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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: 543

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

Functional Analysis

Postings: 266

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

Complex Analysis

Postings: 590

Complex Analysis refers to the study of complex numbers.

Del in cylindrical coordinates

The velocity potential is given by ∅ and obeys the relation: A strem function is given by ψ In polar coordinates one can obtain: The scaling factor r can be wxplained as this: Theese relations also holds between the potential and the stream function: In my book they then do something that I don't get.

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:

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.

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: 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

Complex variables questions

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


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.

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

Finding Values and Domains, Average Rate of Change, and Odd/Even Functions

An even function is defined as f(x) = f(-x), and an odd function has -f(x) = f(-x). The domain of a function is the set of input data that keeps the function defined. Determine if the function f(x) = -2x^2 * absolute value(-6x) is even, odd, or neither. Find the average rate of change for the function f(x) = 4/(x+3) between t

Calculus Help

Consider the function: y = 15/4 - x/2 - x2/4. a) Present the function in the turning point form. b) Find the equation of the axis of symmetry and coordinates of the turning point. Determine whether there is a function maximum or minimum at this point? Substantiate

Calculus Review on Integrals

1. Evaluate the following indefinite integrals: See attached 2. On a dark night in 1915, a German zeppelin bomber drifts menacingly over London. The men on the ground train a spotlight on the airship, which is traveling at 90 km/hour, and at a constant altitude of 1 km. The beam of the spotlight makes an angle θ with the

Motion of an object on the outside of a cylinder

This question considers the motion of an object of mass m sliding on the outside of a cylinder of radius R whose axis is horizontal. The motion occurs in the vertical plane, and the surface of the cylinder is rough — the coefficient of sliding friction is μ'. The diagram below shows the position of the object when it is at an