Creating a Taylor Polynomial
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The document provides an example of creating a Taylor polynomial for the function f(x) = ln(x) centered at x = 2. It finds the first three derivatives of f(x) evaluated at x = 2. It then constructs the first four non-zero terms of the Taylor polynomial one by one using the general formula. The resulting Taylor polynomial with four non-zero terms is given.
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2) Solving quadratic equations through factoring and using the quadratic formula.
3) Graphs of polynomial functions and applying the factor theorem, remainder theorem, and rational zero theorem to polynomials.
4) Determining the domains of functions when combining, composing, or taking inverses of functions.
The document includes examples and step-by-step solutions for applying these algebraic concepts.
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This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
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- Standard formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
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BSC_COMPUTER _SCIENCE_UNIT-1_DISCRETE MATHEMATICS
This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
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- Examples demonstrate calculating specific high-order derivatives, such as the 10th derivative of
Pre-Cal 20S January 20, 2009
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Creating a Taylor Polynomial
- 1. Creating a Taylor Polynomial An Example
- 2. Write the first four non-zero terms of the Taylor polynomial for f(x) = lnx, centered at x = 2. Let’s get to work!
- 3. I suggest starting by finding the first three derivatives of f(x) = lnx and then plugging in the center. 𝑓 𝑥 = ln 𝑥 𝑓′ 𝑥 = 1 𝑥 𝑓′′ 𝑥 = − 1 𝑥2 𝑓′′′ 𝑥 = 2 𝑥3 𝑓 2 = ln 2 𝑓′ 2 = 1 2 𝑓′′ 2 = − 1 4 𝑓′′′ 2 = 2 8 = 1 4
- 4. Then, I recommend constructing the terms one-by-one using the general set up: 𝑓 𝑛 (𝑐)(𝑥−𝑐) 𝑛 𝑛!
- 5. Here We Go! Non-Zero Term #1 (0 Derivatives) 𝑓(2)(𝑥−2)0 0! = ln 2(𝑥−2)0 0! = ln 2 Non-Zero Term #2 (1st Derivative) 𝑓′(2)(𝑥−2)1 1! = 1 2 (𝑥−2) 1 = 1 2 (𝑥 − 2)
- 6. Here We Go! Non-Zero Term #3 (2nd Derivative) 𝑓′′(2)(𝑥−2)2 2! = − 1 4 (𝑥−2)2 2 = − 1 8 (𝑥 − 2)2 Non-Zero Term #4 (3rd Derivative) 𝑓′′′(2)(𝑥−2)3 3! = 1 4 (𝑥−2)3 6 = 1 24 (𝑥 − 2)3
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