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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.

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## Categories within Calculus and Analysis

### Basic Calculus

Solutions: 559

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

### Functional Analysis

Solutions: 287

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

### Complex Analysis

Solutions: 564

Complex Analysis refers to the study of complex numbers.

### Real Analysis

Solutions: 3,818

Real Analysis refers to the study of real valued functions.

### Roots of Polynomial for a Derivative

An elliptic curve can be written as y^2=x^3+ax+b. I need a proof for why x^3+ax+b either have 3 real roots or 1 real root and 2 complex roots. I don't have anything that I know about it prior to asking for help here at Brainmass.

### Fixed Costs, Net Income, Capital Plan, & Utilization Reduction

Fixed Costs, Net Income, Capital Plan, & Utilization Reduction Assume that Valley Forge hospital has only the following three payer groups: Payer # of Admissions Avg. revenue/admission Variable cost/admission PennCare 1,000 \$5,000 \$3,

### Calculus Maxima, Minima and Saddle Points

Show all work 1. Determine the exact value for each of the following limits: a. b. 2. Determine derivatives (with respect to x) for the following: a. b. c. d. Determine for e. Determine the partial derivative with respect to x for 3. Integrate the following: a. b. c. 4. For the fun

### Math Homework

MATH 141 Homework Due Dec 2 Name : Solve the following linear programming problem using a graphical method A company makes two puddings, vanilla and chocolate. Each serving of vanilla pudding requires 2 teaspoons of sugar and 25 fluid ounces of water, and each serving of cho

### Value of each definite integral

Name __________________________ Math 331 Fall 2015 DIRECTIONS: Show as much work as possible within each question as I grade on both the process and the final answer. TI-89's are wonderful calculators, but they don't show me if you know anything about calculus! Show all work. 1. (6 pts each) Determine the following ant

### Trigonometric equations and angle between vectors.

Question #1: You used Pythagorus' theorem to determine whether or not a triangle was a right triangle. The sides of the triangle are: a = sqrt(416), b = sqrt(601), and c = sqrt(1009) so that a2 + b2 did not equal c2. Thus it is not a right triangle. Let the α, β, γ be the angles of the triangle across from sides a, b, c

### The Simplex Method

In the simplex method, how is a pivot column selected? A pivot row? A pivot element? Give an example of all three and define the simplex method.

### The Jacobian Matrix in the Implicit Function Theorem

In the attached problem, I am having trouble showing that the determinants are in the same form in the Lagrangian as the one in the implicit function theorem.

### 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

### Differential and difference equation

Please help with the following problem, providing step by step calculations in the solution. 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 file for equations and full problems.

### Check Validity of L'Hopital's Rule & Use to Find Limit

Task: Graph the f(x) = e^2x - 1/x Find the Limit x→0 f(x)

### 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 with differentiation and integration

This solution shows how to solve for various calculus problems, including differentiation of functions using the product rule, the quotient rule, and the chain rule, as well as how to calculate integrals.

### 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.

### Proving for Compactness and Convergence of Sequences

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.

### 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.

### 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.

### 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.

### Computing angular velocity of the wheels

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

### Math Equations: Trigonometry Equation Problems

See attached problems.

### 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

### Maximizing enclosed volumes

An open topped box is constructed by removing a square from each corner of a flat piece of metal and folding up the resulting flaps. The piece of metal is 40 cm by 25 cm. What size square should be removed from each corner so as to maximize the enclosed volume of the box? What is that volume?

### Local maxima, minimum and saddle points

Please provide detailed answers for all questions as per the attached file. These questions are from ESSENTIAL CALCULUS 2ND EDITION, BY WRIGHT-HURD-NEW.

### Complex number identities

Let z, w E C a) Prove the following identities: i) |z+w|^2 = |z|^2 + 2Re(zw) + |w|^2 ii) |z - w|^2 = |z|^2 - 2Re(zw) + |w|^2 b) Deduce that |z+w|^2 + |z-w|^2 = 2(|z|^2 + |w|^2). c) Use (a)(i) to prove that |z+w| < |z-| + |w| and give necessary conditions for equality to hold. d) Prove that [|z| - |w|] < |z-w|.

### Using Lagrange multipliers to find minimum values

MA312 - CALCULUS-2 (GRANTHAM UNIVERSITY) WEEK-5 - Would like to get the detailed answers to all questions.

### Partial Fraction Proof

See attachment.

### Finding Derivatives of Given Functions

Find the derivatives of the following functions: (i) f(x) = sqrt(x)*(2x^3-4) + 3x^(-1/4) (ii) y(x) = (x(x^2-1))/(x^3-4) (iii) g(u) = (4u^(1/3))*(sqrt(u^3+1)) Please see attached and show step by step, thanks.

### Proving the Joule-Thomson Effect

I wonder how one can use a partial derivative with constrain slope dH= 0 in a general equation with two partial derivatives where dH=0 is not true. The problem is for the Joule-Thomson effect. The question is much better described in the attachment. This is a problem from chemistry but since it is just mathematical I thought I m

### Assorted Calculus Questions

Below are the graphs of four functions. Which function is invertible? Set up the integral for the length of the smooth arc y = e x on [0, 10]. What is the area of the triangle bounded by the lines x = 1, y = x − 1, and y = 3 − x ? Suppo