We did it! Good luck preparing for finals! My office hours next week have been posted in SacCT.

- Miss you already! -

1. Introduction through Example 1
2. Example 3
3. Table 1 on page 768

1. Find an equation of the tangent line at $x=0$ for $\displaystyle e^x$. Compare your answer to the Maclaurin series for $\displaystyle e^x$ (on page 768).


2. Find an equation of the tangent line at $x=0$ for $\displaystyle\sin(x)$. Compare your answer to the Maclaurin series for $\displaystyle\sin(x)$ (on page 768).


3. Find an equation of the tangent line at $x=2$ for $\displaystyle e^x$. Compare your answer to the Taylor series at $a=2$ for $\displaystyle e^x$ (see Example 3 on page 763).


4. Try to generalize your findings. Suppose $f(x)$ has a Taylor series at $x=a$ given by \[f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots\] What is the relationship between this series and the tangent line for $f(x)$ at $x= a$?

1. Examples 5, 6, and 7

1. Use a substitution in the geometric series \[\frac{1}{1-x} = \sum_{n=0}^\infty = 1+ x + x^2 + x^3 + \cdots , \quad\quad -1\lt x\lt 1\] to find a power series representation for $\displaystyle\frac{1}{1+\frac{x}{3}}$. Make sure to find the new interval of convergence.


2. Now use your previous answer to find a power series representation for $\displaystyle\frac{1}{3+x}$. Hint: notice that $\displaystyle\frac{1}{3+x} = \frac{1}{3}\frac{1}{1+\frac{x}{3}}$


3. Modify your answer from the previous part to find a power series representation for $\displaystyle f(x) = \frac{2x}{3+x}.$


4. Make sure that you can write you answer in closed form as well as expanded form.

1. Theorem 2 on page 754 and Notes 1, 2, and 3 that follow it.
2. Examples 4 and 5

1. Find expressions for $f'(x)$ and $f''(x)$ by working with the expanded form of $f(x)$.
2. Show that $f''(x) = -f(x)$.
3. Find $f(0)$ and $f'(0)$.
4. Meditate, hazard a guess as to a simpler formula for $f(x)$.
5. Go to Desmos and graph $\sum_{n=0}^{25} (-1)^n \frac{x^{2n}}{(2n)!}$ to try to confirm your conjecture.

1. Theorem 4 on page 749
2. Examples 1,2,4,5

1. What is $f(0)$?
2. For what values of $x$ does $f$ converge? (In other words, what is the domain of $f$?)
3. What do you think is a formula for $f'(x)$? (You do not need to justify your answer! Remember, $f(x)$ looks like a polynomial.)
4. Based on what you found in the previous parts, hazard a guess as to a simpler formula for $f(x)$.
5. Go to Desmos and graph $\sum_{n=0}^{100} \frac{x^n}{n!}$ to try to confirm your conjecture.

1. 11.7: Read everything (including the examples)! The section is short and helpful.
2. 11.8: Introduction

1. Let $\displaystyle f(x) = \sum_{n=0}^\infty x^n$. Writing $f(x)$ out, we get \[f(x) = 1 + x + x^2 + x^3 + x^4 + x^5 +\cdots\]
   
   • Is $f\left(0\right)$ defined? (That is, does $f(0)$ converge?) If so, what is it?
   • Is $f\left(1\right)$ defined? If so, what is it?
   • Is $f\left(\frac{1}{2}\right)$ defined? If so, what is it?
   • Explain why the domain of $f$ is $-1 < x < 1$.
   • Explain why $f(x)$ is given by the formula $f(x) = \frac{1}{1-x}$ whenever $-1 < x < 1$.

Back to it.

1. Definition 1 on page 737
2. Theorem 3 on page 738
3. Examples 1,2,3

1. Explain why you cannot use the Comparison Test or LCT to compare $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$ to anything.
2. Explain why the Alternating Series Test does not apply to $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$.
3. Show that you can use the Comparison Test to compare $\sum_{n=1}^\infty \left\lvert\frac{\sin\left(3^n\right)}{2^n}\right\rvert$ to a convergent geometric series.
4. What, if anything, does Theorem 3 on page 738 allow you to conclude about $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$?

Break time!!

Good luck with your studying!!

1. Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
2. Explain why you cannot use the LCT to compare $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
3. What's your feeling about the convergence or divergence of $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$? Write this down, with your reasoning, before going to the next part.
4. Look at the Desoms graph here: https://www.desmos.com/calculator/npezpry5vc. The orange dots represent the sequence of terms $\frac{(-1)^{n+1}}{n}$. The black dots represent the sequence of partial sums $s_n = \sum_{i=1}^n\frac{(-1)^{i+1}}{i}$. Did this change or confirm your feeling about convergence or divergence? If you want, you can compare the previous graph to the graph of the terms and partial sums of the harmonic series here: https://www.desmos.com/calculator/3vvosxcslp.

1. The "Limit Comparison Test" (LCT)
2. Example 4

1. Explain why $\frac{1}{n!} \le \frac{1}{n}$ for $n\ge 1$.
2. Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty\frac{1}{n!}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
3. Explain why $\frac{1}{n!} \le \frac{1}{n(n-1)}$ for $n\ge 2$.
4. Show that $\sum_{n=1}^\infty\frac{1}{n(n-1)}$, which is equal to $\sum_{n=1}^\infty\frac{1}{n^2-n}$, converges by comparing it to $\sum_{n=1}^\infty\frac{1}{n^2}$ with the LCT.
5. What can you conclude about $\sum_{n=1}^\infty \frac{1}{n!}$? And why

1. The blue boxes, i.e. the "(Direct) Comparison Test" (DCT) and the "Limit Comparison Test" (LCT)
2. Note 1 and Example 2
3. Note 2 and Example 3

1. Explain why $\frac{1}{n!} \le \frac{1}{n}$ for $n\ge 1$.
2. Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty\frac{1}{n!}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
3. Explain why $\frac{1}{n!} \le \frac{1}{n(n-1)}$ for $n\ge 2$.
4. Show that $\sum_{n=1}^\infty\frac{1}{n(n-1)}$, which is equal to $\sum_{n=1}^\infty\frac{1}{n^2-n}$, converges by comparing it to $\sum_{n=1}^\infty\frac{1}{n^2}$ with the LCT.
5. What can you conclude about $\sum_{n=1}^\infty \frac{1}{n!}$? And why

1. The statement of the Integral Test on pg 721
2. The (important) note between the statement of the Integral Test and Example 1 on pg 721
3. Example 4

1. After reading Example 4, determine if $\displaystyle\sum_{i=0}^\infty \frac{n}{e^n}$ converges or diverges.
• You will probably want to use the integral test, but to do this, make sure to explain why $\displaystyle\frac{x}{e^x}$ is continuous, positive, and (eventually) decreasing.
2. Also, determine if $\displaystyle\sum_{i=0}^\infty \frac{-n}{e^n}$ converges or diverges.

Such a good day. Lot's of doing math and tons of excellent presentations!

- So clear! So great! Thanks to all of the presenters! -

1. The statement of the Integral Test on pg 721
2. The (important) note between the statement of the Integral Test and Example 1 on pg 721
3. Example 4

1. After reading Example 4, determine if $\displaystyle\sum_{i=0}^\infty \frac{n}{e^n}$ converges or diverges.
• You will probably want to use the integral test, but to do this, make sure to explain why $\displaystyle\frac{x}{e^x}$ is continuous, positive, and (eventually) decreasing.
2. Also, determine if $\displaystyle\sum_{i=0}^\infty \frac{-n}{e^n}$ converges or diverges.

1. Theorem 6 through end of section. There is a lot of important information here!!!

1. After reading examples 2-4 and the note in the margin next to $\fbox{4}$, try this variation: find the value of $c$ such that $\sum_{n=1}^\infty e^{nc} = 10$. Hint: $\sum_{n=1}^\infty e^{nc}$ is a geometric series. What is the common ratio?
2. How would your answer change if the equation was $\sum_{n=0}^\infty e^{nc} = 10$?
3. After reading Theorem 8 and Example 11, determine if the following series is convergent or divergent: \[\sum_{n=1}^\infty \frac{2^n+e^n}{\pi^n}\]
4. Meditate on Theorem 8, and be able to explain why the following is true:

   If $\sum a_n$ is convergent and $\sum b_n$ is divergent, then $\sum (a_n + b_n)$ is divergent.

5. Determine if the following series is convergent or divergent: \[\sum_{n=1}^\infty \frac{2^n+\pi^n}{e^n}\]
6. Meditate on Theorem 8 again, and determine if the following is true or false:

   If $\sum a_n$ is divergent and $\sum b_n$ is divergent, then $\sum (a_n + b_n)$ is divergent.

Vote!!

1. Intro through Example 4
2. The comment in the margin next to box $\fbox{4}$ for the geometric series is very important.

1. After reading examples 2-4 and the note in the margin next to $\fbox{4}$, try this variation: find the value of $c$ such that $\sum_{n=1}^\infty e^{nc} = 10$. Hint: $\sum_{n=1}^\infty e^{nc}$ is a geometric series. What is the common ratio?
2. How would your answer change if the equation was $\sum_{n=0}^\infty e^{nc} = 10$?

1. Box $\fbox{9}$ and Definition $\fbox{10}$
2. Examples 9-13 (Example 10 is important!)

1. Consider the sequence $\left\{\frac{n!}{2^n}\right\}$.
   • Write out $a_1$, $a_2$, $a_3$, and $a_n$ without simplifying (as in Example 10)
   • Try to group the terms in your unsimplified expression for $a_n$, and explain why $a_n > \frac{n}{4}$. This is slightly subtle; take your time.
   • Explain why the sequence $\left\{\frac{n!}{2^n}\right\}$ diverges (to infinity).
2. Consider the sequence $\left\{\frac{(2n)!}{(2n+2)!}\right\}$.
   • Write out $a_1$, $a_2$, $a_3$, and $a_n$ without simplifying (as in Example 10)
   • Try to group the terms in your unsimplified expression for $a_n$, and explain why $a_n = \frac{1}{(2n+2)(2n+1)}$.
   • Explain why the sequence $\left\{\frac{(2n)!}{(2n+2)!}\right\}$ converges (to 0).

Started sequences and series today - hooray!

1. Theorem 3 and the limit laws on page 697
2. Examples 4-7

Talked about area in polar, and started arc length. On to sequences and series tomorrow.

Also, Jullien Gordon will be speaking on campus tomorrow (during the 12pm class ☹). You should definitely consider going if you can. There is more information here: theuniversityunion.com/unique/event/jullien-gordon

1. Introduction through Example 3
2. Definition 1

Hearts and roses.

1. Examples 1 through 3

1. Try to graph it yourself. Check yourself on Desmos.
2. Find all values for $\theta$ for which the curve goes through the origin (i.e. when $r=0$)
3. Set up an integral that computes the area of the inner loop of the curve.

Circles and hearts.

1. Introductions through Example 1

1. Graph it on Desmos, and study what happens when $\theta \rightarrow 0^+$.
2. There seems to be a horizontal asymptote. Can you confirm this? Hint: you want to study what happens to the $y$-values as $\theta \rightarrow 0^+$, i.e. you want to study $\displaystyle\lim_{\theta \rightarrow 0^+} y$. Remember that (always) $y=r\sin\theta$ and that for this problem you have $r=\frac{\pi}{\theta}$. Combine this all together, and take the limit.
   
3. Use what you read in the section to compute $\displaystyle\frac{dy}{dx}$ for this curve. You should get an expression in terms of $r$ and $\theta$. Now compute $\displaystyle\lim_{\theta \rightarrow 0^+} \frac{dy}{dx}$. Does this support what you found before?
   
4. And what is happening with the graph when $\theta$ is negative? (Careful: Desmos is not showing you this!) Do you notice any symmetry?

On to polar.

1. The subsection "Tangents to Polar Curves" on pg 663
2. Example 9

1. Graph it on Desmos, and study what happens when $\theta \rightarrow 0^+$.
2. There seems to be a horizontal asymptote. Can you confirm this? Hint: you want to study what happens to the $y$-values as $\theta \rightarrow 0^+$, i.e. you want to study $\displaystyle\lim_{\theta \rightarrow 0^+} y$. Remember that (always) $y=r\sin\theta$ and that for this problem you have $r=\frac{\pi}{\theta}$. Combine this all together, and take the limit.
   
3. Use what you read in the section to compute $\displaystyle\frac{dy}{dx}$ for this curve. You should get an expression in terms of $r$ and $\theta$. Now compute $\displaystyle\lim_{\theta \rightarrow 0^+} \frac{dy}{dx}$. Does this support what you found before?
   
4. And what is happening with the graph when $\theta$ is negative? (Careful: Desmos is not showing you this!) Do you notice any symmetry?

So happy to get back to the discussions - great day with great presentations!

1. Introduction
2. Examples 1, 4, 5, 7

1. Play around with polar equations in Desmos. Look at the following for examples:
   • https://www.desmos.com/calculator/rayzacuktw
   • https://www.desmos.com/calculator/hxmu7mmchq
2. Use Desmos to make a super cool graph using a polar equation. Write everything down, so you can reproduce it in class. (I'll make sure we have time to share some of these this time!)
3. Graph the polar equation $r=\frac{\pi}{\theta}$. Think about what $r$ and $\theta$ represent and be able to explain why the graph looks like it does. Make sure you zoom in enough to see the spiral and zoom out enough to see that it does not continue to spiral. Also, Desmos has a hard time when you zoom in - it should keep spiraling. Try looking at the following graph (on WolframAlpha):
   http://www.wolframalpha.com/input/?i=polar+plot+r+%3D+1%2Fθ,+1+%3C+θ+%3C+500

More calculus with parametric equations

1. Subsections on area and arc length.
2. Example 5

1. If you need to, check out the graph of $\mathcal{C}$ here https://www.desmos.com/calculator/x7cia8bdpz. The graph also illustrates tangent lines.
2. Find the area of the "teardrop" at the bottom of $C$. Hint: to find the area between a parametric curve $x=f(t)$, $y=g(t)$, $\alpha\le t\le \beta$, and the $x$-axis, the book gives the formula $\int_\alpha^\beta g(t)f'(t)\;dt$. Here it might be better to find the area between the curve and the $y$-axis (and multiply by 2), and for this, the formula is $\int_\alpha^\beta f(t)g'(t)\;dt$

Group work on parametric equations.

Midterm.

Hope you had a good review of integration in class today. I missed you all!

- Asia (over there - across the river) wishes you lots of luck on the exam. And so do I!! -

Kept on with the parametric and got started reviewing for the midterm.

Got a little slowed down by the geometry of cones today ☺, but still had some time to think about different ways of representing a collection of points. There was a nice balance in the class of those that found a function to represent the data (e.g. $y=\sqrt{x} + 1$ or $x = (y-1)^2$) and those that found parametric equations to represent the data (e.g. $(x,y) = (n^2,n+1)$). Nice job everyone! And a special thanks to the one that went to the board; it was super clear, detailed, and, of course, right on!

1. Introduction
2. Examples 1, 2, 4

1. Play around with parametric equations in Desmos. Look at the following for examples:
   • https://www.desmos.com/calculator/rnyjcjpouf
   • https://www.desmos.com/calculator/bzn9bmfba9
2. Use Desmos and what you learned in Example 4 to do problem 35 on page 647 of the textbook. Write everything down on paper, so you can reproduce it in class. Hint: each part of the picture will be represented by a different set of parametric equations.
3. Use Desmos to make a super cool graph using parametric equations. Write everything down on paper, so you can reproduce it in class.

1. Introduction
2. Examples 1, 2, 4

1. Play around with parametric equations in Desmos. Look at the following for examples:
   • https://www.desmos.com/calculator/rnyjcjpouf
   • https://www.desmos.com/calculator/bzn9bmfba9
2. Use Desmos and what you learned in Example 4 to do problem 35 on page 647 of the textbook. Write everything down on paper, so you can reproduce it in class. Hint: each part of the picture will be represented by a different set of parametric equations.
3. Use Desmos to make a super cool graph using parametric equations. Write everything down on paper, so you can reproduce it in class.

Finished up arc length, and then started surface area by showing that Gabriel's Horn has an infinite surface area (but finite volume)!

1. The blue boxes labeled $\fbox{4}$ up through $\fbox{8}$
2. Examples 2-3

1. Set up an integral that computes the surface area of Gabriel's Horn. (Recall that Gabriel's Horn is the solid obtained by revolving the graph of $y=\frac{1}{x}$, from $x=1$ to $\infty$, about the $x$-axis.)
2. Show that the integral diverges. Hint: you can try to directly evaluate the integral if you want, but a comparison test with $\int_1^\infty \frac{1}{x}\,dx$ should be much easier.

Arclength today. Many thanks to those who came prepared today and many$^2$ thanks to those that went to the board.

1. Introduction through the blue box labeled $\fbox{4}$
2. Example 1

1. Set up an integral that computes the surface area of Gabriel's Horn. (Recall that Gabriel's Horn is the solid obtained by revolving the graph of $y=\frac{1}{x}$, from $x=1$ to $\infty$, about the $x$-axis.)
2. Show that the integral diverges. Hint: you can try to directly evaluate the integral if you want, but a comparison test with $\int_1^\infty \frac{1}{x}\,dx$ should be much easier.

Finally wrapped up improper integrals.

1. Introduction through Example 2

1. Set up an integral to compute the length of $y=\sin(x)$, $0\le x\le 2\pi$. Can you evaluate this integral? If you can, do it; otherwise, estimate it using a Riemann sum (with not too many rectangles).
2. Set up an integral to compute the length of $y=\ln|\cos(x)|$, $0\le x\le \frac{\pi}{3}$. Can you evaluate this integral? If you can, do it; otherwise, estimate it using a Riemann sum (with not too many rectangles).

Continued making our way through improper integrals. We had several excellent, and highly productive failures in both classes. Many thanks to all those involved!! Unfortunately, I only got a picture of a nonfailure, which nevertheless was also productive ☺

- Crushed. Completely. -

1. Examples 5-7
2. The Warning right after Example 7

1. True or False: $\displaystyle\int_{-1}^1 \frac{1}{x^3}\,dx = 0$. Explain.
   
   
2. True or False: $\displaystyle\int_{-1}^1 \frac{1}{\sqrt[3]{x}}\,dx = 0$. Explain.

- Trig. sub. and I.P (and rainbows) - Love it! -

1. Examples 1-4

Had a fun day of group work with many excellent presentations (though, if you ask me, there were not enough mistakes ☺). Nice job solving hard problems! We'll finish presenting the problems on Monday. Happy Friday!

- "Ugly" integrals and pretty colors -

1. Everything! (It's a review of the integration techniques we've covered in Chapter 7.)

1. Take a look at the last two group work problems, if you haven't already. Here they are:
   • Compute \[\int \frac{\sqrt{1+\sqrt{x}}}{x}\,dx.\]
   • Compute \[\int \frac{x\ln x}{\sqrt{x^2-1}}\,dx.\]
2. Meditate (again) on the next question. (And, be careful!)

   True or False: $\displaystyle\int_{-1}^1 \frac{1}{\sqrt[3]{x}}\,dx = 0$. Explain.

More partial fraction decompositions. We'll finally wrap this up tomorrow with you all working through a few more examples.

1. Continue to think about this one form last time: using what you read (in Example 9), try to compute \[\int \frac{1}{x^2+x\sqrt{x}}\;dx.\] Hint: as in Example 9, you want to start with a $u$-sub. that will turn the integrand into a rational function. Make a similar choice as in Example 9.
2. As a hook into our next topic, meditate on the next question. (And, be careful!)

   True or False: $\displaystyle\int_{-1}^1 \frac{1}{\sqrt[3]{x}}\,dx = 0$. Explain.

Kept working on partial fraction decompositions. More to come...

1. Examples 7-9

1. Be prepared to write out the form of the partial fraction decomposition (as in Example 7) for any big, nasty rational function, but you can assume that the denominator will be easy to factor.
2. Using what you read (in Example 9), try to compute \[\int \frac{1}{x^2+x\sqrt{x}}\;dx.\] Hint: as in Example 9, you want to start with a $u$-sub. that will turn the integrand into a rational function. Make a similar choice as in Example 9.

Wrapped up trig. substitution today and got started with partial fractions. We had several excellent presentations (but unfortunately ran out of time in both classes)! We'll definitely revisit these at the beginning of Wednesday.

- Slaying partial fractions at 10AM! -

- With some small typos corrected - the final answer should also have a swap. (My apologies for rushing the speaker!! I take full responsibility for the typos.) -

- Round 2 at noon -

- Success again! -

1. Reread Example 2
2. Example 3

1. Using what you read in 7.4, try to compute \[\int \frac{x+7}{x^2+x-6}\;dx.\] Here are some steps to follow:
   • Factor $x^2+x-6$ as $x^2+x-6 = (x+3)(x-2)$
   • Write \[\frac{x+7}{x^2+x-6} = \frac{A}{x+3} + \frac{B}{x-2},\] and use the ideas in Example 2 to solve for $A$ and $B$.
   • Now evaluate $\int \frac{x+7}{x^2+x-6}\;dx$ by instead evaluating \[\int \frac{A}{x+3} + \frac{B}{x-2}\;dx.\]

More trig. substitution today. Thanks for all of the help at the board!

- End game. -

1. Introduction through Example 2

1. You know how to compute $\int \frac{1}{x-6}\;dx$, right?
2. Using what you read in 7.4 (Example 1), compute \[\int \frac{x^2+x+1}{x-6}\;dx.\] If you need to review long division of polynomials, try this Khan Academy video: https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/dividing-polynomials-1
3. Remember how to compute $\int \frac{4x+2}{x^2+x-6}\;dx$? Hint: basic substitution.
4. Using what you read in 7.4, try to compute \[\int \frac{x+7}{x^2+x-6}\;dx.\] Here are some steps to follow:
   • Factor $x^2+x-6$ as $x^2+x-6 = (x+3)(x-2)$
   • Write \[\frac{x+7}{x^2+x-6} = \frac{A}{x+3} + \frac{B}{x-2},\] and use the ideas in Example 2 to solve for $A$ and $B$.
   • Now evaluate $\int \frac{x+7}{x^2+x-6}\;dx$ by instead evaluating \[\int \frac{A}{x+3} + \frac{B}{x-2}\;dx.\]

CarTalked, yet again, today. We had very nice presentations in both classes, and finally, with the help of a lot of trig., found an antiderivative for $\sqrt{100 - x^2}$.

1. Make the substitution $x = 2\tan\theta$ (as was mentioned in class). Don't forget to substitute in for $dx$ too.
2. Go through the steps to simplify the resulting integral to the form \[\frac{1}{4}\int\frac{\cos\theta}{\sin^2\theta} \,d\theta\]
3. Evaluate the integral. (Hint: try a $u$-sub.) You should end up with \[-\frac{1}{4\sin \theta} + C\]
4. Build a triangle for $x = 2\tan\theta$ and substitute to get the integral back in terms of $x$.

Exam today. I'll do my best to have it graded quickly.

1. The introduction
2. Examples 1 and 2

More trig. fun today, and a little review for the first midterm. Good luck with your studying! (Remember that the first midterm will be Wednesday. It'll cover up through section 7.1.)

1. Make sure you understand all of the WebAssign problems that you've done. You can look back at each problem to see the answer and a solution if you want.
2. Review the Written Homework problems too. Feel free to talk with me about them, but answers and solutions can also be found on Slader.com.
3. Finally, look over the discussion questions. These should reinforce the concepts and are, of course, always fair game to put on an exam.

Had some fun with trig. today. There will be lots more to come! I promise ☺

Remember that the first midterm will be next Wednesday. It will cover up through section 7.1.

1. Examples 5-7

1. What is an example of a problem that you still find challenging, and what specifically do you find challenging about it? For example, is it the visualization, the set up, the integration,...
2. What is an example of a problem that you have worked hard to understand, and what specifically do you think is the key to understanding the problem?

1. Examples 4 and 5

1. the introduction and formula for integration by parts, i.e. everything before Example 1
2. Examples 1, 2, 3

Wrapped up volumes by disks/wahers today, and found out that Gabriel's Horn, which has an infinite length, holds a finite volume of $\pi$

1. The intro. to 6.3 (again) paying special attention to the pictures in Figure 3
2. Examples 1 and 2

Talked more about volumes today, but ran out of time for Gabriel's Horn. We'll do it first thing tomorrow.

1. Examples 6 in 6.2
2. The intro. to 6.3 up through Example 1

Dissected eggplants today in the name of Mathematics and found their volumes to (approximately be 288.7cm$^3$ and 249.52cm$^3$. Many thanks to all of those who participated!

- And now we know. -

1. Examples 2-5 and the wrap up on page 443.

Talked (mostly me ☹)a bit more about areas today, and had a couple of brave souls in both sections share their work with the class. All were great and most came with a touch of productive failure, which I love! Bonus Points!

1. Introduction through Example 1

Wrapped up our review of Section 5.2 today, as well as the CarTalk problem, for now. But we (and Rich) are still waiting for an actual estimate...anyone?

I really enjoyed the CarTalking today! Thanks to everyone, especially those who presented their ideas at the board and those who commented from the audience. I'm looking forward to hearing from even more of you tomorrow!

1. the introduction through formula 2, and
2. Examples 2, 6, and 7.

It was great to meet you all today. Thanks for all of the participation! (But a few of you still owe me a spirit animal...) See you Wednesday!

- Math Skills. -

1. the definition of the definite integral, and
2. the content in blue boxes, e.g. statements of theorems and properties of the definite integral.