Buch druckenBuch drucken

Demokurs: Approach to the Basics of Calculus - ENGLISH

Website: WueCampus
Kurs: vhb - Demokurs Lehramt
Buch: Demokurs: Approach to the Basics of Calculus - ENGLISH
Gedruckt von: Gast
Datum: Sonntag, 24. November 2024, 08:10

Beschreibung

Democourse

Inhaltsverzeichnis

Content

Here be a line, if not the image is missing

x

Modul I: Sequenzes and Limits

  • Sequenzes - basics
  • Recursive sequences
  • Some important sequences
  • Divergence, convergence, limits
  • Characteristics of difference sequences
  • Exercises


Modul II: Functions

  • Linear und Quadratic Functions
  • Power functions, Polynomials and Power Series
  • The Exponential function and Growth
  • Trigonometric Functions
  • Multivariable Functions
  • Curves and Maps


Modul III: Properties of Functions

  • Limit of a function
  • Limits and Continuity
  • Properties of Continuous Functions
  • Exercises


Modul IV: Differentiation

  • The Derivative
  • Properties of the Derivative
  • Derivatives of Trigonometric Functions
  • The Cain Rule
  • The Exponential Function and the Logarithm
  • Extremal Value Problems
  • Exercises


  

Sequences - Basics

Here be a line, if not the image is missing

This section contains the most important definitions about sequences. Through these definitions the general notion of sequences will be explained, but then restricted to real number sequences.


  
 

Definition 1 (Sequence). Let M be a non-empty set. A sequence is a function:

f : ℕ → M.

Occasionally we speak about a sequence in M .


Note: Characteristics of the set ℕ give certain characteristics to the sequence. Because ℕ is ordered, the terms of the sequence are ordered.


Definition 2 (Terms and Indices): A sequence is denoted ( a 1 , a 2 , a 3 , … ) or for short ( a n ) n instead of f ( n ) . The numbers a 1 ,a 2 ,a 3 , … ∈ M are called the terms of the sequence. Because of the mapping

f : ℕ → M n ↦→ an
we can assign a unique number n ∈ ℕ to each term. We write this number as a subscript and define it as the index; it follows that we can identify any term of the sequence by its index.
n 1 2 3 4 5 6 7 8 9 …

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
a n a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 …

A few easy examples

Random sequence of coloured squares. At any one time only one term of the sequence is displayed.

Example 1: The sequence of natural numbers
The sequence ( a n ) n defined by a n := n, n ∈ ℕ is called the sequence of natural numbers. Its first few terms are:

a1 = 1, a2 = 2, a3 = 3,...

This special sequence has the property that every term is the same as its index.


Example 2: The sequence of triangular numbers
Triangular numbers get their name due to the following geometric visualization: Stacking coins to form a triangular shape gives the following diagram:


To the first coin in the first layer we add two coins in a second layer to form the second picture a 2 . In turn, adding three coins to a 2 forms a 3 . From a mathematical point of view, this sequence is the result of summing natural numbers. To calculate the 10th triangular number we need to add the first 10 natural numbers:


D10 = 1 + 2 + 3 + ...+ 9 + 10


In general form the sequence is defined as:


D n = 1 + 2 + 3 + … + ( n − 1) + n.


This motivates the following definition


Notation and Definition (sum sequence): Let ( a n ) n ,a n : ℕ → M be a sequence with terms a n , the sum is written:

∑n a1 + a2 + a3 + ...+ an−1 + an =: ak k=1

The sign ∑ is called sigma sign here the index k increases from 1 to n.

Sum sequences are sequences whose terms are formed by summation of previous terms.


Thus the n th triangular number can be written as:


n D = ∑ k n k=1

Example 3: Sequence of square numbers
The sequence of square numbers ( q n ) n is defined by: q n = n 2 . The terms of this sequence can also be illustrated by the addition of coins.

Interestingly, the sum of two consecutive triangular numbers is a square number. So, for example, we have: 3 + 1 = 4 and 6 + 3 = 9. In general this gives the relationship:


q = D + D n n n−1

Example 4: Sequence of cube numbers
Analogously to the sequence of square number, we give the definition of cube numbers as

3 an := n .

The first terms of the sequence are: (1 , 8 , 27 , 64 , 125 , … ).



 

If, as in Definition 1, M = ℝ, we obtain the following definition:

Definition 3 (Sequence of Numbers): A sequence of real numbers is a function

f : ℕ → ℝ.


When we speak about a sequence in this part of the course we mean in general a real sequence.


Note: In this course, we only consider sequences in ℕ , ℚ and ℝ.

If we delete arbitrarily many terms from a sequence but retain the order of the remaining sequence, then we obtain a so called subsequence. Mathematically this is defined as follows:


Various ways to represent sequences

 

Definition 4 (Subsequence): Given a sequence ( a n ) n and a strictly monotonically increasing

function

φ : ℕ → ℕ

we call the composition 2

(aφ(n))n

a subsequence of ( a n ) n .

H. Böhm: Relief 6 Punkte ... (1959)

Example of subsequences in art

Example 5: Let ( q n ) n with q n := n 2 be the sequence of square numbers

(1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 … )

and define the function φ ( n ) = 2 n. The composition ( q 2 n ) n yields:
( q 2 n ) n = ( q 2 ,q 4 ,q 6 ,q 8 ,q 10 , … )


= (4 , 16 , 36 , 64 , 100 , … ) .

Definition 5 (Sequence of differences):
Given a sequence ( a n ) n = a 1 , a 2 , a 3 , … , a n , …; then

(an+1 − an)n := a2 − a1, a3 − a2, ...
is called the 1st difference sequence of ( a n ) n

The 1 st difference sequence of the 1 st difference sequence is called the 2 nd difference sequence. Analogously the n th difference sequence is defined.

 

Example 6: Given the sequence ( a n ) n with a n := n2+n-- 2, i.e.


( a n ) n = (1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , … )


Let ( b n ) n be its 1 st difference sequence. Then it follows that

( b n ) n = ( a 2 − a 1 ,a 3 − a 2 ,a 4 − a 3 , … )


= (2 , 3 , 4 , 5 , 6 , 7 , 8 , 9)


A term of ( b n ) n has the general form

bn = an+1 − an (n + 1)2 + (n + 1) n2 + n) = ------------------− -------- 22 2 2 = (n-+-1)-+-(n-+-1)-−-n--−-n- 2 (n2 + 2n + 1) + 1 − n2 = ---------------------- 2n + 2 2 = ------- 2 = n + 1.

New terms:



 

Short Revision Exercises
 

Exercise 1: Illustrate the sequence of square numbers in at least three different ways.


Exercise 2: Let ( π n ) n be the sequence of decimal places of the constant π. Is the finite sequence (1 , 4 , 1 , 5 , 9 , 2) a subsequence ( π n ) n ? Give a reason for your answer.