I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on different ways of computing the limit

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on different ways of solving the contest problem “If , what is ?”

One of my colleagues placed the following problem on an exam for his Calculus II course…

and was impressed by the variety of correct responses that he received. I thought it would be fun to discuss some of the different ways that this limit can be computed.

Method #3. A trigonometric identity. When we see inside of an integral, one kneejerk reaction is to try the trigonometric substitution . So let’s use this here. Also, since , we can change the limit to be :

The following problem appeared on the American High School Mathematics Examination (now called the AMC 12) in 1988:

If , what is ?

When I presented this problem to a group of students, I was pleasantly surprised by the amount of creativity shown when solving this problem.

Yesterday, I presented a solution using a Pythagorean identity, but I was unable to be certain if the final answer was a positive or negative without drawing a picture. Here’s a third solution that also use a Pythagorean trig identity but avoids this difficulty. Again, I begin by squaring both sides.

Yesterday, I used the Pythagorean identity again to find . Today, I’ll instead plug back into the original equation :

Unlike the example yesterday, the signs of and must agree. That is, if , then must also be positive. On the other hand, if , then must also be negative.

If they’re both positive, then

,

and if they’re both negative, then

.

Either way, the answer must be .

This is definitely superior to the solution provided in yesterday’s post, as there’s absolutely no doubt that the product must be positive.

The following problem appeared on the American High School Mathematics Examination (now called the AMC 12) in 1988:

If , what is ?

When I presented this problem to a group of students, I was pleasantly surprised by the amount of creativity shown when solving this problem.

Yesterday, I presented a solution using triangles. Here’s a second solution that I received: begin by squaring both sides and using a Pythagorean trig identity.

We use the Pythagorean identity again to find :

Therefore, we know that

,

so the answer is either or . However, this was a multiple-choice contest problem and was not listed as a possible answer, and so the answer must be .

For a contest problem, the above logic makes perfect sense. However, the last step definitely plays to the fact that this was a multiple-choice problem, and the concluding step would not have been possible had been given as an option.

Amazingly, the integral below has a simple solution:

Even more amazingly, the integral ultimately does not depend on the parameter . For several hours, I tried to figure out a way to demonstrate that is independent of , but I couldn’t figure out a way to do this without substantially simplifying the integral, but I’ve been unable to do so (at least so far).

So here’s what I have been able to develop to prove that is independent of without directly computing the integral .

Earlier in this series, I showed that

Yesterday, I showed used the substitution to show that was independent of . Today, I’ll use a different method to establish the same result. Let

.

Notice that I’ve written this integral as a function of the parameter . I will demonstrate that , so that is a constant with respect to . In other words, does not depend on .

To do this, I differentiate under the integral sign with respect to (as opposed to ) using the Quotient Rule:

I now apply the trigonometric substitution , so that

and

The endpoints of integration change from to , and so

.

I’m not completely thrilled with this demonstration that is independent of , mostly because I had to do so much simplification of the integral to get this result. As I mentioned in yesterday’s post, I’d love to figure out a way to directly start with

and demonstrate that is independent of , perhaps by differentiating with respect to and demonstrating that the resulting integral must be equal to 0. However, despite several hours of trying, I’ve not been able to establish this result without simplifying first.

Amazingly, the integral below has a simple solution:

Even more amazingly, the integral ultimately does not depend on the parameter . For several hours, I tried to figure out a way to demonstrate that is independent of , but I couldn’t figure out a way to do this without substantially simplifying the integral, but I’ve been unable to do so (at least so far).

So here’s what I have been able to develop to prove that is independent of without directly computing the integral .

Earlier in this series, I showed that

I now multiply the top and bottom of this last integral by :

I now employ the substitution , so that . Since , the endpoints of integration do not change, and so

.

This final integral is independent of .

Since is independent of , I can substitute any convenient value of that I wish. For example, I can let without altering the value of :

This provides a considerable simplification of the integral which also opens up additional methods of evaluation.

My wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by and then employing the substitution (after using trig identities to adjust the limits of integration).

But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.

So far, I have shown that

where and (and is a certain angle that is now irrelevant at this point in the calculation).

There are actually a couple of ways for computing this last integral. Today, I’ll lay the foundation for the “magic substitution”

With this substitution, the above integral will become a rational function, which can then be found using standard techniques.

First, we use some trig identities to rewrite in terms of :

Next, I’ll replace by :

.

Second, for the sake of completeness (even though it isn’t necessary for this particular integral), I’ll rewrite in terms of :

Next, I’ll replace by :

.

Third, again for the sake of completeness,

.

Finally, I need to worry about what happens to the :

These four substitutions can be used to convert trigonometric integrals into some other integral. Usually, the new integrand is pretty messy, and so these substitutions should only be used sparingly, as a last resort.

I’ll continue this different method of evaluating this integral in tomorrow’s post.

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

In this series, I’ll explore different ways of evaluating this integral.So far in this series, I’ve shown that

This last integral can be evaluated using a standard trick. Let , so that . We differentiate this last equation with respect to :

Employing a Pythagorean identity, we have

Since , we may rewrite this as

Integrating both sides with respect to , we obtain the antiderivative

We now employ this antiderivative to evaluate :

And so, at long last, we have arrived at the solution for the integral . Surprisingly, the answer is independent of the parameter .

These last few posts illustrated the technique that I used to compute this integral for my wife in support of her recent paper in Physical Review A. However, I had more than a few false starts along the way… or, at the time, I thought they were false starts. It turns out that there are multiple ways of evaluating this integral, and I’ll explore another method of attack beginning with tomorrow’s post.

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Tracy Leeper. Her topic, from Precalculus: verifying trigonometric identities.

Many students when first learning about trigonometric identities want to move terms across the equal sign, since that is what they have been taught to do since algebra, however, in proving a trigonometric identity only one side of the equality is worked at a time. Therefore my idea for an activity to help students is to have them look at the identities as a puzzle that needs to be solved. I would provide them with a basic mat divided into two columns with an equal sign printed between the columns, and give them trig identities written out in a variety of forms, such as on one strip, and written on another strip. Other examples would also include having on one, and on another. The students will have to work within one column, and step by step, change one side to eventually reflect the term on the other side, and each strip has to be one possible representation of the same value. By providing the students with the equivalent strips, they will be able to construct the proof of the identity. I feel that giving them the strips will allow them to see different possibilities for how to manipulate the expression, without leaving them feeling lost in the process, and by dividing the mat into columns, they can focus on one side, and see that the equivalency is maintained throughout the proof. The students would need to arrange the strips into the correct order to prove the left hand side is equivalent to the right hand side, while reinforcing the process of not moving anything across the equal sign.

Trigonometry identities are used in most of the math courses after pre-calculus, as well as the idea of proving an equivalency. If the students learn the concept of proving an equivalency that will help them construct proofs for any future math courses, as well as learning to look at something given, and be able to see it as parts of a whole, or just be able to write it a different way to assist with the calculations. If students learn to see that

,

their ability to manipulate expressions will dramatically improve, and their confidence in their ability will increase, as well as their understanding of the complexities and relations throughout all of mathematics. The trigonometric identities are the fundamental part of the relationships between the trig functions. These are used in science as well, anytime a concept is taught about a wave pattern. Sound waves, light waves, every kind of wave discussed in science are sinusoidal wave. Anytime motion is calculated, trigonometry is brought into the calculations. All students who wish to progress in the study of science or math need to learn basic trigonometric identities and learn how to prove equivalency for the identities. Since proving trigonometric identities is also a practice in logical reasoning, it will also help students learn to think critically, and learn to defend their conjectures, which is a valuable skill no matter what discipline the student pursues.

For learning how to verify trigonometric identities, I like the Professor Rob Bob (Mr. Tarroy’s) videos found on youtube. He’s very energetic, and very thorough in explaining what needs to be done for each identity. He also gives examples for all of the different types of identities that are used. He is very specific about using the proper terms, and he makes sure to point out multiple times that this is an identity, not an equation, so terms cannot be transferred across the equal sign. He also presents options to use for a variety of cases, and that sometimes things don’t work out, but it’s okay, because you can just erase it and start again. I also like that he uses different colored chalk to show the changes that are being made. He is very articulate, and explains things very well, and makes sure to point out that he is providing examples, but it’s important to remember that there are many different ways to prove the identity presented. I enjoyed watching him teach, and I think the students would enjoy his energy as well.