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# Functional Analysis

### Equation for parabola

Vertex(-2,1) and focus (-2,-3)

### Parallel Lines and Equal Length

Please see attached file. If lines A, B and C are parallel and length a=2.8, c=3, b=3.2, what is length d? a. 1.88 b. 3.43 c. 3 d. 3.2.

### Finding Polar Coordinates for the Intersection of Curves

See attached file for the problem.

### Continuity and differentiability

G(x) =ksqrt(x+1) if 0<=x<=3 mx+2 if 3<x<=5 Find the values of k and m so that g'(3) exists.

### distance from fire to observing tower A

From fire tower A, a fire with bearing N 75° E is sighted. The same fire is sighted from tower B at N 49° W. Tower B is 55 miles east of tower A. How far is it from tower A to the fire?

### Convergence Tests Investigated

Using one of the tests for convergence (comparison, limit, integral, nth term, etc.), show whether the following series converges or diverges: ∞ ∑ (e^n)/ 1 + (e^2n) n=1

### Cylindrical and spherical coordinates

Please give a detailed solution to the attached problem.

### Converges Alternating Series Estimated

Show that the alternating series converges and use the partial sum S9 o estimate the error made as an approximation to S of the series.

### Space with a Universal Covering Space

Let X be a space which has a universal covering space. If (X1, p1) is a covering space of X and (X2, p2) is a covering space of X1, then (X2, p1p2) is a covering space of X. See the attached file.

### Finding Conics Given Conic Sections (Ellipses, Hyperbolas and Parabolas) and Polar Coordinates

Please see the attached file for the fully formatted problems.

### Conic Sections (24 Problems)

Please see the attached file for the fully formatted problems. Please do 1-47 odd.

### Coic Sections, Areas and Lengths in Polar Coordinates (20 problems)

Please do problems 1-41 odd. Please see the attached file for the fully formatted problems.

### Functional Analysis

Attachment file. Let X be a normed space and . Show that if for every bounded linear functional f on X , then .

### Graphically Analysis Trigonometric Functions

Please explain step by step how to complete the following: In the following determine graphically whether the equation could possibly be an identity. If it could, prove that it is. 5. sin^4 t - cos^4 t = 2 sin² t -1 ans. - sin^4t - cos^4 t = (sin²2t - cos²t) (sin²t + cos²t) = (sin²t-(1-sin²t)) (1) = 2 sin²t -

### Volume of a solid of revolution about the Y axis

Find the volume of the solid generated by revolving the region described about the Y axis: x=e^y intersecting the x axis at (1,0) and between the points on the y axis (0,0) and (0,ln3) Using the formula V=&#8747;&#960;[R(y)]²dy

### Rectangular Storage Unit: Effect of Changing Dimensions Upon Volume and Number of Solutions

A rectangular storage unit has dimensions 1 m by 2 m by 3 m. If each linear dimension is increased by the same amount a) what increase would create a new storage unit with a volume 10 times the original b) how many solutions are there to this problem? please explain why?

### Graph Parabolas Points

24. Graph the following parabola clearly labeling exact points for the vertx, x-intercepts(s), and y-intercept(s): y=2xsquare+4x-3

### Gradient : Find grad(f). Let f(x,y,z) = e^sinx +xy^2 -(2z + 1)^2 lnx

Find the gradient of f ( grad(f) ). Let f(x,y,z) = e^sinx +xy^2 -(2z + 1)^2 lnx

### Uniform Convergence

Please see the attached file for the fully formatted problems. We are using the book Methods of Real Analysis by Richard R. Goldberg.

### A Discussion On Rolle's Theorem

Please solve the following with as much explanation of each step as possible. Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's

### How to Use Mean Value Theorem to Find Guaranteed Points

Find the points guaranteed by the mean value theorem for F(x) = x^.5 - 2x on the closed interval [ 0 , 4 ].

### Find number of degrees...

Use the equation s=rO for the following: 1. Find the number of degrees in the central angle of a circle with radius of 20 inches if the angle subtends an arc of 13 inches. 2. Find the length of a degree on the equator considering the diameter of the equator to be 7912 miles. 3. A bicycle has a 28 inch wheel. How man

### Outer Measure Functions

Let A = union ( i from 1 to infinity) of M_i, Mi's are disjoint, show that m*(A) = sum (i from 1 to infinity) of |M_i| m*(A) is the outer measure of A, that is, m*(A) = inf sum (i from 1 to infinity) of M_i. PLEASE NOTICE THE = SIGN, A = the union, not a subset of the union.

### Intermediate value theorem

Determine if the following conforms to the intermediate value theorem. Clearly give the conditions and, if appropriate, find an appropriate value for c. f(x)=10/((x^2)+1) [0,1] k=8

### Weak Convergence

Please explain why the following sequence for otherwise is an example of a sequence in such that weakly, but not strongly. Please see the attached file for the fully formatted problem.

### Converting Polar Coordinates to Rectangular Coordinates

Find the rectangular coordinates of the point (r,degrees)= (-5,-55 degrees).

### Volumes of revolution

Find the volume of the object rotated by the given points. 1. y=1/x, y = 0, x = 1, x = 3; about y = -1 2. y = x, y = 0, x = 2, x = 4; about x = 1

### What is the fifth term in the following sequence?

What is the fifth term in the following sequence? asubcript n =n+asubscript n-1. if a1 equals -2, for n greater than or equal to 2.

### Poles & Singularities

Classify the behavior at infinity (analytic, pole, zero, or essential singularity; if a zero or pole, give its order) of the following functions: f1(z) = (z^3 + i)/z f2(z) = e^(tan(1/z))

### Find a horizontal line that divides an area into two equal parts.

Find a horizontal line that divides an area between y=x^2 and y=9 into two equal parts. I need help in approaching this problem. I know how to calculate the area. I divided the area by two and through trial and error I found the horizontal line to be close to y=5.6. I don't know how to get an exact answer.