#### 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 $-{140}^{\circ}\mathrm{C}$.

- What is the average rate of temperature change between the first and second measurements?

Answer:

- The `falling' property of the temperature shows in the fact that the rate of change is

.

- Calculate the average rate of temperature change for the whole measuring period.

Answer:

##### **Exercise 7.6.2 **

##### **Exercise 7.6.3 **

Calculate for the function

- $f:\{x\in \mathbb{R}\hspace{0.5em}:\hspace{0.5em}x>0\}\to \mathbb{R}$ with $f(x):=\mathrm{ln}({x}^{3}+{x}^{2})$ the value of the first derivative $f\text{'}$ at $x$:

$f\text{'}(x)=$

.

- $g:\mathbb{R}\to \mathbb{R}$ with $g(x):=x\xb7e{}^{-x}$ the value of the second derivative $g\text{'}\text{'}$ at $x$:

$g\text{'}\text{'}(x)=$

.

`(x+1)/((x+2)^2)`.##### **Exercise 7.6.4 **

Consider the function $f:\text{}]0;\infty [\text{}\to \mathbb{R}$, $x\to f(x)$ with $f\text{'}(x)=x\xb7\mathrm{ln}x$. On which regions is $f$ monotonically decreasing, and on which regions is $f$ concave? Specify the regions as open intervals $\text{}]c;d[\text{}$ that are as large as possible:

- $f$ is monotonically decreasing on .

- $f$ is concave on .

`infty`for $\infty $ in your answer. The test evaluation will be displayed here!