Math 142 - Week 6 Notes

Monday, September 28

Today we introduced the partial fraction decomposition. This is an algebra technique that helps integrate rational functions. Here is a video example. We applied the technique to solve the following integrals:

  1. \(\displaystyle \int \frac{x+5}{x^2+x-2} \, dx\).

  2. \(\displaystyle \int \frac{1}{P(1-P)} \, dP\).

Partial fractions works for any rational function where the degree of the numerator (top) is less than the degree of the denominator (bottom) and the denominator factors into linear (i.e., degree 1) factors that all have different roots. If the degree of the numerator is bigger than or equal to the degree of the denominator, then use polynomial long division first. Here is a video review of how polynomial long division works.

We did this example in class:

  1. \(\displaystyle \int \frac{x^3 + x}{x-1} \, dx\).

We left one example for next time:

  1. \(\displaystyle \int \frac{x^4-5x^2+4x}{x^2-4} \, dx\). Hint: Since the degree on top is bigger than the degree on bottom, start with polynomial long division.

Wednesday, September 30

Today we switched from integrals to limits. We will be using limits a lot in the next few weeks, so we need a fast way to compute them. One option that works frequently (but not always) is L’Hospital’s rule. Here is a quick video explanation. We did the following examples:

  1. \(\displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x}\).

  2. \(\displaystyle \lim_{x \rightarrow 5^+} \frac{x}{x-5}\). (You can’t use L’Hospital’s rule here! Why not?)

  3. \(\displaystyle \lim_{x \rightarrow \infty} \frac{x^2}{e^x}\).

  4. \(\displaystyle \lim_{x \rightarrow 3^-} \frac{x^2 - 2x - 3}{x^2 - 5x +6 }\).

  5. \(\displaystyle \lim_{x \rightarrow \infty} \frac{\ln x}{1-x}\).

  6. \(\displaystyle \lim_{x \rightarrow \pi^+} \frac{\sin x}{1-\cos x}\). (Watch out, this one isn’t L’Hospital’s rule either!)

In addition to the indeterminant forms \(\frac{\infty}{\infty}\) and \(\frac{0}{0}\), there are other indeterminant forms that come up when working with limits. These include: \[0 \cdot \infty, ~ \infty - \infty, ~ 0^0, ~ 1^\infty, ~ 0^\infty\] Here are some ideas on how to deal with these when they pop up in limit calculations:

  1. \(\displaystyle \lim_{x \rightarrow 0^+} x \ln x\). Hint: re-write the problem as \(\displaystyle \lim_{x \rightarrow 0^+} \dfrac{\ln x}{x^{-1}}\) so you can apply L’Hospital’s rule.

  2. \(\displaystyle \lim_{x \rightarrow \tfrac{\pi}{2}^-} (\sec x - \tan x)\). Hint: combine the secant and tangent into one fraction by switching to sines and cosines.

  3. \(\displaystyle \lim_{x \rightarrow 0^+} x^x\). Hint, let \(A = \lim_{x \rightarrow 0^+} x^x\) and take the natural log of both sides.

Friday, October 2

Today we talked about improper integrals which are integrals involving infinity (either in the bounds, or because of a vertical asymptote). You deal with these just like any other definite integral, except you might have to calculate a limit to find the answer. We did several examples:

  1. \(\displaystyle \int_1^{\infty} \frac{1}{x^2} \, dx\). This one turns out to just be 1, which illustrates a very important idea: The area under an infinite curve can still be finite!

  2. \(\displaystyle \int_2^3 \frac{1}{\sqrt{x-2}} \, dx\). You could integrate this one without even realizing that it is technically an improper integral. But it you look at the graph, you’ll see why it counts as an improper integral.

  3. \(\displaystyle \int_2^\infty \frac{1}{x} \, dx\).

  4. \(\displaystyle \int_{-\infty}^0 x e^{-x} \, dx\).

If the value of the integral is a finite number, we say it converges. Otherwise, it diverges.

  1. For what values of \(p\) does the integral \(\displaystyle \frac{1}{x^p} \, dx\) converge? Answer, \(p > 1\). (This is known as the \(p\)-test).

We also talked about another simple idea known as the comparison test. If \(f(x)\) and \(g(x)\) are nonnegative functions and \(f(x) \le g(x)\), then \(\int f \le \int g\). This has two consequences:

  1. \(\displaystyle \int_1^e \frac{1}{x \ln x} \, dx\). Hint: Let \(u = \ln x\) and use u-substitution.

  2. Compare the previous integral with \(\displaystyle \int_1^e \frac{1}{\ln x} \, dx\). Which integral is bigger? What does that mean about whether this integral converges or diverges?