We introduce Taylor polynomials for functions of several variables.
We have a similar formula for functions .
Basically, given a function , the second degree Taylor polynomial at a point is a polynomial “cooked-up” so that:
- The values are equal: .
- The first partial derivatives are equal: and .
- The second partial derivatives are equal: , , and .
Here’s the plan. Soon we will be trying to find maximums and minimums for functions of two variables. The second degree Taylor polynomial will be the key to developing a “second derivative test” for identifying these extrema.
Computing the Taylor polynomial is not so bad, you just need to get the hang of it.
Now that we have a formula and we (hopefully!) can apply it. Let’s finish by talking about what is really going on. Given a function , the Taylor polynomial is a polynomial “cooked-up” to share the value of the function, meaning and share values of the first derivatives, meaning whenever . The exact same idea is true for functions of several variables. Let’s explain the construction of the Taylor polynomial as an iterative process. Given (and similarly for functions ) the degree zero Taylor polynomial is just the value of the function where is the center of the Taylor polynomial. The degree one Taylor polynomial is just the degree zero polynomial plus the first partial derivatives with respect to multiplied by The degree two Taylor polynomial can be found by adding the degree one Taylor polynomial to one-half of all the second partial derivatives with respect to and multiplied by
- when taking the partial derivative with respect to ,
- when taking the partial derivative with respect to , and
- when taking the partial derivative with respect to and .
Putting this together, we see
The interested reader can (repeatedly) differentiate to see that its value at and the values of the first two derivatives of do indeed match those of .
This final section is for the interested student, and is not required for this course.
Recall that if is a function whose first derivatives exist at . The Taylor polynomial of degree of centered at is
This formula is complex and will take some unpacking. We’ll walk you through this.
First note that
This means for any function , the th degree Taylor polynomial for at is just
Now let’s look at the st degree Taylor polynomial:
Now we ask, what is ? Well, computing the dot product, and to find , we distribute to obtain
This means for any function , the st degree Taylor polynomial for at is just the tangent “plane” for at .
To get our hands on the nd degree Taylor polynomial, we will specialize to functions . Let and let . Write with me:
Now we ask ourselves, what is ? Well, we know that and
Now we use the distributively property and (since we are assuming that all derivatives of exist) we have that so We can now unpack our second degree Taylor polynomial:
and finally we see:
Whew. That was a lot of work. However, as we have said before, all you need to know right now, is how to compute the degree two Taylor polynomial for functions of two variables.