### 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.

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

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

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

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

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.

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

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.

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

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? asubcript n =n+asubscript n-1. if a1 equals -2, for n greater than or equal to 2.

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 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.

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

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

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

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.

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!)

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

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

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

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?

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

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

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

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

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

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

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}.

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 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)