Calculus II · Unit 4B · lesson
Taylor Series Centered at a
Build Taylor series at nonzero centers, track powers of x-a, and compare approximation quality near different centers.
Section overview
Constructing Taylor and Maclaurin seriesWhat this section is building
Build Taylor series at nonzero centers, track powers of x-a, and compare approximation quality near different centers.
Taylor coefficients encode local derivative data as a polynomial of increasing degree.
Choose the center, compute the derivative pattern, divide by factorials, and state whether you need a polynomial or an infinite series.
Assuming every smooth-looking function equals its Taylor series everywhere.
Learning objectives
construct Taylor series about a nonzero center and interpret the variable .
Taylor Series Centered at a
The center is the point where information is measured
A Taylor series centered at uses powers of , because approximation quality is organized by distance from . The coefficients encode derivative data at that center. Changing centers changes every coefficient even though the underlying function remains the same.
Nonzero centers are useful when approximating near a specific input or when a Maclaurin series converges too slowly for the desired calculation. Translating the variable with often makes the pattern clearer. The final series should state both its center and its convergence interval.
Move the center to where the approximation is needed
A Taylor series centered at uses powers of , so its accuracy is organized by distance from . Choosing the center near the target input often makes the expansion variable small and the approximation efficient.
The center also changes which derivative values are convenient. A logarithm near , for example, is naturally expanded in powers of , while trigonometric functions near zero favor Maclaurin series. Center choice is a modeling decision, not merely notation.
Read this graph as text
Taylor accuracy is local to the chosen center. The same curve is shown with two local polynomial approximations, each closely matching near its own center and separating farther away. The same function with Taylor models centered at two different points. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in taylor accuracy is local to the chosen center; color is never the only cue.
Why it matters: The same function with Taylor models centered at two different points.
The same curve is shown with two local polynomial approximations, each closely matching near its own center and separating farther away.
Taylor accuracy is local to the chosen center. The same function with Taylor models centered at two different points.
Taylor series
The Taylor series of centered at is
Expand around
For ,
Thus
for the appropriate interval around .
Approximate ln(1.1) from a center at 1
Let and center at . Since
we obtain
At , using three terms gives
close to the true value because is small.
Keep the expansion variable visible
When centered at , write and manipulate powers of . Expanding them too early often hides the center and creates coefficient errors.
u4b-taylor_series_centered_at_a-01What is the center of a series written in powers of ?
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The center makes the power factor zero.
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Find the quadratic Taylor polynomial for at .
Rewrite the geometric series around for .
Explain why powers of measure distance from the center.
Compare Maclaurin and Taylor terminology.
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