AP Calculus BC · Unit 10: Infinite Sequences and Series
10.11 Finding Taylor Polynomial Approximations of Functions
In short
This topic focuses on constructing Taylor polynomials for a function f centered at x = a by calculating the coefficients using the nth derivative evaluated at a, divided by n!. You must understand that these polynomials can be used to approximate function values near the center x = a, and that increasing the degree of the polynomial often results in the polynomial approaching the original function over a specific interval.
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What else do you need to know about Finding Taylor Polynomial Approximations of Functions?
The content is exclusive to the AP Calculus BC course.
Can you do these?
- I can explain that as the degree of a Taylor polynomial increases, it approaches the original function over some interval.
- I can use a Taylor polynomial centered at x = a to approximate function values near x = a.
How is Finding Taylor Polynomial Approximations of Functions tested on the exam?
You are expected to apply mathematical definitions and theorems to construct Taylor polynomials (Suggested Skill 3.D) and identify re-expressions of mathematical information (Suggested Skill 2.C). On the AP Exam, you may need to approximate the value of a function using a Taylor polynomial and effectively communicate their supporting work. A recurrent theme is using limiting cases to move from discrete approximations to analytic calculations.
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Finding Taylor Polynomial Approximations of Functions: common questions
What do you need to know about Finding Taylor Polynomial Approximations of Functions for AP Calculus BC?
This topic focuses on constructing Taylor polynomials for a function f centered at x = a by calculating the coefficients using the nth derivative evaluated at a, divided by n!. You must understand that these polynomials can be used to approximate function values near the center x = a, and that increasing the degree of the polynomial often results in the polynomial approaching the original function over a specific interval.
How is Finding Taylor Polynomial Approximations of Functions tested on the AP Calculus BC exam?
You are expected to apply mathematical definitions and theorems to construct Taylor polynomials (Suggested Skill 3.D) and identify re-expressions of mathematical information (Suggested Skill 2.C). On the AP Exam, you may need to approximate the value of a function using a Taylor polynomial and effectively communicate their supporting work.
How do I know I have mastered Finding Taylor Polynomial Approximations of Functions?
Check you can do each of these: explain that as the degree of a Taylor polynomial increases, it approaches the original function over some interval; use a Taylor polynomial centered at x = a to approximate function values near x = a.
Where does Finding Taylor Polynomial Approximations of Functions fit in AP Calculus BC?
Finding Taylor Polynomial Approximations of Functions is topic 10.11 in Unit 10: Infinite Sequences and Series of AP Calculus BC.
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