{y=x}{.}

}
{} {}




\inputexercise
{4}
{

Let $f(x)=\frac{x^2+5x-2}{x+1}$ and $g(y) = \frac{y-2}{y^2+3}$. Determine the derivative of the composite $h(x)=g(f(x))$ directly and by the chain rule (Theorem 19.8).

}
{} {}




\inputexercise
{2}
{

Determine the affine-linear map \maabbdisp {\alpha} {\R} {\R } {,} whose graph passes through the two points \mathkor {} {(-2,3)} {and} {(5,-7)} {.}

}
{} {}




\inputexercise
{3}
{

Let
\mathl{D \subseteq \R}{} be a subset and let \maabbdisp {f_i} {D} { \R , \, i = 1 , \ldots , n } {,} be differentiable functions. Prove the formula
\mathdisp {(f_1 { \cdots } f_n)' = \sum_{i=1}^n f_1 { \cdots } f_{i-1} f_{i}' f_{i+1} { \cdots } f_n} { . }

}
{} {}



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