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: 505

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

Functional Analysis

Postings: 227

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

Complex Analysis

Postings: 609

Complex Analysis refers to the study of complex numbers.

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.

Calculating integrals with the trapezoidal method

Consider the integral in the attachment. Using the trapezoidal method with n = 4 and n = 8, estimate the integral numerically. Calculate the integral exactly and compare this with your numerical results. 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.

Line integrals and rectangles

Line Integrals Please see the attached. Please do the problem(s) in detail and show all work. This question requires a line integral around the rectangle defined by the points (1,-1), (1,1), -1,1), (-1,-1) and with the function given. This defines 4 integrals that have to be evaluated as described in the problem.

Angular and Linear Velocity

I need help in answering this question: A car is traveling down a road at 50 miles per hour. The car runs out of gas and drives allows the car to coast to a halt in order to get as close to the nearest gas station as possible. The car travels another 2.3 miles before continue to a stop. a) find the angular velocity of the wh

Angular and linear velocity

In the year 1930s, some trucks used a chain to transmit power from the engine to the wheels. Suppose the drive sprocket had a diameter 6 in., the wheel sprocket had diameter 20in., and the drive sprocket rotated at 300 rev/min. a) Find the angular velocity of the drive sprocket in radians per second. b) Find the liner velocity

Math Equations

Can somebody help to understand the attached equations and show me the way to the solution?

Effects of Data Changes Using Least Squares Method

Watch: and/or view: Consider the following dataset: (20,525) (17,57) (10,19) (9,18) (7,14) (16,41) (3,5) (5,10) (10,23) (12,24) (9,15) (20,571) (18,102) (16,56). Calculate a best fit equation using the Least Squares

Decreasing Rate Calculation Question

I need help with this question: 'A spherical snowball is melting in the sun. Given that the radius of the snowball is decreasing at a rate of 0.5cm per hour, at what rate is the volume of the snowball decreasing when the radius is 6cm?'