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

Exercises

Exercise

Sketch the slope triangle and the secant for the function

$f(x)=x^2-3x+5$

in the points $1$ and $3$.

Exercise

Determine the affine-linear map

$\alpha :ℝ⟶ℝ,$

whose graph passes through the two points

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

Exercise

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

$f:ℝ⟶ℝ,x⟼f(x)=x^3+2x^2-5x+3,$

at any point $a\in ℝ$.

Exercise

Prove that the real absolute value

$ℝ⟶ℝ,x⟼|x|,$

is not differentiable at the point zero.

Exercise

Let $f:ℝ\to ℝ$ be an even function, and suppose that it is differentiable in the point $x$. Show that $f$ is also differentiable in the point $-x$ and that the relation

$f^{\prime }(-x)=-f^{\prime }(x)$

holds.

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

Exercise

Determine the derivative of the functions

$ℝ⟶ℝ,x⟼x^n,$

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

Exercise

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

Exercise

Determine for a 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 approximation (including the remainder function $r(x)$) in the zero point.

Exercise

Show, using limits of functions, that a function $f:D\to ℝ$, which is differentiable in a point $a\in D$, is also continuous in this point.

Exercise

Prove the product rule for differentiable functions, using limits of functions, applied to the difference quotient.

Exercise

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

Exercise

Determine the linear approximation (including the remainder function $r(x)$) for the exponential function $\exp x$ in the zero point.

Exercise

Determine the derivative of the function

$f:D=ℝ\setminus \{0\}⟶ℝ,x⟼f(x)=x^n,$

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

Exercise

Determine the derivative of the function

$f:D=ℝ\setminus \{0\}⟶ℝ,x⟼f(x)=\frac{x^2+1}{x^3}.$

Exercise

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

Exercise

Let

$g,h:ℝ⟶{ℝ}_{+}$

denote differentiable functions, 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

Let

$g_1,g_2,\dots ,g_n:ℝ⟶ℝ\setminus \{0\}$

denote differentiable functions. Prove, by induction over $n$, the relation

${\left(\frac{1}{g_1\cdot g_2\cdots g_n}\right)}^{\prime }=\frac{-1}{g_1\cdot g_2\cdots g_n}\cdot (\frac{{g_1}^{\prime }}{g_1}+\frac{{g_2}^{\prime }}{g_2}+\cdots +\frac{{g_n}^{\prime }}{g_n}).$

Exercise

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

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

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

Exercise

Let

$f,g:ℝ⟶ℝ$

be two differentiable functions and consider

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

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

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

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

Exercise

Determine the derivative of the function

$f:D={ℝ}_{+}⟶ℝ,x⟼f(x)=x^{\frac{1}{n}},$

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

Exercise

Let

$f:ℝ⟶ℝ$

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

We have

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

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

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

Hence,

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

Exercise

Give an example of a continuous, not differentiable function

$f:ℝ⟶ℝ$

fulfilling the property that the function $x\mapsto f(|x|)$ is differentiable.

Hand-in-exercises

Exercise (2 marks)

Determine the affine-linear map

$\alpha :ℝ⟶ℝ,$

whose graph passes through the two points

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

Exercise (2 marks)

Let $f:ℝ\to ℝ$ be an odd differentiable function. Show that the derivative $f^{\prime }$ is even.

Exercise (3 marks)

Let $D\subseteq ℝ$ be a subset and let

$f_i:D⟶ℝ,i=1,\dots ,n,$

be differentiable functions. Prove the formula

${(f_1\cdots f_n)}^{\prime }=\sum _{i=1}^n{f}_1\cdots f_{i-1}{f}_{i^{\prime }}{f}_{i+1}\cdots f_n.$

Exercise (4 marks)

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)

Determine the derivative of the function

$f:D⟶ℝ,x⟼f(x)=\frac{x^2+x-1}{x^3-x+2},$

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

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

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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