Intro to derivatives
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This document introduces derivatives and their rules of calculation. It defines a derivative as the slope of the tangent line to a function's graph at a given point. It shows a graphic demonstrating a tangent line and provides the point-slope formula. It then lists some common derivative rules, such as the power, exponential, constant, and product rules, to calculate the derivatives of basic functions.
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Math 170 notation
some notation commonly seen in calculus 1 courses. Includes the symbol, it's meaning, and sample usage.
intro to Implicit differentiation
This document discusses implicit differentiation, which is a technique for taking the derivative of equations that cannot be solved explicitly for y as a function of x. It explains that when differentiating terms involving both x and y, the derivative of the y term is dy/dx. As an example, it shows the differentiation of xy using the product rule, which yields y + x*dy/dx. The document concludes by applying this technique to differentiate the equation y4 + xy = x3 - x + 2 implicitly with respect to x.
Tower of hanoi problem
This paper looks at the tower of hanoi problem and how to generate the closed formula from the recursive formula using linear algebra and matrix operations.
One sided limits
This document discusses evaluating one-sided limits when the function value at the limit point is undefined, represented as f(a) = 0/0. There are four possible scenarios for the limits as the input approaches a from the left and right: (1) both limits are -∞, (2) the left limit is -∞ and the right is ∞, (3) the left is ∞ and the right is -∞, or (4) both are ∞. The overall limit only exists in cases (1) and (4). To determine if the limit is positive or negative infinity, evaluate the function very close to a on the given side.
Limits as x approaches infinity
This document discusses computing limits involving infinity by focusing on the highest degree terms in the numerator and denominator of a rational function. It provides three examples: 1) if the highest degree is in the numerator, the limit is 1 or -1 depending on the leading coefficient, 2) if the degrees are equal, the limit is the ratio of the leading coefficients, and 3) if the highest degree is in the denominator, the limit is always 0. Key steps are ignoring lower degree terms and constants as x approaches infinity.
Computing limits
The document discusses various methods for computing limits, including:
1) Plugging the value directly into the function f(a) if it is defined
2) Factoring the function if f(a) is undefined to eliminate problems
3) Using the sandwich theorem to evaluate a limit if the function is between two other functions with known limits
It then provides an example using the sandwich theorem to show that limx→0 sin(x)/x = 1 by considering the areas of triangles on a unit circle as x approaches 0.
Introduction to limits
A brief introduction to limits and the formal definition, as well as how to find delta and epsilon to satisfy the definition.
Solving a trig equation
The document solves the trigonometry problem of cos 4θ + cos 2θ = 0 by using trigonometric identities like the double angle formula and factoring techniques. It expresses cos 4θ in terms of cos 2θ using the double angle formula, factors the resulting polynomial, and then solves each factor independently. This results in three solutions for θ: θ = π/2 + k1π, θ = π/6 + k2π, and θ = 5π/6 + k3π, where k1, k2, k3 are integers.
solving a trig problem and sketching a graph example problems
We go through (in great detail) a problem involving solving a trig problem and a problem involving sketching a graph
Finding a polyomial from data
This document covers how to find a polynomial function given certain data like the zeroes, y-intercept, end behavior, and multiplicity.
functions review
A function is a rule that assigns exactly one output to each input. An example of something that is not a function is a rule that assigns two outputs to some inputs. The vertical line test can be used to determine if a relationship represents a function - if a vertical line intersects the graph at more than one point, it is not a function. Polynomial functions have the general form f(x) = Axn + Bxn-1 + ... + Tx + U. To sketch the graph of a polynomial function, one finds the x-intercepts by setting the function equal to 0 and solving for x, determines the end behavior by looking at the highest power term, finds the multiplicity of each x-intercept, finds
Jason format preferences m170
This document provides tips for formatting homework assignments to succeed in Math 170. It recommends (1) reading the textbook, studying, and asking questions; (2) writing each problem and solution on separate lines with space between, showing all work clearly; and (3) providing an example problem solution formatted with each step on a new line and indented continuation lines to improve readability.
Final exam review
The study guide provides 3 methods for proving that the power set of a set A with n elements has 2^n elements: 1) Using the product rule of counting, 2) Applying the binomial theorem, and 3) Using induction and defining a function between subsets of A with and without a particular element x.
Inclusion exclusion principle
This document contains the proof of two problems using the Inclusion/Exclusion Principle (IEP). For the first problem, it is proven that if the size of the union of two finite sets A and B is less than the sum of their individual sizes, then their intersection is empty. For the second problem, a formula is derived for the size of the union of a finite set A and the complement of another finite set B within a universal set U. The formula obtained is the size of U minus the size of B plus the size of the intersection of A and B.
Oral exam structure M189
The document provides tips for preparing for an oral exam on discrete mathematics. It recommends choosing good problems from each of the five sections - logic, set theory, functions, counting, and graph theory - that demonstrate key principles. Students should memorize the problems and be prepared to present three to four proofs during the exam, explaining their work and answering questions. The goal is to show an ability to discuss and present mathematical proofs verbally.
Class mods
This document contains class modifications and scenarios for an online multiplayer game called "Envelopes - Drop Trow Game". The modifications include characters being infected with viruses, equipment malfunctions, receiving injuries, and gaining new abilities. The scenarios in the envelopes result in player characters dying through comedic means, such as being frightened to death, succumbing to dysentery, or being killed by the beards of famous people. The overall tone is silly and nonsensical.
Drop trow game rules
This document outlines the rules and scenario for a 5-person squad airsoft game called "Drop Trow". Squads consist of a grenadier, assassin, rifleman, sniper, and gunman with specific weapon and ability restrictions for each class. The objective is for squads to eliminate other squads by any means necessary, capture enemies for points, and exchange intel found in envelopes and hostages at the bounty board for further points. Various power-ups are hidden around the field that can temporarily modify class abilities. The squad with the most points at the end wins.
Exam 3 Math 189
This document contains solutions to 5 math problems proved using mathematical induction and other techniques:
1) It proves via induction that the sum of the given fractions equals n/(n+1) for all n in the natural numbers.
2) It proves via induction that 5n-1 is divisible by 4 for all n in the natural numbers.
3) It proves that if sets A and B have sizes n and m respectively, with m>n, then no function from A to B can be onto.
4) It proves that the function f(a,b) = (b,a) mapping A x B to B x A is both 1-1 and onto, and therefore the
More from Tyler Murphy (20)
solving a trig problem and sketching a graph example problems
solving a trig problem and sketching a graph example problems
Intro to derivatives
- 1. Intro to Derivatives Tyler Murphy September 15, 2014 1 Introduction First, what is a derivative? To understand the derivative, we must
- 2. rst understand some- thing called a tangent line The tangent line is a line that intersects the graph at only one point. See Figure 1. Figure 1: A Tangent Line Point-Slope form of a line y y1 = m(x x1) infomal de
- 3. ntion The derivative of a function gives the slope of the tangent line. 2 Derivative Rules Name Function (f(x)) Derivative (f0(x)) Power Rule axn naxn1 Exponential ex ex Constant Function a 0 Scalar a f(x) a f0(x) Sum f(x) + g(x) f0(x) + g0(x) Product Rule f(x) g(x) g(x) f0(x) + f(x) g0(x) Quotient Rule f(x) g(x) f0(x) g(x) f(x) g0(x) f(x)2 1