The Fundamental Theorem of Calculus
... Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the xaxis, and the vertical lines x = 0 and x=2 ...
... Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the xaxis, and the vertical lines x = 0 and x=2 ...
Calculus I: Section 1.3 Intuitive Limits
... By examining these tables, we can see that as x comes closer and closer to 1 from above and from below, the value of the function approaches to . Therefore, we conclude that lim ...
... By examining these tables, we can see that as x comes closer and closer to 1 from above and from below, the value of the function approaches to . Therefore, we conclude that lim ...
A Summary of Differential Calculus
... equals the average (mean) rate of change of f on [a, b] ((f (b) − f (a))/(b − a)). You should be familiar with a graphical interpretation of this theorem; it involves two parallel lines. The Mean Value Theorem can be used to prove the following three facts. ...
... equals the average (mean) rate of change of f on [a, b] ((f (b) − f (a))/(b − a)). You should be familiar with a graphical interpretation of this theorem; it involves two parallel lines. The Mean Value Theorem can be used to prove the following three facts. ...
1 Introduction and Definitions 2 Example: The Area of a Circle
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
Generating Functions and the Fibonacci Sequence
... and find the 10th term fairly easily with minimal effort. However, what if we want to find the 20th term or the 50th term? This task would consume much more time and be very monotonous to complete. Wouldn’t it be nice if there was an easier way to find these terms? Luckily, there is a solution to th ...
... and find the 10th term fairly easily with minimal effort. However, what if we want to find the 20th term or the 50th term? This task would consume much more time and be very monotonous to complete. Wouldn’t it be nice if there was an easier way to find these terms? Luckily, there is a solution to th ...
The Method of Partial Fractions to Integrate
... ln |x2 + 1| − arctan x + 2 ln |x − 3| + C. See the text for some examples that are more complicated than these. Weierstrass’s universal t-substitution. This is a substitution that converts and any function built out of trig functions and the four arithmetic operations into a rational function like t ...
... ln |x2 + 1| − arctan x + 2 ln |x − 3| + C. See the text for some examples that are more complicated than these. Weierstrass’s universal t-substitution. This is a substitution that converts and any function built out of trig functions and the four arithmetic operations into a rational function like t ...
Ch 11.10 Taylor Series
... use this to deduce an impossible statement, which will show that the original assumption e = ab is also impossible. Thus, using the Taylor series definition for e, we assume: ...
... use this to deduce an impossible statement, which will show that the original assumption e = ab is also impossible. Thus, using the Taylor series definition for e, we assume: ...
Document
... Types of functions, Graph of a relation, Invertible functions, Finding inverse of a function, Operations on functions (sum, difference, product, quotient, and composition). Sequences (Definition and examples), Arithmetic and geometric sequences and their nth terms, Sequences in Computer programming, ...
... Types of functions, Graph of a relation, Invertible functions, Finding inverse of a function, Operations on functions (sum, difference, product, quotient, and composition). Sequences (Definition and examples), Arithmetic and geometric sequences and their nth terms, Sequences in Computer programming, ...
FABER FUNCTIONS 1. Introduction 1 Despite the fact that “most
... Before analyzing this function, we will first need to establish some background. I will assume that the reader is familiar enough with numerical series and sequences to understand uniform convergence and uniformly cauchy sequences. With that in mind, we will begin by establishing some results about ...
... Before analyzing this function, we will first need to establish some background. I will assume that the reader is familiar enough with numerical series and sequences to understand uniform convergence and uniformly cauchy sequences. With that in mind, we will begin by establishing some results about ...
4.2 Mean Value Theorem
... right? In fact, we could just take this as the definition of increasing couldn’t we? We could say that by definition, f 0 (x) > 0 means a function is increasing. But before we ever knew calculus, we probably knew what increasing meant. It meant that a second value of f was bigger than a first value. ...
... right? In fact, we could just take this as the definition of increasing couldn’t we? We could say that by definition, f 0 (x) > 0 means a function is increasing. But before we ever knew calculus, we probably knew what increasing meant. It meant that a second value of f was bigger than a first value. ...
Multidimensional Calculus. Lectures content. Week 10 22. Tests for
... P sin(k) Example. : We do not know how to investigate this series directly. Its terms are not non-negative, 2k therefore the tests won’t work. We can’t use AST, since the series is not alternating. The necessary condition won’t help either, since ak → 0. Thus we try the absolute convergence and hope ...
... P sin(k) Example. : We do not know how to investigate this series directly. Its terms are not non-negative, 2k therefore the tests won’t work. We can’t use AST, since the series is not alternating. The necessary condition won’t help either, since ak → 0. Thus we try the absolute convergence and hope ...
Expectation of Random Variables
... In this case, two properties of expectation are immediate: 1. If X(s) ≥ 0 for every s ∈ S, then EX ≥ 0 2. Let X1 and X2 be two random variables and c1 , c2 be two real numbers, then E[c1 X1 + c2 X2 ] = c1 EX1 + c2 EX2 . Taking these two properties, we say that expectation is a positive linear functi ...
... In this case, two properties of expectation are immediate: 1. If X(s) ≥ 0 for every s ∈ S, then EX ≥ 0 2. Let X1 and X2 be two random variables and c1 , c2 be two real numbers, then E[c1 X1 + c2 X2 ] = c1 EX1 + c2 EX2 . Taking these two properties, we say that expectation is a positive linear functi ...
AVERAGE VALUES OF ARITHMETIC FUNCTIONS 1. Introduction
... n ∈ N, which maps to a complex number such that f: N → C. Exam ples of arithmetic functions include: the number of primes less than a given number n, the number of divisors of n, and the number of ways n can be represented as a sum of two squares. While the behavior of values of arithmetic function ...
... n ∈ N, which maps to a complex number such that f: N → C. Exam ples of arithmetic functions include: the number of primes less than a given number n, the number of divisors of n, and the number of ways n can be represented as a sum of two squares. While the behavior of values of arithmetic function ...
Math 111 – Calculus I
... The following is a fundamental result involving absolute maximums and minimums of continuous functions. It is called the Extreme Value Theorem. The formal statement of the theorem follows. Theorem 9.5(the Extreme Value Theorem): Assume f is a continuous function on a closed interval [a,b]. Then, f a ...
... The following is a fundamental result involving absolute maximums and minimums of continuous functions. It is called the Extreme Value Theorem. The formal statement of the theorem follows. Theorem 9.5(the Extreme Value Theorem): Assume f is a continuous function on a closed interval [a,b]. Then, f a ...
Prerequisites and some problems
... 4. Basic point-set topology of the real line (you might not have called it topology, but it is!): Least upper bound property, Bolzano-Weierstrass theorem, Cauchy sequences, limit points (also known as accumulation points), open and closed subsets, etc. Now, here are some problems to test your unders ...
... 4. Basic point-set topology of the real line (you might not have called it topology, but it is!): Least upper bound property, Bolzano-Weierstrass theorem, Cauchy sequences, limit points (also known as accumulation points), open and closed subsets, etc. Now, here are some problems to test your unders ...
§ 1-1 Functions
... cannot be solved by simple algebraic means. For instance, if we needed to find the roots of the polynomial , x 3 x 1 0 we would find that the tried and true techniques just wouldn't work. However, we will see that calculus through Newton’s Method gives us a way of finding approximate solutions ...
... cannot be solved by simple algebraic means. For instance, if we needed to find the roots of the polynomial , x 3 x 1 0 we would find that the tried and true techniques just wouldn't work. However, we will see that calculus through Newton’s Method gives us a way of finding approximate solutions ...
Document
... 21. In period 1, a chicken gives birth to 2 chickens (so, there are three chickens after period 1). In period 2, each chicken born in period 1 either gives birth to 2 chickens or does not give birth to any chicken. If a chicken does not give birth to any chicken in a period, it does not give birth i ...
... 21. In period 1, a chicken gives birth to 2 chickens (so, there are three chickens after period 1). In period 2, each chicken born in period 1 either gives birth to 2 chickens or does not give birth to any chicken. If a chicken does not give birth to any chicken in a period, it does not give birth i ...
