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Calculus and Analysis

Calculus III

Please show all the steps to solving this problem.

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

True - False

The following statements are either true or false. a). If x=p is not a critical point of f, then x=p is not a local maximum of f. b). If x=p is not a local maximum of f, then x=p is not a critical point of f. c). If f'(x) is continuous and f(x) has no critical points, then f is everywhere increasing or everywhere decrea

Differential Equations

Hi, I attached one problem. Please help me providing detailed solution with explanation. Thanks

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

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)

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

Power Series

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 -

Differential equation

Please help me to solve attached problem. 1. Determine whether one of the solutions of the differential equation x^2y" + xy' + (x^2 -1/4)y =0 on 0<x<infinity is y(x) = x^-1/2sinx

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

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)


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.