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Calculus and Analysis

Growth and decay models

When a vertical beam of light passes through a transparent medium, that rate at which its intensity I decreases is proportional to I(t), where t represents the thickness of the medium (in feet). In clear sea water, the intensity 3 feet below the surface is 25% of the initial intensity I0 of the incident beam. What is the inten

Steps to solve a limit question are noted.

1. lim (12x^(3) + 12)/5x^(4) + 5x^(3) + 7 x-> infinity 2. lim (12x^(3) + 12)/5x^(4) + 5x^(3) + 7 x-> -infinity 3. lim (3x^(3) - 9x^(1) + 7)/3x^(3) + 7x^(2) - 7 x->infinity 4. lim (3x^(3) - 9x^(1) + 7)/3x^(3) + 7x^(2) - 7 x-> - infinity 5. lim

Solving a Laplace Transform Problem

Solve the following DE using Laplace Transform. Be sure to show your steps and reference the Laplace Transform you will be using. {i.e. - e^at sin bt; b/((s-a)^2 + b^2) (s > a) } (d^2y)/(dt^2) + 2(dy/dt) + 2y = x(t) where x(t) = { 1, 0 < = t < = 1 x(t) = { 2, t > = 2 x(t) = { 0, elsewhere and y(0) = 0;

theoretical limiting concentration

A tank with capacity of 700 gal of water originally contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/ min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/ min. Let Q(t) lb. be the amount of salt in the tank, V(t) gal be the v

maximum height above the ground that the ball reaches

First order differential equation model: A ball with mass m kg is thrown upward with initial velocity 19 m/sec from the roof of a building 22 m high. Neglect air resistance. (a) Find the maximum height above the ground that the ball reaches. (b) Assuming that the ball misses the building on the way down find the time th

Change of Variables

Consider the following Diff Eq: (x^2)(y'') + x(y') - 9y = 0. a) Solve the equation by seeking change of variables of the form x = e^x. b) Show that the DE with non-constant coefficients above becomes a Linear Ordinary Diff Eq with constant coefficents. c) Determine y by solving that Linear Ordinary Diff Eq. d) Find t

Functions: uniform convergence

Note: all of the following _n denote n as a subscript. Suppose {f_n} from n = 0 to infinity, is a sequence of functions converging uniformly to a function f on an interval [a,b]. Also assume that the sequence has a uniform bound i.e. there exists M1 such that for all positive integers n, | f_n(x) | < = M1 for all x belonging

Identify the type of sampling

1: Given the situation described, identify the type of sampling involved. A student doing a report on fashion trends asks all her best friends their opinion on plaid clothing. - Random sampling - Systematic sampling - Convenience sampling - Cluster sampling - Stratified sampling 2: Scripps Hospital surveyed 10 patients

Calculus: Distance between two points

Find the values of a>0 and b>0 such that (a,b) is at distance 5 from (1,2) and the line passing through the points (a,b) and (1,4) is parallel with the line passing through the points (0,0) and (b,a).

Tangents and Normals

Find the tangents and the normals at any point of the following curves: 1) y=1/x^2+2 (1,1/2) 2) y=6/(x^2+1)^2 (1,3/2)

ship movement

Two ships are steaming away from a point O along routes that make a 120°angle. Ship A moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yards). Ship B moves at 21 knots. How fast are the ships moving apart when OA = 5 and OB = 3 nautical miles?

Calculus and the Connecticut River

Needed info: Whenever the Connecticut River reaches a level of 105 feet above sea level, two Northampton, Massachusetts flood control station operators begin a round the clock river watch. Every two hours they check the height of the river using a scale marked off in tenths of a foot, and record the data in a log book. In

Relative extrema, Limits, Minimum average cost per unit

5. Find the open intervals on which f(x) = x - log-base-4(x) is increasing or decreasing, and locate all relative extrema. 6. Find all relative extrema and points of inflection of y = x^2 log-base-3(x). 7. Find the limit of each of the following functions: Please refer to the attachment for details. 8. A farmer p

Pre-Calculus Homework Issue

** Please see the attached file for the complete problem description ** A trainee is hired by a computer manufacturing company to learn to test a particular part of a personal computer after it comes off the assembly line. The learning curve for an average trainee is given by ...(a) How many computers can a trainee be expec

Heine-Borel Theorem for LUB and GLB

Let f:[a,b]--> R be continuous. Prove that the set f([a,b]) has both a least upper bound M and a greatest lower bound m and that there are points u,v in [a,b] such that f(u)=M, f(v)=m.

Demand Function

Research has indicated that the demand for heroin is given by the function q=100p^(-0.17) a) Find E b) Is the demand for heroin elastic or inelastic? Solve this problem, show all steps used to solve the equation, and explain the answer to receive credit for the solution.

arc length and differential equations

Please show all work with explanations. Thank you for your help. 4. Find 6. Find the arc length of the graph of the function over the interval . 18. Solve the differential equation

For what gross sales is Plan B better?

A car salesman has two choices for his salary. Plan A: A salary of $100 per week plus a commission of 3% of gross sales Plan B: A salary of $75 per week plus a commission of 5% of gross sales For what gross sales is Plan B better? Any help is appreciated!

Finding the equilibrium, quantity, and price

The quantity of watches demanded per month is related to the unit price by the equation: p = d(x) = 50/ (0.01x^2 + 1) (1 less than equal x less than equal 20) where p is measured in dollars and x is measured in units of a thousand. The supplier is willing to make x thousand watches available per month when the price

Differential Equation -Fourier transform distribution

Evaluate the fundamental solution of the equation u^(4) - 2u'' + u. More in general, evaluate the fundamental solution of u^(4) - (m+1)u'' + mu, where m>=0. Please note that you are not solving the equation set equal to zero. Since you are looking for the fundamental solution you set the equation equal to the delta distributi

Linear Functions Example Problems

For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly. a. The time is takes to get to work as a function of speed at which you drive. b. The probability of getting into a car accident as a function of the speed a

Properties of an Ellipse

Equation: x^2+9y^2-4x+54y+49=0 1. Identify the conic section represented by the equation. 2. Write the equation of the conic section in standard form. 3. Identify relevant key elements of your conic section such as center, focus/foci, directrix, radius, lengths of major and and or minor axes, equations of asymptotes

Maximum altitude attained by the rocket

1. If f is continuous at x= a and g is differentiable at x=a then: Lim x->a f(x)g(x) = f(a)g(a) (if true explain why true, if false, explain and give an example of why it is false) 2. Let f(x) = 2/3x^3 + x^2 â?" 12x + . Find the values of x for which: A f prime (x) = -12 B f prime(x) =o C f prime (x) = 12 3

Calculating stock price changes due to variations in beta: Example problem

Hard Hat Construction's stock is currently selling at an equilibrium price of $30 per share. The firm has been experiencing a 6 percent annual growth rate. last years earnings per share, was $4.00, and the dividend payout ratio is 40 percent. The risk-free rate is 8 percent, and the market risk premium is 5 percent. If systemati

Secant and Tangent Lines

Consider function f(x) = -x^2 +2x and points P1 = (0, 0), P2 = (1, 1) and P3 = (2, 0). Find equations of: - secant lines to f through every pair of these points (3 pairs), and - tangent lines to f at each of these points.

values of y and theta

The attached diagram shoes a lighthouse, L, 4.2 km from the nearest point, S, on a straight shoreline. The lighthouse beam rotates with a constant period of one complete rotation every 40 seconds. The beam makes an angle of theta with the line LS and the beam hits the shoreline a distance y km from S. At what rate is the beam

Proof involving non-autonomous differential equation

Let A be an n x n matrix. Let f(t; x0) be a solution to the initial value problem x' = Ax; x(0) = x0. Show that f(t2; f(t1; x0)) = f(t1 + t2; x0). Show by example that this is not true for a non-autonomous differential equation.

Arc length and surface area

Please show all your work with helpful explanations. 1. Find the length of the arc of the curve y = x^(3/2) from x=1 to x=2. 2. Find the length of the arc of the curve y = ln(sec x) from x=0 to x=PI/4. What is the length of the curve from x=0 to x=PI/2? 5. Find the surface area of the cone formed by rotating the line y = ax

Lebesque integral

A) Evaluate â?«R^2 xy ? {[0,1]Ã?[0,1]} dm *(x, y), (or if you prefer, â?«[0,1]Ã?[0,1] xydm*(x, y)). b) Evaluate â?«R^2 xy ?{[0,1]Ã?[0,1]â?'Q^2} dm* (x, y) The original problem is sent as PDF file attachment below. It is number 4 on the list.