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# Functional Analysis

### Polar Coordinates of a Point

The polar coordinates of a point (-1, -pi/3), find the rectangular coordinates for that point.

### Equation of Parabola in Standard Forms

Write the following parabola in standard form: x - y + y^2 = 0

### CalcII

The resale value os a certain idustrial machine decreases over a 10 year period at a rate that changes with time, when the machine is x years old, its value is decreasing at a rate of 220(x-10) dollars/year. By how much does the machine depreciate during the second year? Please see attached for full question.

### Convergence or Divergence; Absolute or Conditional

Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find a sum, then do so (see attachment for equations, please - thanks!)

### Area, volume, and expantion

A.) let A be the area of a circle of radius r that is changing w/ respect to time. if dr/dt is a constant, is dA/dt a constant, explain. b.) let V be the volume of a sphere of radius r that is changing w/ respect to time. If dr/dt is constant, is dV/dt constant, explain. c.) All edges of a cube are expanding at a rate of

### Single Residue : Interior to Closed Contour

5. Let the degress of the polynomials {see attachment} be such that m [less than or equal to] n+2. Use the theorem in Sec. 64 {see attachment} to show that if all of the zeros of Q(z) are interior to a simple closed contour C, then {see attachment} Please specify the terms that you use if necessary and clearly explain each

### Cross Section, Bounded Region, and Solid of Revolution

See the attached file. 6. Sketch the region and then find the volume of the solid whose base is the given region and which has the property that each cross section to the x-axis is an equilateral triangle a) the region bounded by the curves {see attachment} 7. Same problem statement as #6 above, except cross-section to the

### Functional Analysis : Continuity, Graphing

Problem: Let f be that function defined by setting (Please see the attached file for the fully formatted problem.) a. Describe graphically f(x). b. At what points is f continuous?

### Bounded Numbers

Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all numbers -x where x E A. Prove that inf A = -sup(-A) (Please see the attached file for the fully formatted problem.) Included in the attachment is a copy of the solution, but please explain in your own words how the proof works; don't j

### Find

Given f (x)= -5/x-2 and g (x)= 3/x+3, find (f+g)(x)

### Polar coordinates

Polar coordinates o a particular point are r=4, 0=pi/3. I need to ind the rectangular coordinates of the point. (x,y)=?

### Converting polar coordinaes to retangular coordinates

Convert the coordinates polar r=15, 0=2.75 (x,y)=?

### Convert a rectangular equation to a polar equation

Convert a rectangular equation y= - 5x + 4 to a polar equation.

### Straight lines and planes...

Describe the 3 possible relationships for 2 straight lines in the same plane.

### Mapping (Quarter-Plane; Half-Line)

Find the image of the quarter-plane {see attachment} under the mapping {see attachment}. Show graphs (shaded regions) in the w-plane and identify the images of the half-lines {see attachment}.

### Continuous and differentiability

Please see the attached file for full problem description. - Prove that f is continuous at 0 - Determine whether f is differentiable at 0, give a careful proof

### Sketch the Region Section

Sketch the region consisting of all points (x, y) such that x2 + y2. You need to include all steps in calculating the resulting region. (See attachment for full question)

### Consider the Shaded Domain D

Problem 14 ONLY 14) Consider the shaded domain D in the attached figure bounded by the simple closed... (see attachment).

### Function Analysis Given

See Attached for equation

### Finding Extrema: Lagrange Multipliers

Use the method of Lagrange multipliers to find the indicated extremum. Let f(x,y) = 8x^2 - 24xy + y^2. Find the maximum and minimum values of the function f(x,y) subject to the constraint 8x^2 + y^2 = 1.

### Minimizing F(x)

Let Q=(0,7) and R=(10,11) be given points in the plane. We want to find the point P=(x,0) on the x-axis such that the sum of distances PQ+PR is as small as possible. To solve this problem, we need to minimize the following function of x. F(x)=?? over the closed interval [a,b] where a=?? and b=??

### Min/Max problem

Find the length of the shortest line from the origin to the line y=1-6x.

### Closed Form Maclaurin Series

See attachment Find the Maclaurin series in closed form of a. b.

### Sec 17.9 #30

#30 Please see the attached file for full problem description.

### Monotonicity Theorem : Concave Up and Concave Down

Use the Monotonocity Theorem to determine where the given function is concave up and where it is concave down. F(x) = (x - 1) / (x^2)

### Volume of a Solid of Revolution Region

Find the volume of the solid formed by revolving the region bounded by the graphs of y=2(x^2), x=0 and y=2 about the y-axis.

### Convergence Series Problem

Please tell me if the following (see attached) series converge. This is a problem from my text book and unfortunately there are no solutions for this problem. I need justification, not just a yes it converges or no it doesn't answer. Determine whether or not the following converge:

### Setting Up Triangle Problems

In the problem we are asked to solve for x and y. I can solve for the triangles but not able to use them to find the answer. can someone guide me.

### Finding Distance and Midpoint

Find the distance between(-2,3) and the midpoint of the segment joining (-2,-2) and (4,3).

### Theta Diagram Functions

I would like some assistance in setting up this problem to solve. In the diagram Theta is 50 degrees and x is the unknown. Thank You