Taylor Series

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.10 – Taylor and Maclaurin Series

• Section 11.11 – Applications of Taylor Polynomials

Power Function

Definition: A power function has the form ${\displaystyle f(x)=k{x}^p}$$\displaystyle f(x)=kx^p$, where $k\ne 0$$k\ne 0$.

Horizontal Translation of a Power Function

• ${\displaystyle k(x-a{)}^p}$$\displaystyle k(x-a)^p$ is a horizontal shift to the right by $a$$a$ units.

• ${\displaystyle k(x+a{)}^p}$$\displaystyle k(x+a)^p$ is a horizontal shift to the left by $a$$a$ units.

Exponentiation Rules

• ${\displaystyle x^{p+q}=x^p{x}^q}$$\displaystyle x^{p+q}=x^px^q$

• ${\displaystyle x^{pq}=x^{p^q}=x^{q^p}}$$\displaystyle x^{pq}=x^{p^q}=x^{q^p}$

• ${\displaystyle x^{-p}=\frac{1}{x^p}}$$\displaystyle x^{-p}=\frac{1}{x^p}$

• ${\displaystyle (xy{)}^p=x^p{y}^p}$$\displaystyle (xy)^p=x^py^p$

Rewrite Exponential Expressions

Use Technology to Use Technology to Rewrite Exponential Expression

NOTE: You may use of technology to rewrite exponential expressions on all Assessments without showing any work.

Mathematica

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(* Example: Rewrite (4x^2)^(2n+1) *)
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Together[(4 x^2)^(2 n + 1)]
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(* Alternatively *)
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PowerExpand[(4 x^2)^(2 n + 1)]
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(* Alternatively *)
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ExpandAll[(4 x^2)^(2 n + 1), x]
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(* Alternatively *)
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Collect[(4 x^2)^(2 n + 1), x]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms or factors in the base or the exponent of the exponential expression.

3. Together[ ] has at least one arguments:

   1. The exponential expression to be rewritten.

4. PowerExpand[ ] has at least one arguments:

   1. The exponential expression to be rewritten.

5. ExpandAll[ ] has at least two arguments:

   1. The exponential expression to be rewritten;

   2. Either the variable $x$$x$ or the parameter $n$$n$.

6. Collect[ ] has at least two arguments:

   1. The exponential expression to be rewritten;

   2. Either the variable $x$$x$ or the parameter $n$$n$.

7. For help on using the Together[ ] function:

   1. In Mathematica, execute the code: $\text{?Together}$$\text{?Together}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Together[ ] function – you will be able to copy-paste code.

8. Remember, correct Mathematica code will be all black except for variables.

9. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Monday, 15 November 2021 - 01:41 (EST) Created: Sunday, 9 April 2023 - 20:21 (EDT)