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# Functional Analysis

### Density and Center of Mass

1) A plate is bounded by the curves y = -x^2, y = x^2, and x = 1. It has density d(x,y)=x. Make a sketch and find its center of mass. 2) A solid occupies a half-cylinder W: x^2 + y^2 is less than or equal to 4, y is greater than or equal to 0, z is greater than or equal to -3 and less than or equal to 3. It has constant

### Mobius Transformations: Example Problem

Suppose T is a Mobius transformation such that the image of the real axis under T is the real axis. Prove that T may be written in the form T(z) = (az+b)/(cz+d) with a, b, c, and d real.

### Turns of a Helix

^ ^ ^ ^ For the helix r = a cos t i + a sin t j + ct k find c ( c > 0) so that the helix will make one complete turn in a distance of 3 units measured along the z - axis.

### Asymptotes

Find all vertical and horizontal asymptotes of f(x) = (x^2+ 4)/(x^2- 4x-12) Please show steps & graph if applicable!

### Singularities and Poles

The function f(z) = zsin(pi/z)/[(z-1)(z-2)^2] has isolated singularities only. Determine the singularities of f(z) and classify each of them as removable, a pole, or an essential singularity. If z0 is a removable singularity, find the value f(zo) that makes f(z) analytic at z0. If z0 is a pole. find the singular part of f(z) at

### Asymptotes

Explain and contrast the types of asymptotes considered for rational functions.

### Cylindrical Coordinates.

Express x^2 + y^2 = z in cylindrical coordinates.

### Polar Equation from Rectangular Equation and Conic Sections

Find the foci of the ellipse x^2 / 7 + y^2/16 = 1. Find the polar equation for the curve that satisfies the rectangular equation 4x+3y=5 Find an equation in x and y for the conic section with polar equation r= 1 +cos &#952;

### Vectors : Divergence, Gradients and Curls

1) V(x, y, z) = (x + y + z)2 i + (x + y)2 j + x2 k. Find div V(3, 2, 4) &#8801; &#8711;? V (3, 2, 4) 2) F (x, y) = xe2y i + y/(x + y) j. Find &#8711; ? F (4, 0) 3) F (x, y, z) = -yz i + xz j - xy k. Find curl F (1, 2, 5) = &#8711;×F ( 1, 2, 5)

### Rectangular to Cylindrical and Spherical Coordinates

Change from Rectangular, Cylindrical and Spherical Coordinates. Problems: 19, 23, 28 See attached file for full problem description.

### Polar Equations: Conic Section

Find an equation in x and y for the conic section with polar equation r=1/1+cos

### Parametric Equations

Find an equation in x and y for the curve given by the parametric equations x=e-t,y=-2t,t in R.

### Equation of an Ellipse

25*y^2 + (10/sqrt2)*y*z + 4*z^2 - 50 = 0 I know when compared with the standard 2nd degree conic equation that this is an ellipse rotated 16.24 degrees. My question is , How do I put this equation in the standard form of an ellipse equation? ie: y^2/a^2 + z^2/b^2 = 1

### Determine the type of conic from a pair of parametric equations.

Eliminate t to determine the type of conic from a pair of parametric equations. See attached file for full problem description.

### Ellipses and Parametric Equations

7. Find the equation of the ellipse whose center is (-3,1); vertex (-3,3) and focus (-3,0). 8. Graph by hand the curve whose parametric equations are given and show its orientation. x=t+3 y=2t^2 0<=t<=5 9. Find the two different parametric equations for the rectangular equation y=x^2 +2

### Holomorphic Map Functions

Suppose z= phi(&) and w=psi(&) are one-to-one analytic maps from the unit disc D(0,1) onto the regions G_1 and G_2. Set phi(0)=z_0 and psi(0)=w_0. Let 0<r<1 and omega_1(r)=phi(D(0,r)), omega_2(r)=psi(D(0,r)). Assume f: G_1->G-2 be holomorphic map with f(z_0)=w_0. Show that f(omega_1(r)) is contained in omega_2(r)

### Green's Theorem Enclosed Curves

Use Green's Thereom to find the area enclosed by the curve: {abs(x)}^(1/2) + {abs(y)}^(1/2) = 1. It is important that you solve this problem using Green's Method because that is how my professor prefers it. I know that you have to make the above statement true and switch to a different set of coordinates. Something al

### Equation of ellipse and hyperbola

1 Find an equation of the ellipse with the center (0,0) , vertical major axis 14 and minor axis 10. 2 Find an equation for the hyperbola with the focus (11,12) and asymptotes 4x-3y=18 and 4x+3y=30. 3 Find the arc length of the curve given by x = sin t - cos t, y = sin t + cos t, pi/1 <= t <= 3pi/4

### Polar and Rectangular Coordinates

1 Express the polar equation r^2 = 2cos2&#920; in rectangular form. 2 Find the total area enclosed by the graph of the polar equation r = 1 + cos2&#920;

### Conformal Mappings

Let O be the upper half of the unit disc D. Find a conformal mapping f: O->D that maps {-1,0,1} to {-1,-i,1}. Find z in O with f(z)=0

### Finding Area of a Region Bounded by a Line and a Curve

Sketch the region bounded by the graph of the functions and find the area of the region 1) f(x) = - x^2+ 4x + 2, g(x) = x + 2 2) f(y) = y(2 - y), g(y) = -y 3) f(x) = 3^x, g(x) = 2x + 1 keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Assume that a small tree will grow to maturity according to

Dh /dt = 2 + 0.5/t^0.5 where t is the time in years (t > 0) and h is the height of the tree in feet. How much does the tree grow between the fourth and ninth years?

### Concave Up Functions

On what interval does the function f(x) = -x^3 + 3x^2 + 24x + 5 concave up? keywords: concave-up

### Finding Absolute Minimum

Determine the absolute minimum of the function f(x) = x^3 - 3x - 1 on the interval [0, 4]. Make sure to show all work that is involved.

### Number Lines Interval Forms

________________(____)________ -5 -4 -3 -2 -1 0 1 2 3 4 5 How would I write this in equality form and/or in interval form? x + 2 =+2x is what I came up with. The graph indicates that the solution is between 0 and 2, where 2 is included in the solution and 0 is not.

### Ellipse: Distance Between the Origin and Vertex

Given an ellipse, could you please explain AND show on a piece of paper exactly why the distance between the origin and vertex equals a? Please don't just show me how to derive the equation for an ellipse. I have that. And please don't just draw a picture without explaining it. Thanks!

### Functional Analysis: Whether a Rule is Metric

Show that a rule is a metric. See attached file for full problem description.

### Real Analysis - finite subsets compact

Prove that every finite subset of a metric space is compact.

### Graph Functions of Inverse

1. If f(x) = -3x + 7, find f(a + 1). 2. Explain how we can obtain the graph of -|x + 4| + 2 from the graph of |x|. 3. If f(x) = 3x - 4 and g(x) = x - 1, find (f - g)(x) and f(g(x)). 4. Find the inverse of the function f(x) = 2x + 8.

### amplitude of resultant wave after superimposition

Two vibrations, x1 = 3 sin(10t + pi/6) and x2 = 2 cos(10t - pi/6) where t is in seconds, are superimposed. determine the time at which the amplitude of the resultant vibration, x1 + x2, first reaches a value of 2.