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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: 551

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

Functional Analysis

Postings: 245

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

Complex Analysis

Postings: 607

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: http://boards.straightdope.com/sdmb/showthread.php?t=720360

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

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: http://mathworld.wolfram.com/TaylorSeries.html 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.

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

Effects of Data Changes Using Least Squares Method

Watch: http://academicearth.org/lectures/least-squares and/or view: http://www.efunda.com/math/leastsquares/leastsquares.cfm 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?'

Complex variables questions

Please answer the attached questions and show the necessary steps for finding the value with your wonderful explanations!

Contours

Please show the steps and the graphs for problems 2 through 6. Thank you.

Graphing Functions and Continuity

Graph the function for -2 < x < 4. Check the continuity of f(x) at x = 0 and x = 2, making sure you address all the conditions for continuity. Please see attached, and if you could show the answer step by step, thanks.

Domains and Interval Notations

When the domain for a given function is not specified, we adopt the conviction that the domain is all numbers for which the function is defined. Using this convention, give the domain for the following functions and write your answers using the interval notation. Be careful when considering the end points of each interval. (i)

Graphing and Labelling Functions

(i) Graph the function f(x) = sqrt(x). (ii) Hence, using translation etc., graph f(x) = 2-sqrt(4-x). Show and label intermediate graphs.

cooling and leaking process mathematical model

1. A heated object is allowed to cool in a room temperature which has a constant temperature of To. a. Analyse the cooling process. b. Formulate mathematical model for the cooling process. 2. At time t= 0 water begins to leak from a tank of constant cross-sectional area A. The rate of outflow is proportional to h, the d