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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: 538

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

### Functional Analysis

Postings: 286

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

### Complex Analysis

Postings: 563

Complex Analysis refers to the study of complex numbers.

### Real Analysis

Postings: 3,819

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 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.

### Net Present Value Decision Analysis

A company which manufactures compact discs has found that demand for its product has been increasing rapidly over the last 12 months. A decision now has to be made as to how production capacity can be expanded to meet this demand. Three alternatives are available: (i) Expand the existing plant; (ii) Build a new plant in an indu

### 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

### 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)

### 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.

### 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

### 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.

### Using Lagrange multipliers to find minimum values

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

### 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

### 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 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

### Vector Calculus and Applications

** Please see the attached file for the complete solution ** 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 integrals

### 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

### Dirichlet Temperature Boundary Conditions

See the attached file. 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 (

### Mountaintop Calculus

** Please see the attached file for the complete problem explanation ** 1. Yon are standing at the point P = (100, 100) on the side of a mountain whose height is given by h = 1/1000 (3x^2 - 5xy + y2) with the x-axis pointing east rind the y-axis pointing north. (a) In what direction should you proceed in order to clamp the

### Vector Calculus, Partial Derivatives, and Coordinates

See the attached file. Vectors 1. The points A(2; -3, 3), B(3,5;4), C(3;8;-2) and D(4;4;6) are vertices of a tetrahedron. Find the volume of the tetrahedron. 2. There are two vectors: a = (2; -6; -4) and b = (3; -4; 2). Calculate: (i) a * b (ii) (2a - 3b) * (a + 2b) ; (a + b)^2 ; (a - b)^2 3. There are two vectors:

### Equations of a Straight Line

Using the applet at Equations of a Straight Line (http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml), plot a line through the points P1(-2,0) and P2(0,1). Select the "show grid" option below the window. Copy the plot, and crop it to show only the area defined on the left and right by X = 5, and on the top an

### Cost and Revenue Calculations using Calculus

1. The cost and the revenue functions (in dollars) for a frozen yogurt shop are given by: C(x)= 400x +400/ x +4 and R(x)=100x Where x is measured in hundreds of units A=Graph C(x) and R(x) on the same set of axes B=What is the break-even point for this shop C=If the profit function is given by P(x), does P(1) represent a

### Elementary Calculus: Integers, Derivatives, 14 Questions

1. Let f:Z->Z , where Z is the set of integers and f(x)=x^5+101 Is f(x) a one-to-one function? 2. Prove that f(x)=4x-3 is one-to-one function 3. The ceiling function maps every real number to the smallest integer greater than or equal to that

### Differential Equations Problem Set

Consider the differential equation dy/dx = -x/y. a) Sketch a direction field for this differential equation. b) Sketch solution curves of the equation passing through the points (0, 1), (1, 1) and (0, -2). c) State the regions of the xy-plane in which the conditions of the existence and uniqueness theorem are satisfied