Math 142 - Week 2 Notes

Monday, February 22

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

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, February 24

Today we started with two important calculus formulas from last class:

• \(\displaystyle \frac{d}{dx} \ln u = \frac{u'}{u}\)
• \(\displaystyle \int \frac{du}{u} = \ln |u| + C\)

Notice the absolute value inside the logarithm in the integral formula. That is what you need if \(u\) ends up being negative. We did these problems:

1. \(\displaystyle \int \frac{2x+3}{x^2 + 3x + 4} \, dx\)

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

Notice that the second integral doesn’t have a logarithm in the answer!

Logarithmic Differentiation

For some functions \(y = f(x)\), it helps to calculate the derivative if you take a natural log of both sides first and then use implicit differentiation to find \(y'\). We did this with these two examples:

1. \(\displaystyle y = \frac{x^2 \sqrt{x+1}}{(x-5)^3}\)

2. \(y = x^x\)

Logarithms to Other 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?

We also pointed out the change of base formula for logs:

• \(\log_b(x) = \dfrac{\ln x}{\ln b}\)

A good exercise is to see if you can prove this using the definition of \(\log_b(x)\).

Friday, February 26

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 \(x=1\).

5. Find the maximum of \(y = x e^{-x}\).

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