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Quadratic Approximations

Recall from the previous section that for a function, we can approximate it at a point by creating another function that is just a tangent plane at that point, known as a linear approximation. Sometimes that is not sufficient, so we introduce another order in the approximation, known as a quadratic approximation.

Table of Contents​

Introduction​

By adding a quadratic term to the linear approximation, we can create a quadratic approximation. This is more accurate than a linear approximation due to the increased degree of freedom.

A quadratic approximation looks graphically like a paraboloid. Essentially, if you slice the paraboloid along any line, you get a parabola.

Recall that the linear approximation of a function has the form:

This is an affine function, which is a linear function with a constant term. For a quadratic approximation, we add quadratic terms to the linear approximation:

Preliminaries - From Linear to Quadratic​

In the previous section we computed the linear approximation of a function as:

If we plug into the linear approximation, we get:

This is an important fact, because it tells us that the linear approximation is tangent to the function at the point .

Recall that the next two terms ensure the rate of changes of the linear approximation match. This can be seen by taking the partial derivatives of the linear approximation:

With all of this in mind, we can set up an equation for the quadratic approximation:

(The reason we use and is because when we plug in , the terms with and will cancel out, making everything easier to deal with.)

Now all we need to do is find the coefficients , , and .

Finding the Coefficients​

For the purposes of this section, assume that the second partial derivatives of are continuous. This is a reasonable assumption, as most functions we deal with are continuous. This assumption fulfills Schwarz's theorem, meaning the mixed partial derivatives are equal.

Let's establish what we want:

Let's compute the second partial derivatives of :

Therefore, we have:

Substitute these values back into the equation for to get the quadratic approximation:

Example Problem: Constructing Quadratic Approximations​

Construct the quadratic approximation of the function at the point .

(Source)

The first step is to find the first partial derivatives of :

Next, find the second partial derivatives:

Now, plug in the values at :

Finally, plug these values into Equation :

Thus, the quadratic approximation of at is:

A graph of the function and its quadratic approximation is shown below:

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Approx point x
Approx point y

Summary and Next Steps​

In this section, we learned how to construct quadratic approximations of functions.

Here are the key points to remember:

  • A quadratic approximation is a function that approximates a function at a point using a quadratic polynomial.

  • The coefficients of the quadratic approximation can be found by taking the second partial derivatives of the function at the point.

  • The quadratic approximation is more accurate than the linear approximation due to the increased degree of freedom.

  • The full quadratic approximation is given by Equation :

Notice that this is quite a complex formula with a lot of terms. In the next section, we will see how to simplify this formula using the Hessian matrix.