- , so the point is on the line.
- , so the slope is found by noting .

Thus, the equation of the tangent line is

If we “zoom in” on the graphs of the function and its tangent line at , denoted by , we see the following picture.

We can approximate sufficiently differentiable functions by polynomials.

When is given by an explicit formula in terms of , the point is found by evaluating the at , and the slope is found by evaluating the derivative at . By taking advantage of the point-slope form of a line

an equation for the tangent line is found. Let’s explore this in the context of an example.

Suppose that we want to approximate near . We pick as the point off of which our
approximation will be based, and find the equation of the tangent line to at
.

- , so the point is on the line.
- , so the slope is found by noting .

Thus, the equation of the tangent line is

If we “zoom in” on the graphs of the function and its tangent line at , denoted by , we see the following picture.

From tangent line approximation, we can approximate values of near . Visually, we can see this since the graphs are quite close. Computationally, we obtain the approximations by plugging -values into the equation of the tangent line; for instance, we can approximate by noting

The actual value of to three decimal places is , so the simple arithmetic needed to estimate using the tangent line produces a reasonable approximation. As we “zoom out” to a larger viewing window, however, the graphs start to become quite different.

Since evaluating polynomials involves only arithmetic operations, we would like to be able to use them to give better results than the tangent line approximation. Also, polynomials are easy to integrate and differentiate, so it would be nice to use polynomial approximations in applications that involve these operations. This will require that we try to extract the idea from the tangent line approximation in a way that allows us to generalize it.

Let’s look for a first degree polynomial of the form

where and are constants that must be determined.

What do we want out of our
approximation?

- We certainly want the function and the approximation to agree at the
-value off of which the approximation is based; that is, we want
In our example, we note

- .
- .

Thus, the requirement gives us that .

- We also want to make sure that the function and the approximation
change at the same initial rate so that both graphs head in similar
general directions. Since the derivative measures the rate of change, we
require
In our example, we note

- , so .
- , so .

Thus, the requirement gives us that .

Our approximation is thus , which matches the equation of the tangent line at .

While this should not be too surprising, it does allow for us to think of conditions that will allow for higher degree polynomial approximations. Suppose that we want to use a quadratic polynomial of the form

for making estimates. Note that is clearly not linear; in fact, it is concave-up on its domain. Note that by drawing tangent lines at different points near , the slopes are different, which is roughly what concavity quantifies. Slopes of tangent lines are found from the first derivative, so in order to measure how these slopes are changing, we should look at the derivative of the first derivative. This is really nothing new; we know already that concavity is measured using the second derivative.

We’ll keep the previous two conditions - that and - and also require that . We thus look for look for a polynomial whose coefficients are found by the requirements

By following the previous example, the reader can (and should) verify that we still have and . To find , note that

- , so .
- , so and thus .

Thus, the requirement gives us that , or .

The quadratic approximation is thus

Let’s now explore our approximations. Geometrically, we can interpret the effectiveness of the approximations by looking at their graphs.

We can also explore the approximations quantitatively for a given -value. For instance, if we want to approximate , we note that , so . We thus approximate by evaluating the polynomials at .

By noting that the actual value to three decimal place is , we can see that the quadratic approximation is better!

We can continue to look for higher degree polynomial approximations. Note that our approximations above require that the function be sufficiently differentiable at the point at which we wish to base the approximation.

Let be a function whose first derivatives exist at . The -th order Taylor polynomial
centered at is the polynomial

whose coefficients are found by requiring for each .

We will develop a more computationally efficient method for computing Taylor Polynomials in the next section, but we conclude this section with a question that explores the ideas put forth so far.

Suppose that is a function for which .

Which of the following that *could* be the second degree Taylor polynomial centered
at for ?

We must find the polynomial for which . Of the polynomials listed, only has this
property.

Let .

Is there enough information to determine what is?

Yes No

Is there enough information to determine what is?

Yes No

Since is the second degree Taylor polynomial centered at for , we only know for
sure that , , and . We can use to approximate at other -values, but there is no
guarantee that and will agree at any -value other than . The curious reader may
inquire whether this would provide a reasonable approximation, and this will be
discussed in a subsequent section.