Precalculus 4: Exponentials And Logarithms
Precalculus 4: Exponentials And Logarithms
Last updated 6/2023
MP4 | Video: h264, 1920x1080 | Audio: AAC, 44.1 KHz
Language: English (US) | Size: 62.70 GB | Duration: 50h 7m
[center]Mathematics from high school to university
What you'll learn
How to solve problems concerning exponentials or logarithms (illustrated with 239 solved problems) and why these methods work.
Applications of exponential and logarithmic functions in finance, engineering, and natural sciences (some of them in my videos, some as reading material).
Binomial Theorem with two proofs (a combinatorial one and one by induction), Pascal's Triangle, and how to apply them.
Definitions of Euler's number e, how to approximate it, and how to prove that this number is irrational.
Definitions and computational rules for powers with various types of exponents (natural, integer, rational, real).
Definition of logarithms in relation to exponents, with computational rules (these will be related to the rules for powers).
Exponential functions, their properties and graphs.
Logarithmic functions, their properties and graphs.
Power functions, their properties and graphs; interactions between power functions and exponential functions.
Graph transformations for exponential and logarithmic functions, and for some power functions.
Solving exponential equations and inequalities.
Solving logarithmic equations and inequalities.
Requirements
High-school mathematics, mainly arithmetic; solving simple equations (linear and quadratic) with one unknown; some (but very few) polynomial equations will be solved during the course, but you can skip them if you want to (no problem at all; it will not affect your understanding of the main subjects of the course)
Precalculus 1: Basic notions (mainly the concept of function and related concepts; sets; logic)
You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.
Description
Precalculus 4: Exponentials and logarithmsMathematics from high school to universityS1. Introduction to the courseYou will learn: you will get a very general introduction to exponential and logarithmic functions: how they look and what they describe; you will learn about their simplicity in some aspects, and about some extraordinary complications you don't necessarily have to think about (but you should have a general idea about them); exponential and logarithmic functions are (next to polynomials, rational functions, power functions, trigonometric and inverse trigonometric functions you have learned about in the previous courses in the Precalculus series) examples of elementary functions: these functions are the building blocks for all the functions we will work with in the upcoming Calculus series.S2.71828. The noble number e, the Binomial Theorem, and Pascal's TriangleYou will learn: about number e, both in an intuitive way (by images) and in a more formal way. In order to be able to perform some formal proofs, you will need the Binomial Theorem: theorem telling you how to raise a sum of two terms to any positive natural power; you will learn about factorial, about binomial coefficients, and Pascal's Triangle (all this will come back in the course in Discrete Mathematics, and then you will get much more practice and combinatorial problems to solve; now we just need the Binomial Theorem as a tool for dealing with e).S3. Powers with various types of exponentsYou will learn: about powers with natural, integer, and rational exponents and the computation rules holding for them (the product rules, the quotient rules, the power rule); you will also get an explanation of a more serious stuff: how we can be sure about existence of nth roots of numbers. We will gradually get more and more understanding about the topic of plotting exponential functions. You will also get plenty of exercise to get comfortable with the topic of powers with various types of exponents, and power-related computations.S4. Power functions, their properties and graphsYou will learn: about power functions and their properties: monotonicity for positive arguments, monotonicity for negative arguments, and how the curves y=x^⍺ and y=x^β are situated in relation to each other for various pairs of ⍺ and β (with some cool interactions between power functions and exponential functions); you will also get a glimpse into the world of derivatives, to illustrate the problem of monotonicity of y=x^⍺ for any real non-zero ⍺ and positive arguments x, and to explain the cusps and rounding in some graphs. S5. Exponential functions, their properties and graphsYou will learn: about exponential functions f(x)=a^x with a>1, and f(x)=a^x with 0
