#### Chapter 7 Differential Calculus

Section 7.6 Final Test

# 7.6.1 Final Test Module 7

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##### Exercise 7.6.1
In a container at 9 a.m. a temperature of $-{10}^{\circ }\mathrm{C}$ is measured. At 3 p.m. the measured temperature is $-{58}^{\circ }\mathrm{C}$. After a period of 14 hours, the temperature has fallen to $-{140}^{\circ }\mathrm{C}$.
1. What is the average rate of temperature change between the first and second measurements?
2. The `falling' property of the temperature shows in the fact that the rate of change is
.
3. Calculate the average rate of temperature change for the whole measuring period.

##### Exercise 7.6.2
A function $f:\left[-3;2\right]\to ℝ$, $x\to f\left(x\right)$ has a first derivative $f\text{'}$ whose graph is shown in the figure below. The function values of $f$ between $-3$ and $0$
 are constant, increase by $3$, decrease.

At the point $x=0$ the function $f$ has
 a jump, no derivative, a derivative of $1$.

##### Exercise 7.6.3
Calculate for the function
1. $f:\left\{x\in ℝ : x>0\right\}\to ℝ$ with $f\left(x\right):=\mathrm{ln}\left({x}^{3}+{x}^{2}\right)$ the value of the first derivative $f\text{'}$ at $x$:
$f\text{'}\left(x\right)=$
.
2. $g:ℝ\to ℝ$ with $g\left(x\right):=x·e{}^{-x}$ the value of the second derivative $g\text{'}\text{'}$ at $x$:
$g\text{'}\text{'}\left(x\right)=$
.
Bracket the terms for clarification, e.g. enter $\frac{x+1}{\left(x+2{\right)}^{2}}$ as (x+1)/((x+2)^2).

##### Exercise 7.6.4
Consider the function $f:\text{}\right]0;\infty \left[\text{}\to ℝ$, $x\to f\left(x\right)$ with $f\text{'}\left(x\right)=x·\mathrm{ln}x$. On which regions is $f$ monotonically decreasing, and on which regions is $f$ concave? Specify the regions as open intervals $\text{}\right]c;d\left[\text{}$ that are as large as possible:
1. $f$ is monotonically decreasing on .
2. $f$ is concave on .
Open intervals can be entered in the form $\left(a;b\right)$, closed intervals are entered as $\left[a;b\right]$, $a$ and $b$ can be arbitrary expressions. Do not use the notation $\right]a;b\left[$ to enter open intervals. Enter infty for $\infty$ in your answer.

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