You need to know everything for Calc AB and more!!!
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.
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.
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)
Three part definition of continuity; differentiability; one-sided limits and differentiability; points at which a function is continuous but not differentiable.
Mean Value Theorem (MVT); applying MVT to functions on a given interval; solving MVT problems with your calculator; MVT problems and tabular data.
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.
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!
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.
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.
Function analysis using derivatives (increasing/decreasing/concavity/First and Second Derivative Tests) and the Candidates Test.
These notes do not exist…but if they ever do, they’ll be about optimization. The thing is, these types of optimizattion problems aren’t really on the AP exam anymore, so we don’t spend a lot of time on them in class.
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.
Antiderivatives; Power Rule for Integrals; u-substitution; strange u-substitutions; “simple” initial value problems. Here’s a link to a file that has a lot of u-substitution problems grouped so you can good at doing them in your head.
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.
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.
Applying the FTC; understanding the FTC; u-substitution and changing the bounds; definite integrals of horrible integrands (calculator!).
The Second FTC; 2nd FTC and critical points; 2nd FTC and limits; typical 2nd FTC and graph problems.
Average value; sketching the rectangle with equal area; average value vs. average rate of change. Here’s a link to a GeoGebra sketch to help visualize average value.
Motion and antiderivatives; the position function as an accumulation function; working with tabular data.
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.
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 sketch (not by me) is extremely helpful in visualizing volumes with known cross sections.
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…)
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’s a link to an online version on GeoGebra.org of a sketch that creates slopefields to play around with. 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.
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).
These are not a part of Calc BC, so I’m not including them here. You’ve got enough to learn that’s actually in the curriculum! These notes are about volume by shells and you can find them on the Calc AB notes page if you want to check them out. (Scroll down to Notes 25 when you get there!)
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…
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.
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.
Improper Integrals; the types of improper integrals; proper notation; practice problems.
Partial fraction decomposition again! (leading to integrals, of course!); non-repeating linear factors; repeated linear factors; quadratic factors. (If you mastered it the first time, these are skippable…the course I actually teach, Calc CD, starts with this.)
Introduction to Taylor polynomials; error of Taylor polynomials for approximations. Not Yet Here’s a link to a GeoGebra file that shows a Taylor Polynomial with variable center and order.
Sequences; some properties of sequences; limits of sequences; linking sequences to series
Introduction to series; geometric series; telescoping series; properties of convergent series; nth term test for divergence
Series Tests: Integral test; p-series
Series Tests: Direct comparison test; limit comparison test.
Series Tests: Ratio Test (best test ever!); Root Test.
Alternating Series; Alternating Series Test; Error for Alternating Series; Absolute Convergence vs. Conditional Convergence
Power Series; Interval of Convergence; new series from old series via integration and differentiation
Taylor Series; working with the known series for e^x, sin(x), cos(x), and 1/(1-x); applications of series; Lagrange Error Bound; identifying the function to which a series converges.
(This is not a topic that’s esssential for Calc BC. If you’re running low on time, just skip it entirely.) Integrals of powers of trig functions; saving a power; using Pythagorean Identities.
Arc Length for functions
Euler’s Method. Not Yet Here’s a link to a GeoGebra file that demonstrates Euler’s Method (change the initial point, the size of dx, and the number of steps; also view the solution curve). Several interesting/useful videos in the playlist.
Logistic Differential Equation. Not Yet Here’s a link to a GeoGebra file of a logistic curve with a slope field and some things you can manipulate. Here’s a fun activity that generates a nice logistic curve (we do it in class, you don’t need to print it). Here’s the spreadsheet we’ll use to track the activity.
Notes 15 are not covered on the AP exam, so I’m not including them here. They cover trig substitution and you can find them on the Calc CD Notes page if you want to see them. Scroll to Notes 15 when you get there.
Calculus of parametric equations; vector-valued functions; calculus of vector-valued functions. Not Yet Here’s a link to a GeoGebra file that shows a curve, dx/dt, dy/dt, dy/dx, and the second derivative of the parametric equations. You might want to review Notes 15 from Math Analysis. The videos for those notes are here.
Calculus of polar coordinates. First you need to really get good at non-calculus related polar coordinates. You might want to review Notes 16 from Math Analysis for basic, non-calculus stuff. Here’s the videos for those. In these notes: common graphs, tangent lines, interpreting derivatives dy/dx, dy/dt, dx/dt, and dr/dt, polar areas.
Here’s a link to a Google Doc where I’ve organized most of my review videos. Start studying now! Start with AB material, then continue with BC. If you’d rather just look through an enormous playlist of all the videos, here’s a link to that!