Sections 3.8, 4.6, and 6.1 Numerical Techniques

For our last PowerPoint of the year we will take a quick tour of some numerical techniques that we have skipped along the way. First, we go all the way back to section 3.8 (which starts on page 229) to discuss a technique for approximating roots of a function. This is called Newton’s Method.

Sections 3.8, 4.6, and 6.1 Numerical Techniques Newton’s Method is a recursive technique that relies on an initial guess (hopefully one that is reasonably close to a true value) and on the construction of of tangent lines. The idea here is that the x - intercept of a tangent line formed at our first guess will serve as the next x -guess for the root of the function.

Sections 3.8, 4.6, and 6.1 Numerical Techniques What feels silly about this technique is the fact that we will rely on our calculators to check these estimates and, of course, our calculator can very quickly give us an estimate. Let’s look at a visual here: hod/Newton'sMethodProof/Links/Newton'sMethodP roof_lnk_2.html

Sections 3.8, 4.6, and 6.1 Numerical Techniques Your text provides an excellent (I think) summary of this technique on page 229 with examples on page 230. We will use the list capability of our calculator to build these tables. The example we will use is to find a real zero of the function

Sections 3.8, 4.6, and 6.1 Numerical Techniques The next section we will revisit is section 4.6 which discusses another approximation technique for finding the area under a curve. We have already discussed inscribed rectangles (underestimates),circumscribed rectangles (overestimates), and midpoint estimates.

Sections 3.8, 4.6, and 6.1 Numerical Techniques However, no matter what we choose as our height for a rectangle, we will always have a noticeable amount of space left over. Trapezoids are a much more efficient way to carve up that space. The technique is identical to our rectangular approximations other than the fact that we need to use our trapezoid area formula.

Sections 3.8, 4.6, and 6.1 Numerical Techniques This section also introduces a technique that uses a parabola approximation to estimate areas under a variety of curves. This technique is called Simpson’s Rule and it is described on page 312. This is definitely another historical snapshot of the development of Calculus

Sections 3.8, 4.6, and 6.1 Numerical Techniques The final technique we will look at is a very new idea in the teaching of and learning of Calculus. The application is related to looking at classes of solutions to differential equations. We are looking for a visual representation of the class of curves that can solve a given differential.

Sections 3.8, 4.6, and 6.1 Numerical Techniques This idea is described in section 6.1 of your text and my students in the past have found this to be a quick concept to master. Let’s look at a good visual link: elds.html elds.html