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# Calculus and Analysis

### 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

### Advanced Calculus: series of functions

Show that the given series of functions converges uniformly on the given set D. series n=1 to infinity (-1)^(n-1)* x^n = x- x^2+ x^3 - x^4+.......D=[-1/2, 1/2] (Do not rely on a general fact about power series.)

### Polynomials and Differentials: The Cauchy Riemann Equation

** Please see the attached document for the complete problem description ** Please help solve this attached problem that involves polynomials and differentials. It involves an application of the Cauchy Riemann equation.

### 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

### Calculating the Amount of Medicine Needed for a 210 lbs Patient

In the hospital pharmacy, Michiko sees that a Medicine is to be given at the rate of milligrams for every 50 pounds of body weight. How much medicine should be given to a patient who weighs 210 pounds?

### 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

### Buoyancy : Equilibrium Position of a Floating Cylinder

I am trying to work through some example problems from Schaum's Outlines: Differential Equations. I'm not sure how to come up with the solution and need help. Based on the attached file the question is: Find an expression for the motion of the cylinder described in previous problem if it is disturbed from its equilibrium

### Differential Equations - Systems of Equations, Laplace Transform System of Equations.

I am looking for help with review problems for my final exam. I narrowed down the problems I am have the most difficulty with and need help with. I need the problems worked out so I can practice the appropriate steps for success on the final next week. 1. A 32 pound weight stretches a spring 2 Feet. The mass is then released

### Property of solution of a system of differential equations

We answer the following question, The system of differential equations: u'=v, v'=-u, A. Does not have any nonzero solution. B. Has only periodic solutions. C. Has only unbounded solutions. D. Has only solutions that decay to zero at infinity. E. Has some unbounded solutions.

### Properties of the equilibrium price for a demand function

The demand function for a certain good is D(p,m) and the supply function is S(p). For a given m, the equilibrium price p* is given by p* = f(m). a) Show that f'(m) = [dD(p*,m)/dm] / [S'(p*) - dD(p*,m)/dp*] b) Verify this result when S(p) = 2p and D(p,m) = 6m^2p^(-1) + m c) By using the implicit function rule, or oth

### Production function, returns to scale and Taylor expansion

Consider a production function of the form F(K,L)=(K^(-a)+L^(-a))^(-1/a). (a) Is this function homogeneous? (b) Does it display increasing, constant or decreasing returns to scale? (c) Let G be a differentiable function. Find an expression for G(K+g,L+h) by taking a first-order Taylor expansion of G about (K,L). (d)

### Calculus: Factorials and Solving Formulas

The seventh term of (a - b)^30 (Hint: 7 = r + 1; r = 6 and n = 30) Possible answers: (1) [n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)]/6! [(a^n-6)(-b)^6] (2) [(30)(29)(28)(27)(26)(25)]/6! (a^24 b^6) (3) (30! - 15!)/6! [a^n-6 (-b)^6] (4) 585,762 a^24 b^6

### Step by Step Explanation of Differential Equations

I am asking for the step-by-step workings for all of the attached problems. ** Please see the attached file for complete problem description ** 1st problems. Please find the general solution of: (1) dy/dx = y/sin(y) - x (2) dy/dx = y + cos(x)y^2010 In the process of finding the solutions for the problems make use of both

### Concentration and Differential Equations

See the attached file. A polluted river with a nutrient concentration of 110g/m^3 is flowing at a rate of 130m^3/day into an estuary of volume 1000m^3. At the same time, water from the estruary is flowing into the ocean at 120m^3/day. The initial nutrient concentration in the estuary is 35g/m^3. (i) Let N(t) be the amount

### Population Change and Growth Rate

Suppose a country's population changes due to births, deaths, and immigration. The annual birth rate is 5.6 births per 100 people, the death rate is 4.3 deaths per 1000 people, the immigration rate is 7.8 per 1000 people, and the emigration rate is 2.4 per 1000 people. Find the growth rate as a %.

### What is the value of given endowment in today's dollars?

A wealthy philanthropist has established the following endowment for a hospital. The details are as follows: a cash deposit of \$ 8 M one year from now; an annual cash deposit of \$3M per year for the next five years. The first \$3M will start today; at the end of 5 years, the hospital will also receive a lump sum payment of \$18M.

### Calculus - Finding the Area of a Specified Region

Please help with the following questions: 1. Use the limit process to find the area of the region between the graph of the function and the axis over the given interval. Sketch the region. y = 4 - x^2, [-2,2] 2. Find the area of the region bounded by the graphs of the equations. y = x^3 + x, x = 2, y = 0

### Set up linear equations for the word problem.

A gourmet coffee shop has a weekly budget for 2 imported coffee beans. Sixty dollars per week is allotted for Italian beans and Kenyan beans. Italian beans cost \$10/lb and Kenyan beans cost \$15/lb. Write a formula for the number of pounds of Kenyan beans, K, the gourmet coffee shop can buy as a function of the number of Ital

### Business Calculus and Average Cost Function

1. Average Cost. A company manufacturing snowboards has fixed cost of \$200 per day and total cost of \$3800 per day at a daily outputs of 20 boards. a) Assuming that the total cost per day c(x) is linearly related to the total output per day x, write an equation for the cost function. b) The average Cost per board for an out

### Forming a Basis in R4

Find a basis for each of the following subspaces of R4. a) Find all vectors that satisfy the equation x1 - 3x2 + 6x3 - 4x4 = 0. b) Find all vectors whose entries satisfy both equations x1 + x2 + x3 + x4 = 0 and x1 - x3 = 0.