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Calculus and Analysis

Calculus is a branch of Mathematics which examines change. It has two major disciplines: differential and integral calculus, with one being concerned with rates of change, while the other focuses on the accumulation of quantities. Thus, it can be seen that the applicability of this study extends into economics, engineering as well as any science.

Although Calculus does not stand apart from Algebra, both of these branches can be used to solve different problems. Algebra deals with structures utilizing letters and symbols to represent specific relationships between each other. However, since the relationship is fixed, it may not be applicable to use algebra to solve problems dealing with continuously changing relationships. Thus, calculus in this context can be a very useful as there are many non-theoretical relationships which rarely stay the same.

Exploring the basic terminology, a derivative is a measure of how the output of a specific function, which is not limited to y or f(x), changes as the input changes. An integral, also known as an antiderivative, is a function F whose derivative is the given function f. Both of these form the basic tools of calculus with numerous applications in everyday life. Thus, understanding basic Calculus may prove to be a practical tool for anyone.

Categories within Calculus and Analysis

Basic Calculus

Postings: 555

Basic Calculus refers to the simple application of both differentiation and integration.

Functional Analysis

Postings: 247

Functional Analysis refers to the study of vector spaces and their properties.

Complex Analysis

Postings: 624

Complex Analysis refers to the study of complex numbers.

Period of a Fraction

From what I have seen, the longest length of a repeating sequence for an irrational number is c-1 for a=b/c. This occurs when c is a prime. How does one prove this? Can you give mathematical proof for this? Here is a link to the problem being discussed:

Differential and difference equation

Solve the differential equation subject to y(0)=2. An Euler approximation to y(x)=2. An Euler approximation to y(x) is given by setting h=x/h, solving the difference equation: See attached With initial condition y0=2. The approximation is then y(x)=yn, and show that if n is large, this approximates y(x).

Limit of inverse functions

Hello, I need some help with the following question. Let f be a real-valued one-to-one function with domain (a-1, a+1). Let lim(x->a) f(x) = L. Prove or disprove: lim(y->L) f^(-1)(y)=a.

Calculus: Graph the Function f

Graph the function ƒ(x) = x3 - 4x + 2. Let ƒ represent the position of an object with respect to time that is moving along a line. Identify when the object is moving in the positive and negative directions and when the object is at rest, showing all work.

Application of L'Hopital's Rule

Task: Graph the f(x) = e^2x - 1/x Verify the Limit x→0 f(x) meets the criteria for applying L'Hopital's Rule Find the Limit x→0 f(x) Explain why L'Hopital's Rule cannot be used to find the limit of Lim x→0 e^2x/x

Calculus Problems

Please find the problems attached. I have some understanding of calculus but come from an engineering background as apposed to a mathematics one. I would like to ensure my workings and answers are correct before submitting my assessment. Any help is greatly appreciated.

Prove Compactness of a Closed Subset

Prove that any closed subset of compact metric space is compact by using Theorem 2. Theorem 2: A subset of S of a metric space X is compact if, and only if, every sequence in S has a subsequence that converges to a point in S.

Prove the Maximum Theorem

Prove the maximum theorem: Let C be a compact set in a metric space and f: C-->R a continuous function. Then f is bounded in C and attains both its maximum in the set.

Prove Compactness

Prove that [0,1]^n is compact for any number (n e N) by using theorem 2. (see attached file) Theorem 2: A subset S of a metric space X is compact if, and only if, every sequence is S has a subsequence that converges to a point in S.

Interpolation polynomial

Hi, I have attached the full question. Please help me prove that the coefficients of the interpolating polynomial are unique.

Algebra: fraction decomposition

I have been working on a general appraoch to partial fractions. And I wanted a proof for why the normal way of doing partial fractions always gives a consistent equation system for the constants in the partial fraction. Question is illustrated with an example and explained more in detail in the document.

Two integrals

Please see attached and show step by step, thank you.

Solutions for intervals

Find the solutions for the attached integrals Please see attached and show step by step, thanks.

Finding the solution to integrals

Find the solution to the three integrals. Make sure you use calculus and show all your working. Please see attached and provide detailed solution.

Definite integral and Taylor Series

View: Give an example of a definite integral that cannot be integrated directly and derive the Taylor Series that represents this integral. Finally, explain how you would find the value of the original integral using this series.

General solutions and summation

Please find the attached file. I am requesting detailed answers for all the questions. All of the questions are calculus-based and include solving for general solutions and solving summation problems.