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Calculus and Analysis

Differential equations of Circles

Differential Equation (XV) Formation of Differential Equations by Elimination Find the differential equations of all circles of radius (whatever their radii or positions in the plane xOy).

Differential equations of all parabolas

Differential Equation (XII) Formation of Differential Equations by Elimination Find the differential equations of all parabolas whose axes are parallel to the axis of y.

Differential Equations of Exponential Functions

Differential Equation (IX): Formation of Differential Equations by Elimination Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

Equations of Tangent Lines and Intercepts in Terms of a Variable

Let f(x) = a(7-x^2) where a is not equal 0 (a) find, in terms of a, the equation of the line tangent to the curve at x = -1 (use point slope) (b) find, in terms of a, the y intercept of the tangent line at x = -1 (c) find the x intercept of the tangent line at x=-1

Differential Equation - Initial Value Problem

11. Consider an electric circuit like that in Example 5 of Section 8.6. Assume the electromotive force is an alternating current generator which produces a voltage V(t) = E sinwt , where E and w are positive constants (w is the Greek letter omega). EXAMPLE 5. Electric Circuits. Figure 8.2(a), page 318, shows an electric circu

Population Dynamics

I think the solution is probably a simple modification to the predation equations: dN/dt = rN - cNP is the growth rate for the prey population and dP/dt = -dp + gcNP is the growth rate for predator population, where: N= prey density P = predator density d= death rate g = conversion efficiency of prey to predators r = p

Modeling Data for Linear Functions and Maximizing Profit

1960 88 1970 121 1980 152 1990 205 1997 217 a) Model the data with two linear function. Let the independendt variable represent the number of years after 1960. b) With each function found in part a), predict the amount of maunicipal solid waste in 2005. c) Which of the two models

Sand falls from an overhead hopper to form a right circular cone. If the cone formed has an angle theta find the rate of change of volume with respect to height? If the height is changing at 2cms per minute, what volume of sand is falling from the hopper when the height of the cone is 3 meters? (The volume of a cone is (1/3)pir2h where h is the height and r is the radius.)

Sand falls from an overhead hopper to form a right circular cone. If the cone formed has an angle theta find the rate of change of volume with respect to height? If the height is changing at 2cms per minute, what volume of sand is falling from the hopper when the height of the cone is 3 meters? (The volume of a cone is (1/3)pir2

Length of a curve

Suspension bridge. The cable of the suspension bridge shown in the accompanying figure hangs in the shape of a parabola with the equation , where x and y are in meters. What is the height of each tower above the roadway? What is the length of z for the cable bracing the tower?

A corner lot has dimensions 25 by 40 yards

A corner lot has dimensions 25 by 40 yards. The city plans to take a strip of uniform width along the two sides bordering the streets in order to widen these roads. How wide should the strip be if the remainder of the lot is to have an area of 844 square yards?

Intersection and distance

1. Find the intersection point of the line (x-1)/2=(y+1)/3=z-2 and the plane 2x+y-z=17. 2. Find the distance from point Q(1,-2,3) to the plane 2x-y-z=6. Need steps and solutions.....Thanks!

Spherical polar coordinates

Please help...this is a revision question very likely to appear in my exam next month but I do not understand it! (See attached file for full problem description)

Finding a particular solution of a differential equation

I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differential equation. Not too difficult - see attachment. Please use equation editor if possible. Thank you. --- Given that: dMS/dt = m(MN - MS) - pMS¬

Radius of convergence

(See attached file for full problem description) --- Find the radius of convergence of the following series...(see attachment for equation) ---

Differentibility of Functions and Limits

Prove that if f is differentiable at x, then (see attachment) Also, show that for some functions that are NOT differentiable at x, this limit still exists. --- (See attached file for full problem description)

Derivatives and Application of the Derivative

F(t) = 10,000 / 10 + 50e ^-0.5t HOW do I obtain the derivative? What is the "e" portion of the problem? I know the derivative = 250,000e^-0.5t/ (10+50e ^-0.5t)^2 Please describe in detail the steps taken to arrive at this answer. For example, Why is the top of the equation 250,000e^-0.5t? Why is the bottom (10+50e^-0

Differential Equations: Solution to Heat Equation

Consider the heat equation delta(u)/delta(t) = (delta^2)(u)/delta(x^2) Show that if u(c, t) = (t^alpha)psi(E) where E = x/sqrt(t) and alpha is a constant, then psi(E) satisfies the ordinary differential equation alpha(psi) = 1/2 E(psi) = psi, where ' = d/dE is independent of t only if alpha = - 1/2. Further, show th

Differential equation explanation

I need to know how to find a particular solutions to an initial conditions. i know how to find the explicit solution of a differential equation but cannot remember how to find particular solutions. please show your working to help me understand. thanks Problems (also attached): Given the attached information: 1) So how

Solve and Explain Three Integrals

∫ (x^5)/[(1+x^3)^(3/2)] dx ∫ {square root of [(x+3)/(x+1)]}dx ∫ (cot^3 v)[(csc v)^(3/2)] dx (I will use the $ sign for the integral sign) Problem #1: $ (x^5)/[(1+x^3)^(3/2)] dx the power 3/2 in the denominator is throwing me off greatly, as is the greater power (x^5) in the numerator. attempt 1

Damping and Resonance Problem : Forces and Displacement

A seismograph is a scientific instrument that is used to detect earthquakes. A simple model of a seismograph is shown below. It consists of a particle of mass m to which a pointer is attached. The particle is suspended by a spring of natural length lo and stiffness k and a damper of damping constant r from a platform of height d

Evaluating the Integral with Limits

Please could you solve the following question showing every stage as simply as possible to get to the correct answer. Where any calculus rules are used could you please explain. Integrate sec^2(3t) - cosec^2(5t) between the limits t = 0 and t = pi/4 Please see attached for a more clear version.