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∘ C is measured. At 3 p.m. the measured temperature is - 58∘ C. After a period of 14 hours, the temperature has fallen to - 140∘ C.
  1. What is the average rate of temperature change between the first and second measurements?
    Answer:
  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.
    Answer:

Exercise 7.6.2
A function f:[-3;2]→ℝ, x→f(x) has a first derivative f' 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:{x∈ℝ : x>0}→ℝ with f(x):=ln( x3 + x2 ) the value of the first derivative f' at x:
    f'(x)=
     .
  2. g:ℝ→ℝ with g(x):=x·e-x the value of the second derivative g'' at x:
    g''(x)=
     .
Bracket the terms for clarification, e.g. enter x+1 (x+2 )2 as (x+1)/((x+2)^2).

Exercise 7.6.4
Consider the function f: ]0;∞[ →ℝ, x→f(x) with f'(x)=x·lnx. On which regions is f monotonically decreasing, and on which regions is f concave? Specify the regions as open intervals ]c;d[ 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 (a;b), closed intervals are entered as [a;b], a and b can be arbitrary expressions. Do not use the notation ]a;b[ to enter open intervals. Enter infty for ∞ in your answer.

 
        

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