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# Calculus and Analysis

### Arc length of graph of function for interval.

Arc length of the graph of function y=x (to the 2/3)-1 interval[0,4]

### Calculating the length of a curve.

Find the length of the curve y=e^-x, 0<_x<_1

### Proving that an equation is harmonic.

Prove that U(x,y)=e^(-x)[xsiny-ycosy] is harmonic.

### Finding the limit of an expression.

Find the limit of following {1-cos(x)}/x^2 as x->infinity

### Finding the maximum of a function.

Find the maximum of the following: f(x)=x^3-2x^2+1, -2<=x<=2

### Using De Moivre's theorem.

If tan(A) = 1/2, find the value of tan(5A) Hint: Use De Moivre's theorem.

### Differential Equation with a Natural Log

Find a general solution for this differential equation. Please see the attached file for the fully formatted problem. dy/dx + 2xy = xe^(-x2+2)

### Differential Equation: System of Equations

For this problem state the method you used and show the work required to obtain the answer. Find the general solution for this system: this is a matrix x'= 3y+z y'= x+z+2y z"= 3y+x

### Differential Equations

Hi, For this problem please state the method you used and show the work required to obtain the answer. Find the complete solution to this equation problem. y^3 - 2y^2 + y^1 = 2- 24e^(x) + 40e^(5x)

### Finding the minimum and maximum with base e.

Find the complete solution of this equation problem y^4 - y^2 = 4x + 2xe^(-x)

### A problem dealing with volume of revolution

A bowl is shaped like a hemisphere with radius R centimeters. An iron ball with radius R/2 centimeters is placed in the bowl and water is poured in to a depth of 2R/3 centimeters. How much water was poured in?

### Volume of water poured in to bowl with iron ball.

A bowl is shaped like a hemisphere with radius R centimeters. An iron ball with radius R/2 centimeters is placed in the bowl and water is poured in to a depth of 2R/3 centimeters. How much water was poured in?

### Surface area of revolution

How do I find the surface area obtained by revolving the curve y=x-1 from x=1 to x=4 about the line x= -1?

### Arc length of a Curve

How does the length of the curve x=t^2, y=t^3, [0,2] come out to be 3/4 + (ln2)/2?

### Volume maximization : Finding the dimensions of a cylinder given the surface area.

Find the dimensions of a cylinder with a surface area of 300 cm^2 with a maximum volume.

### Vertex, focus, and directrix of a parabola.

1. 20x=y^2 2. (x-3)^2 =1/2(y+1) 3. y2+14y+4x+45=0 Find the vertex, focus, and directrix of the parabola described by the above equations.

### Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

1. 20x=y2 2. (x-3)squared =1/2(y+1) 3. y2+14y+4x+45=0 Find an equation of the parabola that satisfies the given conditions Focus F(0-4), directrix y=4 Find the vertices, the foci and the equations of the asymptotes of the hyperbola. 1.y2divided by 49 minus x2 divided by sixteen =1 2.x2-2y2=8 Find an equat

### Differential Equations Reduction of Order Methods

Please see the attached file for the fully formatted problems. (i) Consider the differential equation: x. = x^2 , x(0) given x(0)>0 Find the solution of x(t) of this equation in terms of x(0) and show that there is a T, which depends on x(0), such that lim x(t) = infinity t --> T- (ii) Find the solution of the

### Working with growth and decay rates and decay rate expressions.

A crude-oil refinery has an underground storage tank which has a fixed volume of 'V' liters. Due to pollutants, it gets contaminated with 'P(t)' kilograms of chemical waste at time 't' which is evenly distributed throughout the tank. Oil containing a variety of pollutants with concentration of 'k' kilograms per liter enters

### Arc length of the curve defined by a parametric system

Find arc length of the curve defined by the following parametric system: x=cos^-1(t) (inverse cosine) y= ln t where t is less than/equal to 1, greater than/equal to (1/sqrt 2)

### Differential Equations and Newton's Law of Cooling

At 4:30 PM on Monday, a Virginia criminalist was called to the scene of a homicide. She noted that the body temperature of the deceased was 85.5 deg. while the air temperature was 78 deg. Thirty minutes later, the deceased's body temperature was 82 deg. Assuming the air temperature stayed constant, what is the estimated time of

### Implicit Differentiation Functions

Find y" for x^2/a^2 - y^2/b^2 = 1

### Interval of Convergence Power Series

I am trying to find the interval of convergence for the attached power series (attached as a gif). I am also supposed to check the endpoints for convergence. I'm not that good with power series and the format of this power series is really throwing me off. So I am looking for the steps to find the interval of convergence (also c

### Power Series Representation

I am trying to figure out a powers series representation for the function f(x) = e^x, centered at c = -2. I also need to find the interval of convergence.

### Finding the tangent to a curve. Picture included.

Find an equation of the tangent line to the curve Y= x3 - 3x2 + 5x that has the least slope.

### Tangent Line to a Curve

Find an equation of the tangent line to the curve, Y = x^3 - 3x^2 + 5x that has the least slope. Make sure to show all of the required steps.

### Rectilinear motion: Calculation of velocity

A billiard ball is hit and travels in a line. If s centimeters is the distance of the ball from its initial position at t seconds, then s=100t2 + 100t. If the ball hits a cushion that is 39cm from its initial position, at what velocity does it hit the cushion?

### Simple Vector Cross Product Proof

Create a proof to show that the following is true. a x (b+c) = a x b + a x c

### Find the maximum area of a window.

A special window has the shape of a rectangle surrmounted by an equilateral triangle. If the perimeter is 16 feet, what dimensions will admit the most light? (hint: Area of equilateral triangle = the square root of 3/4 times x squared.)

### Sum of a Series Trigonometric Functions

I am stuck on how to solve the sum of the series that I have attached in a word document.