- , 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.
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.
In our example, we note
Thus, the requirement gives us that .
In our example, we note
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
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.
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.
Is there enough information to determine what is?
Is there enough information to determine what is?