Topics 10.1, 10.2, 10.5
Understand that higher degree polynomials can approximate a function with greater accurancy than a tangent line approximation.
Understand how the terms of a Taylor polynomial are constructed.
Write and use an nth degree Taylor polynomial centered around a certain x-value to approximate a function value.
Quick Lesson Plan
Today students are introduced to the idea of Taylor polynomials. So far in this course students have approximated functions with tangent lines (Topic 4.6) and Euler’s method (Topic 7.5). In this activity students explore how approximations can be improved by upgrading from a linear function to a higher order polynomial function.
Students are given the graph of a familiar function f(x)=cos x and use the tangent line at x=0 to approximate values of f(x). Questions 1-4 focus on creating, using, and examining the validity of a linear approximation for a function. This sets the stage for considering how we might improve the approximation. Students should see that a straight line (which happens to be horizontal in this case) does not take into account the curvature of the graph. Question 5 has students consider what else they know about the behavior of f at x=0. It is possible that students will have additional things to say besides the value of the derivative. They may note that f has a relative maximum at x=0 or changes from increasing to decreasing at x=0 or is concave down at x=0. The goal of the rest of the activity is to use this information to improve the model.
In question 6 students are asked to consider a quadratic model. This feels intuitive to students since the cosine function looks quite like a downward opening parabola near x=0. Parts a-e scaffold the process of determining the parameters of the quadratic function (which we’re calling T(x) - hint, hint!). The big takeaway is that a quadratic model can incorporate more information about the behavior of f, since it includes not only the correct y-value and correct slope, but also the correct concavity. A cubic function can incorporate even more information.
A helpful analogy is thinking about how platforms like Netflix and Spotify give music recommendations. The more information you give them when setting up your profile, the more accurately they will be able to pinpoint your preferences and make good recommendations. The more information put in up front, the more customized the final experience is to you! The more information put in about a function (i.e. its derivatives), the more customized the Taylor polynomial is to the actual function.
Note that the scaffolding of question 8 is greatly reduced. We’re confident that students will be able to generalize their approach from question 6 to find the coefficients of the cubic function or at least get started on it!
What do you think a tangent parabola would look like?
Why is the parabola closer to the cosine curve than the tangent line?
What patterns do you notice as you’re solving for the parameters?
How is each parameter (coefficient) in the polynomial connected to the derivatives of f?
Do you think this parabola would do a good job of predicting values of f(x) near x=2? Why or why not?
Could we use a function that is even more accurate than a cubic function to approximate f? How?
The debrief should focus on generalizing the pattern in the coefficients of the polynomial. Help students see why the coefficients can be written using factorials by showing the derivatives in an unsimplified form.
Since we are evaluating the Taylor polynomial at x=0, only the constant term of each derivative is nonzero. In the QuickNotes, help students see that this will happen regardless of where the Taylor polynomial is centered since (x-c)^n is always zero when x=c.
One thing that needs to be emphasized is that the Taylor polynomial does not provide a good approximation for all values of x. You don’t need to get into anything formal about intervals of convergence but students should understand that the Taylor polynomial is good at approximating values near x=c. Why? Because the information being used to create the Taylor polynomial is based on the derivatives of f at x=c, not the general derivative at any x-value.
Taylor polynomials and Taylor series are a big topic on the Calc BC exam, often making an appearance on question 6 of the FRQ section. Students should feel comfortable writing a Taylor polynomial of degree n by understanding how the coefficients of each term are created.When asking for a Taylor polynomial approximation, the exam rarely asks for anything above a degree 4 polynomial. Be prepared that information about the derivatives at x=c may be given in various representations. The third Check Your Understanding question is based on a released FRQ where the point and slope were shown on the graph and the higher order derivatives were given in a table.