
We introduce Taylor polynomials for functions of several variables.

Recall the definition of a Taylor polynomial:
Let $f(x) = \sin (x)$. Compute the degree $7$ Taylor polynomial centered at $x=0$.
Let $f(x) = \cos (x)$. Compute the degree $7$ Taylor polynomial centered at $x=0$.
Let $f(x) = e^x$. Compute the degree $7$ Taylor polynomial centered at $x=0$.

We have a similar formula for functions $F:\R ^n\to \R$.

As you can see, it is more complex than the formula for a single variable. Good news everyone: In this class, we will only compute the degree two polynomial for functions of two variables. In this case $P_2$ is:

Basically, given a function $F:\R ^2\to \R$, the second degree Taylor polynomial $P_2$ at a point $\vec {c}$ is a polynomial “cooked-up” so that:

• The values are equal: $P_2(\vec {c}) = F(\vec {c})$.
• The first partial derivatives are equal: $P_2^{(1,0)}(\vec {c}) = F^{(1,0)}(\vec {c})$ and $P_2^{(0,1)}(\vec {c}) = F^{(0,1)}(\vec {c})$.
• The second partial derivatives are equal: $P_2^{(2,0)}(\vec {c}) = F^{(2,0)}(\vec {c})$, $P_2^{(0,2)}(\vec {c}) = F^{(0,2)}(\vec {c})$, and $P_2^{(1,1)}(\vec {c}) = F^{(1,1)}(\vec {c})$.

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.

Try it, you might like it

Computing the Taylor polynomial is not so bad, you just need to get the hang of it.

Compute the degree $2$ Taylor polynomial for: centered at $(0,0)$.
Start by making a table of partial derivatives along with their value when evaluated at $(0,0)$.
Compute the degree $2$ Taylor polynomial for: centered at $(-1/2,1/8)$.
Start by making a table of partial derivatives along with their value when evaluated at $(-1/2,1/8)$.
Compute the degree $2$ Taylor polynomial for: centered at $(1,2)$.
Start by making a table of partial derivatives along with their value when evaluated at $(1,2)$.

In other words

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 $f:\R \to \R$, the Taylor polynomial is a polynomial “cooked-up” to share the value of the function, meaning and share values of the first $d$ derivatives, meaning whenever $0\le i\le d$. 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 $F:\R ^2\to \R$ (and similarly for functions $F:\R ^n\to \R$) the degree zero Taylor polynomial is just the value of the function where $\vec {c}=\vector {c_1,c_2}$ 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 $x_i$ multiplied by $(x_i-c_i)$ 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 $x$ and $y$ multiplied by

• $(x-c_1)^2$ when taking the partial derivative with respect to $x$,
• $(y-c_2)^2$ when taking the partial derivative with respect to $y$, and
• $2(x-c_1)(y-c_2)$ when taking the partial derivative with respect to $x$ and $y$.

Putting this together, we see

The interested reader can (repeatedly) differentiate $P_2(x,y)$ to see that its value at $\vec {x}=\vec {c}$ and the values of the first two derivatives of $P_2(x,y)$ do indeed match those of $F(x,y)$.

Unpacking the general formula

This final section is for the interested student, and is not required for this course.

Recall that if $F:\R ^n\to \R$ is a function whose first $d$ derivatives exist at $\vec {x}=\vec {c}$. The Taylor polynomial of degree $d$ of $F$ centered at $\vec {x}=\vec {c}$ is

This formula is complex and will take some unpacking. We’ll walk you through this.

The degree zero Taylor polynomial

First note that

This means for any function $F:\R ^n\to \R$, the $0$th degree Taylor polynomial for $F$ at $\vec {x}=\vec {c}$ is just

The degree one Taylor polynomial

Now let’s look at the $1$st degree Taylor polynomial:

Now we ask, what is $(\vec {a}\dotp \grad )F$? Well, computing the dot product, and to find $(\vec {a}\dotp \grad )F$, we distribute $F$ to obtain

So

This means for any function $F:\R ^n\to \R$, the $1$st degree Taylor polynomial for $F$ at $\vec {x}=\vec {c}$ is just the tangent “plane” for $F$ at $\vec {x}= \vec {c}$.

The degree two Taylor polynomial

To get our hands on the $2$nd degree Taylor polynomial, we will specialize to functions $F:\R ^2\to \R$. Let $\vec {c}=\vector {c_1,c_2}$ and let $\vec {x} = \vector {x,y}$. Write with me:

Now we ask ourselves, what is $(\vec {a}\dotp \grad )^2 F(\vec {x})$? Well, we know that and

Now we use the distributively property and (since we are assuming that all derivatives of $F$ exist) we have that so We can now unpack our second degree Taylor polynomial:

as

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.