Partial fraction decomposition (leading to integrals, of course!); non-repeating linear factors; repeated linear factors; quadratic factors.
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.
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.
Integration by Trig Substitution; integrating lots of things with radicals of a certain form; introducing a radical to make it work; drawing reference triangles. (This is not actually a topic that’s on the AP Calculus BC Exam, but I really think you should know it!)
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.
Nothing after this point will be on the AP Calculus BC Exam! ***
3D vectors; distance, midpoint, spheres; dot product; components and projections; cross products; distance from a point to a line; scalar triple product; parallelepipeds. Here’s a proof about a relationship between sine and the cross product.
vector, parametric, and symmetric equations of a line in 3-dimensions; angle between lines; planes; normal vectors to planes.
The various quadratic surfaces and their traces; cylindrical coordinates (basically polar); spherical coordinates. If you follow along with the videos about spherical coordinates, you can find the problems, some work, and some things to type in here.
Intersections of surfaces; domains; limits; derivatives; tangent lines; arc length parameterizations; curvature
Domains; partial derivatives; mixed partial derivatives; the Chain Rule; Direction Derivative and Gradient; tangent planes and normal lines. See the videos for a bunch of examples. Here’s a write up of directions on how to look at partial derivatives graphically using GeoGebra 3D.
Critical points; Second Derivative Test; Global extrema (absolute max/min); constrained optimization; Lagrange Multipliers.
Riemann sums leading to double integrals; evaluating iterated integrals; changing the order of integration; determining the bounds; polar coordinates; triple integrals; cylindrical coordinates; spherical coordinates. Here’s a video of how to use your TI-Nspire CAS to help you find double (or triple) integrals. Here’s a video example of how to reverse the order of integration. Here’s a link to a GeoGebra sketch that tries to show Riemann Sums for double integrals in rectangular on GeoGebra.org. (It is very slow to update as you change things because it is doing a lot of work!)
Gradient vector fields and potential functions; conservative vector fields; curl; line integrals; work; Fundamental Theorem of Line Integrals; path independence. Here’s a video example of a line integral of a scalar function. Here’s a video example of a line integral of a vector field.
This is an ongoing project of mine, but the videos are all in this playlist.