AP Calculus BC / Calculus 2 Tutoring Series Welcome to our AP Calculus BC / Calculus 2 Tutoring Series! NOTE: More sessions to be added. This series HIGHLY RECOMMENDED for high school and college students who already have an understanding of calculus 1 and want to increase their calculus skills. Prerequisites: This series is designed for students who have a foundation in limits, derivatives, and integrals and are ready to dive deeper into more advanced topics. This series will NOT cover topics already in AP Calculus AB or Calculus I syllabus. The goal is to provide clear, easy-to-understand explanations and practical examples to help you excel in your studies. Here's what we'll cover: 1. Techniques of Integration- Learn various methods to tackle complex integrals, including integration by parts, partial fractions, trigonometric integrals, and more. These techniques will expand your toolkit and make solving integrals more manageable. 2. Applications of Integrals - Discover how integrals are used in real-world applications. We'll explore areas such as calculating areas between curves, finding volumes of solids of revolution, work done by a force, and more. 3. Parametric Equations and Polar Coordinates - Understand how to work with curves represented by parametric equations and polar coordinates. We'll cover graphing, analyzing motion, and converting between different coordinate systems. 4. Sequences and Series- - Dive into the world of sequences and series, including arithmetic and geometric series, convergence tests, power series, and Taylor series. You'll learn how to determine the behavior and sum of infinite series. 5. Vectors and 3D Space - Explore the fundamentals of vectors and their applications in three-dimensional space. Topics include vector operations, dot and cross products, equations of lines and planes, and analyzing motion in space. Throughout the series, we'll use a variety of teaching methods, including step-by-step problem-solving, visual aids, and interactive discussions to ensure you grasp each concept thoroughly. Join us and take your calculus skills to the next level!

Hey everyone, it's finally here! Calculus 3 (the equivalent of the first semester of a full-year course). NOTE: REQUIRED READING OF APPROXIMATELY 25 PAGES A WEEK + WEEKLY HOMEWORK (with feedback) + 1 TEST PER UNIT (in-class and graded with feedback) Prerequisites: - Calculus 1 and 2 (the equivalent of Calc BC) CONTENT COVERED: (UNIT I) Geometry of vectors and projections 1. Vector geometry in R^n 2. Planes in R^3 3. Span, subspaces, and dimension 4. Basis and orthogonality 5. Cross Products (UNIT II) Multivariable functions and optimization 6. Multivariable functions, level sets, and contour plots 7. Partial derivatives and contour plots 8. Maxima, minima, and critical points 9. Gradients, local approximations, and gradient descent 10. Constrained optimization via Lagrange multipliers (UNIT III) Geometry and algebra of matrices 11. Linear functions, matrices, and the derivative matrix 12. Linear transformations and matrix multiplication 13. Matrix algebra 14. Multivariable Chain Rule 15. Matrix inverses and multivariable Newton’s method for zeros (UNIT IV) Further matrix algebra and linear systems 16. Linear independence and the Gram–Schmidt process 17. Matrix transpose, quadratic forms, and orthogonal matrices 18. Linear systems, column space, and null space 19. Matrix decompositions: QR-decomposition and LU-decomposition (UNIT V) Eigenvalues and second partial derivatives 20. Eigenvalues and eigenvectors 21. Applications of eigenvalues: Spectral Theorem, quadratic forms, and matrix powers 22. The Hessian and quadratic approximation 23. Application of the Hessian to local extrema 24+ (special topics - TBD)

I got a 5 on the AP Calculus BC exam, and you can too. Having previously taught Calculus 1 (the AB curriculum) and now seeing the staggering amount of demand for further maths, I've decided to host this intensive, yet comprehensive series on Calculus 2, including a FULL work-through of a real past exam. The first two sessions will serve as a review of differential calculus. Below are what we will cover: INTEGRATION AND APPLICATIONS (UNIT 1) 1. Introduction to accumulation of change 2. Left and right Riemann sums; over and under-approximation 3. Midpoint and trapezoidal sums 4. Definition of indefinite integral using as the limit of Riemann sums 5. The Fundamental Theorem of Calculus (including the proof), antiderivatives, and definite integrals 6. Interpretation of accumulation functions: negative definite integrals, definite integrals over a single point; graphical interpretation and evaluation. 7. Integrating sums of functions, switching the bounds of integration, and with functions as bounds. 8. The reverse power rule, u-substitution, and integration by parts 9. Antiderivatives of all the previously mentioned functions 10. Improper integrals 11. Finding the average value of a function on an interval 12. Connecting position, velocity, and acceleration functions using integrals 13. Using accumulation functions and definite integrals in applied contexts 14. Finding the area between curves expressed as functions of x 15. Finding the area between curves expressed as functions of y 16. Finding the area between curves that intersect at more than two points 17. Volumes with cross sections: squares and rectangles 18. Volumes with cross sections: triangles and semicircles 19. Volume with disc method: revolving around x- or y-axis 20. Volume with disc method: revolving around other axes 21. Volume with washer method: revolving around x- or y-axis 22. Volume with washer method: revolving around other axes 23. The arc length of a smooth, planar curve and distance traveled 24. Calculator-active practice DIFFERENTIAL EQUATIONS (UNIT 2): 1. Modeling situations with differential equations 2. Verifying solutions for differential equations 3. Sketching slope fields 4. Reasoning using slope fields 5. Approximating solutions using Euler’s method 6. Finding general solutions using separation of variables 7. Finding particular solutions using initial conditions and separation of variables 8. Exponential models with differential equations 9. Logistic models with differential equations PARAMETRIC EQUATIONS AND VECTOR-VALUED FUNCTIONS (UNIT 3): 1. Defining and differentiating parametric equations 2. Second derivatives of parametric equations 3. Finding arc lengths of curves given by parametric equations 4. Defining and differentiating vector-valued functions 5. Solving motion problems using parametric and vector-valued functions 6. Defining polar coordinates and differentiating in polar form 7. Finding the area of a polar region or the area bounded by a single polar curve 8. Finding the area of the region bounded by two polar curves 9. Calculator-active practice INFINITE SERIES (UNIT 4): 1. Defining convergent and divergent infinite series 2. Working with geometric series 3. The nth-term test for divergence 4. Integral test for convergence 5. Harmonic series and p-series 6. Comparison tests for convergence 7. Alternating series test for convergence 8. Ratio test for convergence 9. Determining absolute or conditional convergence 10. Alternating series error bound 11. Finding Taylor polynomial approximations of functions 12. Lagrange error bound 13. Radius and interval of convergence of power series 14. Finding Taylor or Maclaurin series for a function 15. Representing functions as power series

We will go through every unit of the AP Calculus AB Exam. There will be both lectures and practice tests to prepare for both the school-year and the AP exam. It will be a fast-paced class, in order to try to cover everything.

This series is all about fractional Calculus. Fractional calculus is a very narrow branch within analysis that deals with the order of differentiations. For example: d^(1/2) / dx^(1/2) or d^i/dx^i. Hop into the session and let's have some fun together!

This is a full first course on proof-based linear algebra. Sessions are 1h30m daily, so the course is at an accelerated pace. Prerequisites: -Algebra 2 with trig (with Calculus preferred but not required) Syllabus: Unit 1 1. Basic vector operations in R^n Unit 2 2. Vector Spaces and Subspaces 3. Linear independence and Gram-Schmidt 4. Projections and Fourier's Formula Unit 3 5. Linear Transformations, Properties, and Kernel 6. Matrix operations, special matrices, and matrix properties Unit 4 7. Determinants, Linear Systems, and Inverse Matrices 8. LU decomposition 9. Row, Column, and Null Space, Rank and Nullity Unit 5 10. Change of Basis and Matrix Multiplication 11. Eigeneverything and Diagonalizability Unit 6 12+ Special Topics We will use Bronson's "Linear Algebra - Algorithms, Applications, and Techniques" as our main text[](http://ndl.ethernet.edu.et/bitstream/123456789/24629/1/Richard%20Bronson.pdf) in addition to Stanford's Math 51 for supplemental reading (fragments taken under fair use law which will be posted gradually). Note: Required reading of 25 pages a week + Weekly homework (with feedback) + 1 test per unit (with review in-class and the tests graded with feedback).

Are you interested in learning Calculus 3? Well, this course is the equivalent of a second-semester course, a continuation of my last series. Don't worry though if you weren't there. This series can be taken by anyone who has completed the only prerequisite: Calculus 1 (with integration) and the rest of the necessary background information will be covered or reviewed in the initial sessions. NOTE: Required reading of about 15 pages a week + weekly homework (with feedback) + 1 test per unit (in-class and graded with feedback) Sessions are between 1h15m to 1h30m. Content Covered: (UNIT I) Single-Variable Integration and Applications 1. Physical interpretations of the integral and the average value of a function over an interval 2. Finding the area between curves expressed as functions of y 3. Finding the area between curves that intersect at more than two points 4. Volumes with cross sections: squares, rectangles, triangles, and semicircles 5. Volume with the disc and washer method: revolving around x-, y-, and different axes 6. Arc length, distance traveled, and unit calculator practice (UNIT II) Multivariable Integration and Applications 7. Formulating Riemann sums 8. The Double Integral and Iterated Integral 9. Applications of The Double Integral 10. Triple Integrals 11. The Jacobian and uv-substitution 12. Integration in Polar, Cylindrical, and Spherical Coordinates 13. Applications of Triple Integrals (Unit III) Single-Variable Vector Analysis 14. Defining and differentiating parametric vector-valued functions 15. Finding arc lengths of curves given by parametric equations and solving motion problems 16. Defining polar coordinates and differentiating in polar form 17. Finding the area of a polar region bounded by one and multiple polar curves, and unit calculator practice. (Unit IV) Multivariable Vector Analysis 18. Scalar-valued Line Integrals 19. Vector-valued line integrals 20. Path Independence 21. Exact Differentials 22. Divergence and Green's Theorem 23 Curl, Stokes' Theorem, and parameterizing surfaces. 24. Flux and Divergence Theorem 25+ (applications to Electrodynamics and other special topics?)

This series will be treated as close to a formal class as possible; the only difference being that it follows a mastery learning method (meaning that it allows ALL students to be caught up with everyone else at once) as opposed to traditional learning (where everyone must follow the same pace as everyone else). This means that several quizzes and tests will be assigned throughout the series to secure a 5 as best as possible on the exam. And in order to execute this idea of a formal yet mastery learning class, I will only allow us to advance to next concepts if everyone scores a 90% or higher on these assessments to make the series as easy sailing as possible. Studying for an AP exam requires a lot discipline, and so you are expected to be prepared to attend the 5 sessions a week (45 minutes each session), although it's fully expected that you'll miss a few sessions here and there. Because of this mastery learning method, you will be withdrawn if you have too many difficulties scoring over a 90% on an assessment (here, "assessment" collectively refers to quizzes, tests, midterms, and finals). If this does happen, i might make another series dedicated to those who've been withdrawn from the series so that you can catchup on your own pace without the guilt of slowing everyone else down. I can always schedule a Catchup/Office hours session by request if you were absent for a session/need extra help with a topic.