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Calculus and Analysis

Solution of Differential Equation

Verify that y_p , where y_p(x) = sin 2x, is a solution of the differential equation: y'' - y = -5 sin 2x. Use this fact to find the general solution of the equation

Differential equation

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. Find the general solution of the differential equation: y''' + 2y'' + y' + 2y = 0

Differential Equation

For the following differential equation find all numbers r, real and complex, such that e^(rx) is a solution; find two linearly independent real solutions: (D^2 - 3D + 2) y = 0

Differential Equation : Basis

Verify that the given functions form a basis for the space of solutions of the given differential equation: x^2 y'' - 2xy' + 2y = 0, f_1(x) = x, f_2(x) = x^2, x > 0

Differential Equation

Most drugs are eliminated from the body according to a strict exponential decay law. Here are two problems that illustrate the process. 1. The drug Valium has a half-life in the blood of 36 hours. Assume that a 50-milligram dose of Valium is taken at time t=0. Let m(t) be the amount of drug in the blood in milligrams t hours

Parametric Equation

Professional golfer Nancy Lopez hits a golf ball with a force to produce an intinial velocity of 175 feet per second at an angle of 35 degrees above the horizontial. She estimates the distance to the hole to be 225 yards. A) Write the position of the ball as a pair of parametric equations. (I know that it is x=175tcos(35)and

Vector Problem : Force up Ramp

What would be the force required to push a 100-pound object along a ramp that is inclined 10 degrees with the horizontal?

LAPLACE TRANSFORMS

Please see attachment. Require problems solving, also explanations etc for better understanding.

Calculus 3 and 4

Show all work. Please DON'T submit answers back to me as an attachment. Thank you. The dimensions of a closed rectangular box are found by measurement to be 10 cm by 15 cm by 20 cm, but there is a possible error of 0.1 cm in each. Use differentials to estimate the maximum resulting error in computing the total surface area

Calculus 3 and 4

Show all work. Please DON'T submit answers back to me as an attachment. Thank you. Determine whether the function is homogenous. If it is, state the degree: f(x, y)=5x^2 + 2xy

Calculus

Find the order of the differential equation and determine whether it is linear or nonlinear: y^(4) + 3(cos x)y''' + y'=0

Minimization of a triangle area

Find the altitude to the base of an isoceles triangle with a side length of 4, such that the area of the triangle is maximal.

Oscillating Inflow Concentration

Make a conjecture, on the basis of physical reasoning, as to whether you expect the amount of salt in the tank to reach a constant equilibrium value as time increases. In other words, will lim(t) -> infinite Q(t) exist? (see attachment for full question)

Functions : Tangent, Increasing or Decreasing and Area under a Curve

2. Let f be a function defined on the closed interval -3≤x≤4 with f(0) = 3. The graph of f', the derivative of f, consists of one line segment and a semicircle. a) On what intervals, if any, is f increasing? Justify your answer. b) Find the x-coordinate of each point of inflection of the graph of f on t

Distance travelled around box problem

An ant is walking around the outside of the cube in "straight" paths (where we define a straight path in this case as one formed by the edges of a cross section created by a plane slicing through the cube). For example, to get from point Q to point R in the picture above on the right, the ant walks along the red path. There are

Geometric series?

Determine if the following series converges and if possible give its sum 2/3 + 2/9 + 2/27 + 2/81 + ...

Comparing Graphs : First and Second Derivatives

X -1.5 -1.0 -0.5 0 0.5 1.0 1.5 f(x) -1 -4 -6 -7 -6 1.0 -7 f'(x) -7 -5 -3 0 3 5 7 Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its derivative f' for selected points x in the closed interv

Graphs : Rectangular and Parametric Forms

Graph x=(t^2+2t+1)^(1/2) y=(t^3+2t^4)/t^2 a. Graph on the interval [0,3] b. Convert to rectangular form. c. Adjust the domain of the rectangular form to agree the parametric form.

Proportionality and Rate of Change Word Problem

A container has the shape of an open right circular cone. The height of the container is 10cm and the diameter of the opening is 10cm. Water in the container is evaporating so that its depth h is changing at the constant rate of -3/10 cm/hr. Show that the rate of change of the volume of water in the container due to evaporati

Parametric Equations : Circle and Ellipse

Create parametric equations for: a. A circle of radius 2 centered at (3,1). b. An elipse with a horizontal major axis of length 5 and a verticle minor axis of length 4.

Maximum and Minimum Distance Word Problem

Johnny Steamboat wants to sail from his island home to town in order to purchase a book of carpet samples. His home island is 7 miles from the nearest point on the shore. The town is 35 miles downshore and one mile inland. If he can run his steamboat at 12 mph and catch a cab as soon as he reaches the coast that will drive 60

Limitation

Let X(n)=Sum{1/(n+i), i=1->n}, find the limit of X(n) as n tends to infinity

The Convergence of Darbox Sums and Riemann Sums

1. Let k >= 1 be an integer, and define Cn = SIGMA (1/(n+i)) as i=1 to kn (a)Prove that {Cn} converges by showing it is monotonic and bounded. (b)Evaluate LIMIT (Cn) as n approach to the infinity

Work done to pump water out of tank

Would like second opinion or other way to solve problem, hopefully using Disk method - OTA #103642 answered last time. Perhaps OTA #103642 could send me email regarding this formula? Problem - A tank is in the shape of an inverted cone (pointy at the top) 6 feet high and 8 feet across at the base. The tank is filled to a de