Math 142 - Week 2 Notes

Monday, August 31

There is a problem with the power rule for integrals: \(\displaystyle \int x^p \, dx = \frac{1}{p+1} x^{p+1} + C\) only makes sense if \(p+1 \ne 0\). In other words, the power rules doesn’t work if \(p = -1\). On the other hand, \[y = \frac{1}{x}\] is a perfect good function, and it is actually very useful to have a formula to represent the area under \(1/x\). It turns out that the antiderivative of \(1/x\) is a new function that is very very useful. It is the natural logarithm.

From the definition, we can see these properties of the natural logarithm:

1. \(\ln(1) = 0\).
2. \(\ln x\) only makes sense when \(x > 0\) and it is a continuous function.
   
3. \(\frac{d}{dx} \ln x = \frac{1}{x}\).
4. \(\ln x\) is always increasing.
   
5. The second derivative of \(\ln x\) is always negative, so it is always concave down.

These are all nice facts, but the most important thing about logarithms is that they convert multiplication to addition! In fact, we have these three properties:

Logarithm Properties

1. \(\displaystyle \ln (ab) = \ln a + \ln b\)
2. \(\displaystyle \ln \left(\frac{a}{b} \right) = \ln a - \ln b\)
3. \(\displaystyle \ln (a^n) = n \ln a\)

One more consequence of these properties, is that the logarithm has no upper bound. If you want to make a bigger y-value, just raise the inside to a higher power (assuming the inside is bigger than 1)!

Examples

We did the following examples in class.

1. Simplify \(\displaystyle \ln \frac{(x^2+5)^4 \sin x}{\sqrt{x}}\).

2. Simplify \(\ln a + \tfrac{1}{2} \ln b\).

3. \(\displaystyle \frac{d}{dx} \ln (x^3+1)\).

4. \(\displaystyle \frac{d}{dx} \ln (\sin x)\).

5. \(\displaystyle \ln \left( \frac{x+1}{\sqrt{x+2}} \right)\).

The Integral of \(\tan x\)

Since \(\tan x = \dfrac{\sin x}{\cos x}\), you can use the substitution \(u = \cos x\) to find the integral of the tangent:

\[\int \tan x \, dx = \int \frac{\sin x}{ \cos x} \, dx\] Let \[u = \cos x ~~~~~~ du = - \sin x \, dx\] Then \[\int \tan x \, dx = - \int \frac{1}{u} \, du = - \ln \cos x + C.\]

More Integral Examples

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

7. \(\displaystyle \int \frac{4x}{x^2+1} \, dx\).

Wednesday, September 2

Today, we started with this activity about logarithmic scales.

Then we talked about logarithms to different bases.

The natural logarithm function \(\ln(x)\) is the same as the base-\(e\) logarithm \(\log_e(x)\), where \(e\) is Euler’s number.

No matter what base we have, all logarithms satisfy the three logarithm laws:

1. \(\displaystyle \log (ab) = \log a + \log b\)
2. \(\displaystyle \log \left(\frac{a}{b} \right) = \log a - \log b\)
3. \(\displaystyle \log (a^n) = n \log a\)

Base-10 logarithms are especially easy to understand because they represent orders of magnitude, and can be estimated pretty easily. They also work well with scientific notation.

To practice working with logarithms, we solved the following exercises in class:

1. Compute \(\log_2(16^3)\) without a calculator.

2. Simplify \(\log_3(\tfrac{4}{3}) - \log_3(12)\).

3. Solve \(\log_x(2) = 3\).

4. Solve \(e^{3x+5} = 7\).

5. Solve \(10^x = 5(3^x)\).

6. If the population of a town grows at 6% per year, how long until the population doubles?

Thursday, September 3

Today we mostly went over the homework problems. We spend a little more time on logarithmic differentiation, using this example:

1. Find \(\displaystyle \frac{d}{dx} \frac{(\sin x)\sqrt{x+1}}{x}\).

We also did a few other examples that were similar to the homework problems. For example, this is similar to problem 2:

2. Find \(\displaystyle \frac{d}{dx} \ln ( \sin ( \ln x) )\).

The only new material we covered was how to integrate \(y = 1/x\) when the x-values are negative. I stated the following correction of the formula for the integral of \(1/x\):

\[\int \frac{1}{x} \, dx = \ln |x| + C \]

Those absolute values are only important if the x-values inside are negative, but that can happen. For example, if you want to calculate:

\[\int_{-8}^{-2} \frac{1}{x} \, dx = \ln|-2| - \ln|-8| = \ln(2) - \ln(8) = \ln (\tfrac{1}{4}) = - \ln 4.\]

Friday, September 4

Today we talked about the exponential function \(e^x\). It is the inverse of \(\ln x\), so \[\ln (e^x) = x ~~~~\text{and}~~~~ e^{\ln x} = x.\]

We reviewed the rules for exponents:

1. \(\displaystyle e^{x+y} = e^x e^y\)
2. \(\displaystyle e^{x-y} = \tfrac{e^x}{e^y}\)
3. \(\displaystyle (e^x)^y = e^{xy}\)

We also showed how to differentiate and integrate \(e^x\):

1. \(\displaystyle \tfrac{d}{dx} e^x = e^x\)
2. \(\displaystyle \int e^x \, dx = e^x + C\)

With these rules in hand, we attacked the following practice problems:

1. Differentiate \(y = e^{\tan x}\)

2. \(\displaystyle \frac{d}{dx} e^{-4x} \sin x\)

3. \(\displaystyle \int x^2 e^{x^3} \, dx\)

4. Find the area under \(y = e^{-3x}\) from \(x=0\) to \(x=1\).

We also talked about the change of basis formulas for exponentials and logarithms:

\[ a^x = e^{x \ln a} ~~~~ \text{and} ~~~~ \log_b(x) = \frac{\ln(x)}{\ln(b)}.\]

We used those two formulas on these exercises:

1. \(\displaystyle \frac{d}{dx} 2^{\cos x}\)

2. Simplify \(3^{2/\ln 3}\)

3. \(\displaystyle \frac{d}{dx} \log_{10}(x^2)\)