- We have a quiz today, over derivatives of logs and exponentials, and limits at infinity.
- Please continue your review of integration (Vol 2, 1.1: Approximating Areas -- Vol 2, 1.3: The Fundamental Theorem of Calculus).
- We will be moving on to Vol 2, Sec 3.1: Integration by Parts for Monday.
- Any questions about L'Hopital's rule before we move on? One more for the road:
Let's show that any positive power of \(x\), \(x^p\) where \(p>0\), grows faster than \(\ln(x)\) as \(x\to \infty\). Consider \[ \lim_{x\to\infty}\frac{\ln(x)}{x^p} \]
You have presumably seen all of this material, so I don't feel compelled to dwell on anything in particular. However we want to make sure that we have a solid foundation under our feet before moving forward.
Some of the key ideas discussed in the first three sections of Volume 2 of our textbook include
- Approximating
areas with rectangles (evaluating integrals geometrically,
as well as physically)
Note the use of the following in this animation:
- their use of summation notation,
- Riemann sums, and
- The floor function.
These areas often have a physical significance, as was the case in that 100-meter dash for which Noah Lyles received the gold medal:
- Conditions for integrability (continuity suffices -- but is not necessary -- on closed and finite intervals: give an example of a function which is integrable but not continuous on a closed interval.)
- The two "halves" of the FTC: Part
1,
\[
F \left(\right. x \left.\right) = \int_{a}^{x} f \left(\right. t \left.\right) d t
\]
shows us how to construct anti-derivatives, concluding that
\(F'(x) = f(x)\), with \(F\) the particular anti-derivative
that has the particular value \(F(a)=0\).
and Part 2: \[ \int_{a}^{b} f \left(\right. x \left.\right) d x = F \left(\right. b \left.\right) - F \left(\right. a \left.\right) \]
Basically integration and differentiation are mirror processes: and you can see the properties of one in the other, through the looking glass.
- Evaluating integrals algebraically
Because we have recently encountered some new derivative functions, we can now solve some integrals that we perhaps couldn't in the past. For some examples, consider the following:
- $\displaystyle \int^{e}_{1} \frac{3}{x^2}dx$
- $\displaystyle \int^{e}_{1} \frac{3}{x}dx$
- $\displaystyle \int^{\frac{\sqrt{2}}{2}}_0 \frac{x}{\sqrt{1-x^2}}dx$
- $\displaystyle \int^{\frac{\sqrt{2}}{2}}_0 \frac{1}{\sqrt{1-x^2}}dx$
- $\displaystyle \int^{10}_{0} \frac{2}{1+x^2}dx$
- $\displaystyle \int^{10}_{-10} x^{3001} dx$
- $\displaystyle \int^{0}_{-2} x e^{x^2} dx$
- $\displaystyle \int^{10}_{1} \frac{\ln(x)}{5x}dx$
- $\displaystyle \int^{e}_{1} \frac{3}{x^2}dx$
- Using integrals to compute mean values of functions (mean
value theorem)
The proof of this theorem invokes a couple of our old friends, a few other theorems: let's have a look at that. \[ m \left(\right. b - a \left.\right) \leq \int_{a}^{b} f \left(\right. x \left.\right) d x \leq M \left(\right. b - a \left.\right) \]
Draw a picture....
- Try out the animation which is mentioned in the text. I like Desmos, and certainly hope that you'll find it useful.
- Here's a L'Hopital worksheet for you to try (a bit of "review" of the inverse trig functions, too).
- Mathematica on-line is an option, if you are at a computer without Mathematica installed. I've tried it and it worked pretty well!