Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 14/refcontrol

Exercises

Exercise Create referencenumber

Sketch the slope triangle and the secant for the function

${}f(x)=x^{2}-3x+5\,$

in the points ${}1$ and ${}3$.

Exercise Create referencenumber

Determine the affine-linear map

$\alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ,$

whose graph passes through the two points

${}(2,5)$ and ${}(4,9)$.

Exercise Create referencenumber

Determine directly (without the use of derivation rules) the derivative of the function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{3}+2x^{2}-5x+3,$

at any point ${}a\in \mathbb {R}$.

Exercise Create referencenumber

Prove that the real absolute value

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,$

is not differentiable at the point zero.

Exercise Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be an even function,MDLD/even function (R) and suppose that it is differentiableMDLD/differentiable (R) in the point ${}x$. Show that ${}f$ is also differentiable in the point ${}-x$ and that the relation

${}f'(-x)=-f'(x)\,$

holds.

Please try to solve the following exercise in a direct way aswell as with the help of derivation rules.

Exercise Create referencenumber

Determine the derivative of the functions

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

for all ${}n\in \mathbb {N}$.

Exercise Create referencenumber

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

Exercise Create referencenumber

Determine for a polynomialMDLD/polynomial

${}f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\,$

the linear approximationMDLD/linear approximation (R) (including the remainder function ${}r(x)$) in the zero point.

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Show, using limits of functions,MDLD/limits of functions (R) that a functionMDLD/function (R) ${}f\colon D\rightarrow \mathbb {R}$, which is differentiableMDLD/differentiable (R) in a point ${}a\in D$, is also continuousMDLD/continuous (R) in this point.

Exercise Create referencenumber

Prove the product rule for differentiable functions,MDLD/differentiable functions (R) using limits of functions,MDLD/limits of functions (R) applied to the difference quotient.MDLD/difference quotient (R)

Exercise Create referencenumber

Show that the exponential functionMDLD/exponential function (R) ${}\exp x$ is differentiableMDLD/differentiable (R) in every point ${}a\in \mathbb {R}$, and determine its derivative.MDLD/derivative (R)
Hint: Apply the definition about the limit of functions to the fraction of differences. The function equation for the exponential function is helpful.

Exercise Create referencenumber

Determine the linear approximationMDLD/linear approximation (R) (including the remainder function ${}r(x)$) for the exponential functionMDLD/exponential function (R) ${}\exp x$ in the zero point.

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Determine the derivative of the function

$f\colon D=\mathbb {R} \setminus \{0\}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{n},$

for all ${}n\in \mathbb {Z}$.

Exercise Create referencenumber

Determine the derivative of the function

$f\colon D=\mathbb {R} \setminus \{0\}\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {x^{2}+1}{x^{3}}}.$

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

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Let

$g,h\colon \mathbb {R} \longrightarrow \mathbb {R} _{+}$

denote differentiable functions,MDLD/differentiable functions (R) and set

${}f(x):={\frac {g(x)}{h(x)^{n}}}\,,$

${}n\in \mathbb {N} _{+}$. Show that the derivative of ${}f$ can be written as a fraction, with ${}h^{n+1}(x)$ as denominator.

Exercise Create referencenumber

Let

$g_{1},g_{2},\ldots ,g_{n}\colon \mathbb {R} \longrightarrow \mathbb {R} \setminus \{0\}$

denote differentiable functions.MDLD/differentiable functions (R) Prove, by induction over ${}n$, the relation

${}{\left({\frac {1}{g_{1}\cdot g_{2}\cdots g_{n}}}\right)}^{\prime }={\frac {-1}{g_{1}\cdot g_{2}\cdots g_{n}}}\cdot {\left({\frac {g_{1}'}{g_{1}}}+{\frac {g_{2}'}{g_{2}}}+\cdots +{\frac {g_{n}'}{g_{n}}}\right)}\,.$

Exercise Create referencenumber

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.

Exercise Create referencenumber

Let ${}f(x)={\frac {x^{2}-3}{x+2}}$ and ${}g(y)={\frac {y+4}{y^{2}-5}}$. We consider the compositionMDLD/composition ${}h(x):=g(f(x))$.

1. Compute ${}h$ (the result must be in the form of a rational function).
2. Compute the derivativeMDLD/derivative (R) of ${}h$, using part 1.
3. Compute the derivative of ${}h$, using the chain rule.

Exercise Create referencenumber

Let

$f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$

be two differentiable functions and consider

${}h(x)=(g(f(x)))^{2}f(g(x))\,.$

a) Determine the derivative ${}h'$ from the derivatives of ${}f$ and ${}g$. b) Let now

$f(x)=x^{2}-1{\text{ and }}g(x)=x+2.$

Compute ${}h'(x)$ in two ways, one directly from ${}h(x)$ and the other by the formula of part ${}a)$.

Exercise Create referencenumber

Determine the derivative of the function

$f\colon D=\mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{\frac {1}{n}},$

for all ${}n\in \mathbb {N} _{+}$.

Exercise Create referencenumber

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be a bijective differentiable function with ${}f'(x)\neq 0$ for all ${}x\in \mathbb {R}$, and the inverse function ${}f^{-1}$. What is wrong in the following "Proof“ for the derivative of the inverse function?

We have

${}{\left(f\circ f^{-1}\right)}(y)=y\,.$

Using the chain rule, we get by differentiating on both sides the equality

${}f'{\left(f^{-1}(y)\right)}{\left(f^{-1}\right)}'(y)=1\,.$

Hence,

${}{\left(f^{-1}\right)}'(y)={\frac {1}{f'{\left(f^{-1}(y)\right)}}}\,.$

Exercise Create referencenumber

Give an example of a continuous, not differentiable function

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

fulfilling the property that the function ${}x\mapsto f{\left(\vert {x}\vert \right)}$ is differentiable.

Hand-in-exercises

Exercise (2 marks) Create referencenumber

Determine the affine-linear map

$\alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ,$

whose graph passes through the two points

${}(-2,3)$ and ${}(5,-7)$.

Exercise (2 marks) Create referencenumber

Let ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ be an oddMDLD/odd (real function) differentiable function.MDLD/differentiable function (R) Show that the derivativeMDLD/derivative (R) ${}f'$ is even.MDLD/even (real function)

Exercise (3 marks) Create referencenumber

Let ${}D\subseteq \mathbb {R}$ be a subset and let

$f_{i}\colon D\longrightarrow \mathbb {R} ,\,i=1,\ldots ,n,$

be differentiable functions. Prove the formula

${}{\left(f_{1}\cdots f_{n}\right)}'=\sum _{i=1}^{n}f_{1}\cdots f_{i-1}f_{i}'f_{i+1}\cdots f_{n}\,.$

Exercise (4 marks) Create referencenumber

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

Exercise (3 marks) Create referencenumber

Determine the derivative of the function

$f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {x^{2}+x-1}{x^{3}-x+2}},$

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

Exercise (7 (2+2+3) marks) Create referencenumber

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.

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