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Functional Analysis

4 Questions

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.

Length of Line

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


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

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


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.

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.

Functional Analysis

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

Please explain in detail the following:

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 -

Please explain in detail

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

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

Mean value theorem

Find the points guaranteed by the mean value theorom 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

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

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

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.

Polar coordinates

X and y represents rectangular coordinates. What is the given equation using polar coordinates (r, theta). x^2 = 4y

Polar coordinates

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