MAT 125 – Lecture 2. Basic Functions and Key Properties

Published on: Fri Jan 29 2010

Second day of work and school. Today went a bit smoother; I did not miss the train on my way home from school. Work was very rushed feeling. I felt like I was finishing tasks left and right but not really making much progress. A big part of today was implementing new procedures to smooth the workflow along. I have very little patience for repetitive tasks. Macros are being created for anything the slightest bit repetitive. On the train ride out to school I worked on the first Physics homework. I am making a file called “Notes and Concepts” which corresponds to the homework. In it I write down the various concepts the homework intended to encourage thought upon. So onto Math lecture #2! For starters I sat in the middle right near the front. I was able to see the board better, and I just dealt with the lack of left-handed desks. It started out with a review of basic functions, and repeated some of what we saw last week. The interesting fact this time, where KEY PROPERTIES. The professor also took time to point out the graph of almost every function he touched upon. The graphs are very important to us right now. I went through a ton of paper, and really I need to keep my paper notes as well, since it is difficult to translate so many nice drawings into PC graphics. Particular Case Polynomial Functions Degree of 1 = Linear function = Straight line graph increasing graph = positive slope = a > 0 decreasing graph = negative slope = a < 0 horizontal graph = no slope = a = 0 Quadratic Equations Smiling parabola graph = positive Frowning parabola graph = negative General Case Polynomials KEY PROPERTIES Graph is 1. Continuous 2. Smooth 3. if x goes to infinity, then f(x) is going to infinity 3. In Math: IF x → ±∞ THEN f(x) → ±∞ Power functions f(x)=xª 1. Smooth 2. Continuous 3. Worry about the domain Symmetric about origin = odd Symmetric about y = Even Even: f(x) = f(-x) Odd: f(x) = -f(-x) Other Examples of Power functions (Need to Specify Domain) RADICAL f(x) = x^½ = ²√x x > 0 f: 0 → R (unsure) FRACTIONAL f(x) = x^-2 = 1/x^2 x ≠ 0 f: ? RADICAL/FRACTIONAL f(x) = x ^ (-3/2) = 1 / x ^ (3/2) = 1 / sqrt(x^3) Look at what happens as the denominator approaches 0. f(x) gets closer and closer but never touches. x=0 is a Vertical Asymptote. That is, f(x) gets closer and closer but never touches the Vertical Asymptote. This is a continuous function, since the Asymptote is Vertical. Continuity is only concerned with jumps in y (that is f(x) ). Rational Functions f(x) = P(x) / Q(x) Ex: f(x) = x / x²-4 Ask Yourself: Are there Asymptotes? What happens near the Asymptotes? In Both directions. What happens near infinity? ALGEBRAIC FUNCTIONS 1 + 2 = 3 Nice, Except worry about domains Non-Algebraic Functions (Nice in Calc 1, Not Nice in Calc 2) Trigonometric Functions 1. Are Wavy 2. Sin, Cos are Periodic (2π) sin(x + 2π) = sin x cos(x + 2π) = cos x 3. sin²x + cos²x = 1 4. -1 ≤ sin x ≤ 1 & -1 ≤ cos x ≤ 1 5. Tan x , Cot x are Periodic (π) & worry about undefined Exponential Functions f(n) = xⁿ Log Functions f(x) = ln(x) Composition of Functions cos(ln(eª) + eª) h ◦ g ◦ f(x) Start with the INNER f(x) = eª = g(y) g(y) = g(ln(y) + y) = h(z) h(z) = cos(z) It felt like we covered quite a lot this class.