Overview
At long last, we start to look at Taylor polynomials! These are polynomials that approximate a function, in much the same way that tangent lines act as “local linear approximations” for functions. These will quickly lead us to Taylor series, which are infinite series (involving a variable \(x\)) that give us new formulas for old functions.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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(Review, Chapter 1) Write a formula for the tangent line to a function \(f\) at \(x = a\).
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State the general formula for the \(n\)th order Taylor polynomial centered at \(x = a\), \(P_{n}(x)\).
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Describe some key features and purposes of a Taylor polynomial in words and in symbols. (For example: What is special about \(P_{n}'(a)\)? \(P_{n}''(a)\), etc.?)
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Be able, verbally and in writing, to identify the meaning of every part of the formula for the \(n\)th term of a Taylor polynomial.
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Given a function \(f(x)\) and a center \(x = a\), write a formula for the \(n\)th degree Taylor polynomial \(P_{n}(x)\) centered at \(x = a\). This includes correctly evaluating the derivatives \(f^{(k)}(a)\) and writing all other parts of the formula correctly.
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Make estimates for values of common functions by using a Taylor polynomial.
To prepare for class
Use these resources to become proficient with the basic objectives (see above) before class:
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Read the beginning of Section 8.5, up until (but not including) section 8.5.2 on “Taylor Series”).
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Watch the following videos:
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Optional Cal 1 Review: If you would like some review on tangent lines and local linear approximations, here are some videos from Cal 1:
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
After class
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Watch the beginning (stop at 14:25) of the following video by Grant Sanderson (“3Blue1Brown”):
