Episode 64 - eigenvalues and eigenvectors

Today (Episode 64), I continued with Program 5 from Frame 47, Page 509. I explained the concept of eigenvalues and eigenvectors and demonstrated how to obtain the characteristic matrix, characteristic polynomial/determinant, characteristic root (eigenvalues), and their corresponding eigenvectors. I concluded the chapter by discussing Cayley Hamilton's theorem.


https://www.youtube.com/watch?v=fHPjpylcUDY

Episode 65


https://www.youtube.com/watch?v=2QA5jUaxyUM

In today's episode (Episode 65), I commenced the next Program (P2, Program 6, page 520) with the topic of vectors. I introduced vectors as quantities possessing both magnitude and direction and discussed how vectors can be represented on a straight line. In the second part of the video, I discussed types of vectors, including the concepts of equal vectors and the triangular laws of vectors.

Episode 67 - Cross product


https://www.youtube.com/watch?v=uLH5Ie5LSyM

Episode 68


In today's episode (Episode 68), the author continued his explanation of differentiation (remember, the topic was introduced as part of the foundational topics in Program F12 (Episode 42)). In this new program (Program 7, page 544), following the author's exposition, I reviewed derivatives and some standard derivatives, as well as methods of differentiating functions in their various forms. I concluded the video with an explanation of logarithmic functions - an easier method to differentiate more than two functions when the product/quotient rule might be longer to use.


https://www.youtube.com/watch?v=P7OFHm2LG84

https://www.youtube.com/watch?v=31zgoo4gbHw

In tonight's episode (Episode 69), I continued my explanation from Program 7, Frame 26, page 555, focusing on the topic of implicit function differentiation. I explained the difference between this type of function and the one we have been working with (explicit) and how to deal with functions of these types. In the second part of the class, I covered parametric equations and how to differentiate them, marking the end of Program 7.

Episode 70 - KA Stroud Engr Math 70 - Differentiation of inverse trigonometric and hyperbolic functions

In tonight's episode (Episode 70), I began the next program (Program from page 562 of the book, which focuses on differentiation applications. The first two topics covered in this program are the differentiation of inverse trigonometric and hyperbolic functions. I derived the derivatives of these functions and applied them to problems.


https://www.youtube.com/watch?v=xfrxpRC17Hc

Episode 72 - Sequences

Tonight, I began the next program (Program 10, page 605) covering the topic of sequences. I reviewed the meaning of sequences, discussing how they relate to and differ from sets. Additionally, I demonstrated how to graph sequences and explained some special sequences along with methods to obtain their nth terms.


https://www.youtube.com/watch?v=Clf4WfI27M0

Episode 73 - recursive prescription

Tonight, I continued with Program 10, page 611, covering the topic of sequences (recursive prescription). I explained how to obtain a recursive prescription for a sequence and from there derive a difference equation. I concluded the video by solving a first-order difference equation.


https://www.youtube.com/watch?v=f1C7WtLSXz0

Episode 74- difference equations


https://www.youtube.com/watch?v=EckvpQO7u-Y

Tonight, I continued with Program 10 (Frame 21) on page 618, covering the topic of difference equations. I revised recursive prescriptions and demonstrated how to solve first- and second-order homogeneous difference equations.

Tonight (Episode 75), we covered the limits of sequences (frame 35, page 626). In the first part of the session, I started by explaining the concepts of limits and infinity, convergence and divergence. Additionally, I explained the properties of limits as well as indeterminate forms marking the end of Program 10.


https://www.youtube.com/watch?v=Ztj7RShZNpM

Episode 76 - Series 1


https://www.youtube.com/watch?v=LV3XT3fvuj8

Tonight (episode 76), I commenced Program 11 (page 640) of the book with the topic of series. In this session, I explained how to manipulate arithmetic series, geometric series, and series of natural numbers. These concepts will help us understand how to determine the limiting values of arithmetic and geometric series and apply various convergent tests to infinite series, which we will cover in the upcoming session.

Episode 77 - Series of powers of the natural numbers

In this episode, starting from page 645 (Program 11, frame 15), I continued with the derivation of formulas for the series of powers of natural numbers and demonstrated how to manipulate them to evaluate various series. I concluded the video by discussing convergence and divergence, along with the tests used to assess them.


https://www.youtube.com/watch?v=nZ-HhdZnWFw

This week (Episode 78), starting from page 652, Program 11, frame 33, I continued with other tests for convergence, including the comparison test, D'Alembert's ratio test, and absolute convergence. I demonstrated how to apply these tests to determine whether a series converges or diverges. This episode concludes Program 11.


https://www.youtube.com/watch?v=MqIUnx5euok

This week (Episode 79), I began the next program—Program 12 (starting from frame 1, page 663)—which serves as an extension of the topic on series. I covered power series and explained the mechanics of expressing a function as a power series, specifically focusing on the Maclaurin series, where the terms are derived from the function's derivatives.


https://www.youtube.com/watch?v=fYl_L26zv_8