1 min read•december 14, 2020
Welcome to AP Calc 10.11! In this lesson, you’ll learn how to approximate a function over at a point.
This theorem states that for a function , it’s Taylor series approximation at is…
This can be rewritten as…
where is the deriviative of the function and . The -order Taylor polynomial is the partial sum of the infinite series.
Taylor series centered at are common and are called Maclaurin series.
Taylor series look very daunting when you first approach them. Let’s define each portion and build a table that will help you tackle problems of this type!
Now, let’s try a practice problem using this table to walk through it step by step.
Find the third-degree Maclaurin polynomial for .
Solution: First, let’s build our table. Remember that a Maclaurin series is just a Taylor series where !
Now, we just put the terms in our final column together as a full formula. The third-degree Maclaurin polynomial for is:
Now it’s your turn to apply what you’ve learned!
Start by building your table and filling in the values:
Putting it all together, we get that the fifth-degree Maclaurin polynomial for is
Keep on building your tables! This time, our column will be a bit more complicated.
We then find the polynomial to be equal to:
One more table!
If we put this all together, we get:
We can simplify a few of these terms a bit more using exponent rules to get:
This is the fourth-degree Taylor polynomial centered at for .
Great work! Taylor polynomials may seem daunting at first, but when in doubt, break it down with a table and you’ll be sure to master them!
Function Approximations
: Function approximations are mathematical techniques used to estimate the value of a function at a particular point or within a certain range. They involve using simpler functions, such as polynomials, to closely mimic the behavior of the original function.p-Series
: A p-series is a series of the form Σ(1/n^p), where n starts from 1 and goes to infinity, and p is a positive constant. It converges if the value of p is greater than 1, and diverges if the value of p is less than or equal to 1.Power Series
: A power series is an infinite series that represents a function as an infinite polynomial expression.Tangent Line Approximation
: Tangent line approximation, also known as linear approximation or tangent line estimation, is a method that uses the equation of a tangent line at a specific point on a curve to approximate the value of the function near that point. It provides a close estimate when dealing with small intervals.Taylor Polynomial
: A Taylor polynomial is a polynomial approximation of a function centered around a specific point. It is used to estimate the value of the function at nearby points.1 min read•december 14, 2020
Welcome to AP Calc 10.11! In this lesson, you’ll learn how to approximate a function over at a point.
This theorem states that for a function , it’s Taylor series approximation at is…
This can be rewritten as…
where is the deriviative of the function and . The -order Taylor polynomial is the partial sum of the infinite series.
Taylor series centered at are common and are called Maclaurin series.
Taylor series look very daunting when you first approach them. Let’s define each portion and build a table that will help you tackle problems of this type!
Now, let’s try a practice problem using this table to walk through it step by step.
Find the third-degree Maclaurin polynomial for .
Solution: First, let’s build our table. Remember that a Maclaurin series is just a Taylor series where !
Now, we just put the terms in our final column together as a full formula. The third-degree Maclaurin polynomial for is:
Now it’s your turn to apply what you’ve learned!
Start by building your table and filling in the values:
Putting it all together, we get that the fifth-degree Maclaurin polynomial for is
Keep on building your tables! This time, our column will be a bit more complicated.
We then find the polynomial to be equal to:
One more table!
If we put this all together, we get:
We can simplify a few of these terms a bit more using exponent rules to get:
This is the fourth-degree Taylor polynomial centered at for .
Great work! Taylor polynomials may seem daunting at first, but when in doubt, break it down with a table and you’ll be sure to master them!
Function Approximations
: Function approximations are mathematical techniques used to estimate the value of a function at a particular point or within a certain range. They involve using simpler functions, such as polynomials, to closely mimic the behavior of the original function.p-Series
: A p-series is a series of the form Σ(1/n^p), where n starts from 1 and goes to infinity, and p is a positive constant. It converges if the value of p is greater than 1, and diverges if the value of p is less than or equal to 1.Power Series
: A power series is an infinite series that represents a function as an infinite polynomial expression.Tangent Line Approximation
: Tangent line approximation, also known as linear approximation or tangent line estimation, is a method that uses the equation of a tangent line at a specific point on a curve to approximate the value of the function near that point. It provides a close estimate when dealing with small intervals.Taylor Polynomial
: A Taylor polynomial is a polynomial approximation of a function centered around a specific point. It is used to estimate the value of the function at nearby points.© 2024 Fiveable Inc. All rights reserved.
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