Jose Roberto Cossich G

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

Sessions are twice a week and last 1h10m. Review and practice key SAT Math concepts before your upcoming exam with the help of a 780 scorer. The sessions will be divided into each of these three components: - 🔍 Review: Refresh your memory and solidify your understanding of key math topics across all four domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry. - 💡 Tips: Receive valuable tips on math concepts, effective use of [Desmos](https://www.desmos.com/calculator), and strategies to help you excel on the SAT. - ✏️ Practice: Complete difficult practice problems to build your endurance and improve your problem-solving skills for the SAT. - This series is designed for those who are currently scoring in the 600+ range and are looking to improve on some of the hardest math topics that appear on the SAT. All are certainly welcome to join! Just keep in mind that we’ll be working quickly through difficult level problems. - Credit to @Joe H for helping me organize the series

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).