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    Calculus and Analysis

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    acceleration, velocity and position of an object

    An object moves along the x-axis with initial position x(0)=2. The velocity of the object at time t is greater than or equal to 0 is given by v(t)=sin((pi/3)t). a.) What is the acceleration of the object at time t=4? b.) Consider the following two statements. Statement I: For 3<t<4.5, the velocity of the object is increa

    Schwarzschild Radius for Sun

    The Schwarzschild radius describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. R = 2 G M / c 2 where G = gravitational constant 6.7x10 -11 M= mass of the object C = speed of light 3x10 8 The sun has M = 2x10 30 . What is the Schwarzschild radius

    Differential Equations Boundary Conditions

    Consider the following problem for u = u(x,t): , , a) Seeking a solution of this problem of the form , show that f and g satisfy the coupled system ; where and ; , ; , . b) Eliminating g between the differential equations in a), show that f satis

    Differential Equations Solving

    Please give all possible explanations. See attached. Solve the following differential equation using the complex number approach.

    Deriving the equation of motion of a projectile shot vertically upward considering the effects of air resistance and solving the first order differential equation obtained.

    Consider a projectile of mass m which is shot vertically upward from the surface of the earth with initial velocity V. Assume that the gravitational force acts downward at a constant acceleration g while the force of air resistance has a magnitude proportional to the square of the velocity with proportionality constant k>0 and a

    Interval of Convergence and Reimann Sum

    Find the Interval of Convergence (include work for endpoints) for ∑∞k=0 [1/(sqrt(k^2 + 1))] * x^k This is the Riemann sum from 0 to infinity (see attached)

    Mean Value Theorem Calculus

    Let f(x) = (x-5)/x Use calculus theorems/techniques to show that the equation has exactly one solution in the interval [1, 8]. Explain carefully. Confirm that f satisfies the hypotheses of the Mean Value Theorem on the interval [1, 8] and find all numbers c in this interval which satisfy the conclusion of the theorem. Show w

    First find a general solution of the differential equation.

    Please see the attached file. dy/dx = 3/y First find a general solution of the differential equation. Then find a particular solution that satisfies the initial condition y(0) = 5. ******************* A bacteria population is increasing according to the natural growth formula and numbers 100 at 12 noon and 156

    Interval where function is increasing and decreasing

    Find the interval where the function is increasing and the interval where it is decreasing. (If you need to enter - or , type -INFINITY or INFINITY. If there is no interval where the function is increasing/decreasing, enter NONE in those blanks.) ( , ) (increasing) ( , ) (decreasing) 2. [TanApCalc7 4.1.0

    Solve the initial value problem

    My text book has to be the worst ever. I can do some of the more basic things but these are throwing me off. Please see the attached file.

    Power Series Values Defined

    Please see the attached file for the fully formatted problems. Let f(x) = 1 + (a) For what values of x is f defined? (b) Find a power series for xf'(x). Justify your answer. (c) Hence show that: f(x) = 1 -

    A small selection of Calculus questions

    1. Give the value of c in the interval [0,3] that satisfies the conclusion of the mean value theorem for -2*sqrt(x)-2 2. Use differentials to estimate the fourth root of 80. 3. Use differentials to estimate the sine of 44 degrees. 4. Use one iteration of Newton's method to approximate a solution to x^3+x-352=0 with initia

    Polar Equations: Rose Curve and Ceva's Trisectrix

    Please see the attached file for the fully formatted problems. E. r = square root theta, 0 <= theta <= 16pi r = theta^2, 0 <= theta <= 16pi r = cos(theta/7) r = 2 + 3cos(theta) r = 2 + sin(2theta) r = 2 + 3sin(4theta) F. r = square root theta, 0 <= theta <= 16pi r = theta^2, 0 <= theta <= 16pi r = cos(theta/7) r =

    Two standard proofs

    Text Book: - Taylor & Menon 1.) Prove that a compact set is bounded. 2.) Prove that if the sequence {xn} converges, than the sequence {IxnI} also converges. Is the converse true as well?

    Fundamental Set of Solutions to an ODE

    Suppose that p and q are continuous on some open interval I, and suppose that y1 and y2 are solutions of the ODE y'' + p(t)y' + q(t)y = 0 on I. (a) Suppose that y1, y2 is a fundamental set of solutions. Prove that z1, z2, given by z1 = y1 + y2, z2 = y1 &#8722; y2, is also a fundamental set of solutions. (b) Prove that i

    Theorems on Limiting Contours

    If a and b are real and positive, show that cos(ax)/(b+x) dx = (xe^-abx)/(1 + x^2) dx which of these would be easier to calculate numerically and why? (See Hildebrand, Advanced Calculus for Applications, 2nd Ed., p. 612, #98 for a similar problem and some hints.)

    A solution of the 2d Laplace equation

    Solve Laplace of u = 0 subject to the conditions: u(x,0) = f1(x) u(0,y) = 0 u(x,b) = 0 u(a,y) = 0 0<x<a 0<y<b (The question attachment contains a slightly different question. The question is restated correctly in the solution attachment)

    L'Hospital's Rule

    It is essential to show all steps by hand!! Also if a method is prescribed use only that method! 8)justifying its use, use L'Hospital's Rule to evaluate the following limit: lim ln(x^8) / 8x^8-8 x->1

    Upper and lower control limits for /x- and R-charts

    Find the upper and lower control limits for /x- and R-charts for the width of a chair seat when the sample grand mean (based on 30 samples of 6 observations each) is 27.0 inches, and the average range is 0.35 inches.