## Calc AB Notes

These are the notes that we use for my AP Calculus AB classes. They're pretty close to what I want them to ultimately look like, but definitely good enough to run with!

**If you find any mistakes anywhere in the notes,**Just click here to do that!*please*let me know!## Notes 01

Limit definition of the derivative; tangent & secant lines; derivative rules; numerical derivatives and finding zeros on calculators; derivatives of the six trig functions; derivatives of e^x and ln(x); some new calculus-style questions.

## Notes 02

The Chain Rule is pretty much the last differentiation rule (you've still got techniques to learn, but those will just use the rules you've already learned). Practice problems with using the chain rule on functions, with tables, with graphs.

## Notes 03

Motion along a line (rectilinear motion); position, velocity, acceleration; distance vs. displacement; speed; finding position functions from velocity functions. After these notes we do old FRQs from 1989 (AB3, BC6) and 1990 (AB1, AB2, BC1)

## Notes 04

Three part definition of continuity; differentiability; one-sided limits and differentiability; points at which a function is continuous but not differentiable.

- Playlist of full solutions and explanations of the notes.
- Problem Sets for Notes 04.
- Videos for Notes 04.

## Notes 05

**Mean Value Theorem (MVT); applying MVT to functions on a given interval; solving MVT problems with your calculator; MVT problems and tabular data.**

## Notes 06

Implicit Differentiation; common mistakes; second derivative of implicitly defined function; horizontal and vertical tangent lines; derivatives of inverse trig functions; derivatives of exponential functions; review of some geometry-type stuff. After these notes we do a number of old FRQs from 1973, 1978, 1980, 1992, and 1994, all of which featured implicit differentiation.

## Notes 07, Notes 07a, Notes 07b

**Print all three! These notes are about related rates. We cover all the classics: falling ladders, cones, etc. Additionally the mastermathmentor.com notes (16) are really good and there are a lot of old FRQs that have great problems. The ones we do: 1970AB4, 1972AB5, 1976AB4, 1977AB6, 1982 AB4, 1984 AB5, 1985 AB5, 1987 AB5, 1988 BC3, 1990 AB4, 1991 AB6, 1994 AB5, 1995 AB5. Notes 07a is the problems, which you have to be in my class to see, but are freely available on the internet, as well, if you google around. Notes 07b is space for doing the FRQs. Also...the shadow problem! Make sure you know and understand that problem!**

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 07.****Videos for Notes 07.**

## Notes 08

Understanding where L'Hopital's Rule comes from; how to evaluate limits that give 0/0 or infinity/infinity; repeated use of L'Hopital's Rule; an example where L'Hopital's Rule gets caught in a loop; some review problems.

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 08.****Videos for Notes 08.**

## Notes 09

Finding the value of the derivative of the inverse to a function without being able to find the actual inverse function. Sadly a lot of people lose points for this on the AP Exam when it's just not that difficult. Also a few review questions.

## Notes 10

Function analysis using derivatives (increasing/decreasing/concavity/First and Second Derivative Tests) and the Candidates Test.

## Notes 11

These notes do not exist...but when they do, they'll be about optimization.

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 11.****Videos for Notes 11.**

## Notes 12

Rate times time equals total; definite integrals; displacement; position from velocity graph; tons of definite integrals that can be done geometrically; u-substitution and definite integrals; definite integrals of functions interpreted as area; definite integral properties; some forward looking review problems.

## Notes 13

**Antiderivatives; Power Rule for Integrals; u-substitution; strange u-substitutions; “simple” initial value problems. Here's a file that has a lot of u-substitution problems grouped so you can good at doing them in your head.**

## Notes 14

**Approximating definite integrals with sums; Left, right, and midpoint sums; trapezoidal sums; Riemann sums and tabular data; Riemann sums given a function; tons of practice.**

## Notes 15

**Writing summations; using the summation formulas; generalizing a right-Riemann sum; limits of Riemann sums; the Fundamental Theorem of Calculus (FTC); understanding summations and their limits.**

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 15.****Videos for Notes 15.**

## Notes 16

**Applying the FTC; understanding the FTC; u-substitution and changing the bounds; definite integrals of horrible integrands (calculator!).**

## Notes 17

**The Second FTC; 2nd FTC and critical points; 2nd FTC and limits; typical 2nd FTC and graph problems.**

## Notes 18

**Average value; sketching the rectangle with equal area; average value vs. average rate of change. Here's a GeoGebra sketch to help visualize average value.**

## Notes 19

**Motion and antiderivatives; the position function as an accumulation function; working with tabular data.**

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 19.****Videos for Notes 19.**

## Notes 20

**Area of regions bounded by curves; Riemann sums to approximate area; top minus bottom; right minus left; using a calculator; slicing up regions into sub-regions; find a vertical line that divides a region; find a horizontal line that divides a region; a little more on average value.**

## Notes 21

**Approximating volumes with Riemann sums; the most common cross sections; volumes by plane slices; volume from tabular data; volumes of solids of revolution (disks and washers).**This video explains a little about volumes of solids with known cross sections (hard to picture, easy to calculate!). This GeoGebra file might help you visualize the solids of known cross sections.

## Notes 22

**Density problems. If you're a teacher pressed for time you could probably skip this and not put your students at a serious disadvantage. I like to cover the concept, especially radial density. Here's a way to look at radial density functions. (Not super self-explanatory, but you'll work it out...)**

## Notes 23

**Slope fields; what to aim for; reading slope fields (what to look for); types of solutions and general trends; how to do them on your calculator.**Here is the GeoGebra file for creating slope fields that the last page talks about. Here's a link to an online version on GeoGebra.org. See the playlist for videos on creating slope fields on the TI-Nspire (CAS/non-CAS...doesn't matter) and how to deal with slope field matching problems.

## Notes 24

**Basic differential equations you already know how to solve; what is a solution to a differential equation; verifying solutions through differentiation and substitution; separation of variables; domains of solution curves; exponential growth models; working with general solutions.**This video goes over a few examples that lead to natural logs (and talks about solving for C).

## AP Free Response Video Solutions.

Not really notes, but a spreadsheet that links to each of my AP Free Response video solutions. You might find yourself consulting that frequently as you study for the exam. Make sure to always try the problems first! Watching the videos can be counter productive if you haven't yet done the problems.

## AP Calculus AB Review Videos

Click the link to go to a Google Doc I'll try to maintain that just has videos that will get you up to speed on particular topics that are on the AP Calculus AB exam. This list is more focused than the playlists linked with each set of notes and is more for when you realize you really need to cram for one topic on the exam.

## Everything beyond this point is for after the AP Calculus AB Exam...

## Notes 25

**Volumes by shells. Not a lot going on here, really, because it's very straight forward. Think about tree rings, cylinders, and**

- Playlist of full solutions and explanations of the notes.
**Problem Sets for Notes 25.****Videos for Notes 25.**

## Notes 26

**Integration by Parts (IBP); when dv = dx; repeated IBP; the table method (rapid repeated IBP); IBP that loops.**Here, here, and here are three videos of examples of pretty much each type. For some reason the second video is the most popular...also, the integral of secant cubed is about as hard as an "easy" integral gets and it uses integration by parts. Here's a video for that too...

- Playlist of full solutions and explanations of the notes.
- Problem Sets for Notes 26.
- Videos for Notes 26.

## Notes 27

**Partial fractions; non-repeating linear factors; cover-up method; repeated linear factors; quadratic factors; using expand on your TI-Nspire CAS.**Here's a video about non-repeating linear factors. Here's a video about repeated linear factors. Here's a video about quadratic factors.

- Playlist of full solutions and explanations of the notes.
- Problem Sets for Notes 27.
- Videos for Notes 27.

## Notes 28

L'Hopital's Rule and the additional cases in which it can be applied; inverting one of the factors; taking natural logs; don't forget to exponentiate the answer! Here's a video about 0 times infinity. Here's a video about the other indeterminate forms.

- Playlist of full solutions and explanations of the notes.
- Problem Sets for Notes 28.
- Videos for Notes 28.

## Notes 29

Improper Integrals; the types of improper integrals; proper notation; practice problems.

- Playlist of full solutions and explanations of the notes.
- Problem Sets for Notes 29.
- Videos for Notes 29.