# 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.

Last updated on Aug 28, 2013

#### Categories within Calculus and Analysis:

Basic Calculus

• Postings:
• 375

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

Functional Analysis

• Postings:
• 88

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

Complex Analysis

• Postings:
• 554

Complex Analysis refers to the study of complex numbers.

Real Analysis

• Postings:
• 4,121

Real Analysis refers to the study of real valued functions.

Assorted Calculus Questions

Please answer the following 23 questions in the file, there are 5 pages because I made the questions large.

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

Separation of variables and Laplace equation

See attached file and answer the questions that cover the concepts of separation of variables and Laplace equation. Please show steps

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

Average Rate of Change of f(x)

1) Find the average rate of change of f(x)=x^3 - 9x+9: a) from -8 to 2 b) from -1 to 1 c) from 1 to 4

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

Finding equation of the conic

Find an equation of the conic with the vertices (-2,3) and (6,3) and length of the minor axis of 8

Cartesian to Polar Equations

Find a polar equation for the curve represented by the given Cartesian equation. a) 4 * y^2 = x b) x * y = 4

Polar Coordinates to Cartesian coordinates Equations

Identify the curve by finding a Cartesian equation for the curve. a) r = 4*sec(theta) b) r = tan(theta) * sec(theta)

determining the general solution of a differential equation

Determine the general solution, in closed form, of the differential equation x d^2 y/dx^2 + 2y = 0

Evaluate the series in closed form

Please evaluate the attached series in closed form (Attached in MS Word).

Calculus: Plotting Functions

See attached file, and please show your workings for the following questions regarding plotting various functions. Concepts covered include contour plots, sine and cosine.

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

Vector Calculus & Applications

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

MULTIPLE CHOICE CALCULUS QUESTIONS

Choose from the alternative that best completes the statement or answers the question attached below in the word document. Show all calculations leading to the chosen correct answers.

Vector Calculas and Applications

We wish to determine whether the following integral is path-dependent: I = f_c - 2ycos2xdx - sin2xdy In the practice problems, you must: - Determine if statement is correct - Calculate the Jacobian of transformation - Evaluate triple intergrals Please see attachment for following problems.

Cauchy criterion in calculus

Assume that un (sub n) belongs to the complete real normed vector space (V, ll.ll) for each n E N and that the series from n=1 to infinity - abs value( un+1 - un) converges. Show that the sequence {un} n=1 to infinity converges. Hint: make use of the Cauchy criterion

Calculus: Estimating Volume Increase

The radius of a sphere increases from 2.00 to 2.02 cm. Estimate the increase in volume.

Related Rates of Change

A cylindrical tank has a radius of 50 cm. How fast does the water level drop, when the tank is drained at a rate of 3 litres/second? [1 litre = 1000 cm3].

Functions when a variable goes to infinity

V_cell = (I_cell)(R_esr) + (I_cellR_p)(1 - exp(-t/((R_p)(C_t))) Calculate the equation for V_cell when R_p > infinity Is the new equation the same as below? If yes, show how you arrive at this: V_cell = (I_cell)(R_esr) + (I_cell)(t/C_t)

Closed Metric Spaces

Prove that a subset W of a metric space (M, d) is closed if and only if W contains all its accumulation points. Note: A set is closed if and only if it contains all its accumulation points

Integration of rational functions by partial fractions

Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients.

Series: S = 1/(1*2*3) + 1/(3*4*5) + 1/(5*6*7) + 1/(7*8*9) + ...

Evaluate the series in closed form. Give an exact answer not a decimal: S = 1/(1*2*3) + 1/(3*4*5) + 1/(5*6*7) + 1/(7*8*9) + ...

Calculus: minimize a rectangle

Minimize the fence needed to contain 200 Smurfs in 50 square feet, supposing that this is a rectangular fence sharing one border with your house. What are the fence dimensions?

Differential Equations: Populations

The population sizes of a prey, X, and a predator, Y (measured in thousands) are given by x and y, respectively. They are governed by the diﬀerential equations ẋ = −pxy + qx and ẏ = rxy - sy (where p, q, r and s are positive constants (p ≠ r). In the absence of species Y (i.e. y = 0), how would I ﬁnd a solution

Evaluating A Series (Closed Form)

Evaluate the series in closed form: f(x) = 1+x+x^2/2!-x^3/3!-x^4/4!-x^5/5!+x^6/6!+x^7/7!+x^8/8!........

Calculus: Rotation and Volumes

Consider the area bounded by the curve y = 2x(1-x) = 2x - 2x^2 and the x-axis 1. Draw a neat sketch of the area 2. This area is rotated about the y-axis. Calculate the volume of solid of revolution.

Dirichlet Problems

1. Consider the Dirichlet Problem where the temperature within a rectangular plate R is steady-state and does not change with respect to time. Find the temperature u(x,y) within the plate for the boundary conditions below and where (See attached) 2. Solve the Dirichlet problem for steady-state (constant with respect to t