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Real Analysis

Integrals and Convergence

Determine whether the integral dx / x^2 which has a an upper limit of 3 and lower limit of -2 converges or diverges. Evaluate the integral if it converges.

Quality Control : Control Limits and p-Charts

Hunter Nut Company produces cans of mixed nuts, advertised as containing no more than 20% peanuts. Hunter Nut Company wants to establish control limits for their process to ensure meeting this requirement. They have taken 30 samples of 144 cans of nuts from their production process at periodic intervals, inspected each can, and

Real Analysis Integer Proofs

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

Real Analysis : Real Numbers

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. If you have any suggestion or question to me, please let me know. Thank you. Please see attached file for full problem description. I. Three real numbers and have the property that . Prove that at least

Interchanging Limits

Suppose that f_k -> f uniformly on (0,1). Let 0 < x < 1. Suppose that lim f_k(t) = A_k for k=1,2,... Show that {A_k} converges and lim f(t) = LIM A_k. That is show lim LIM f_k(t) = LIM lim f_k(t). Where lim represents the limit as t approaches x and LIM represents the limit as k approaches infinity.

Piecewise Functions, Derivatives and Limits

1. A piecewise function is given. Use the function to find the indicated limits, or state that a limit does not exist. (a) lim is over x gd - f(x), (b) lim is over x gd + f(x), and (c) lim is over xgd f(x) f(x) = { x^2 - 5 if x < 0 } { -2 if x >= 0 } : d = -3 (a) -5 (b) -2 (c) does not exist

Derivatives, Integrals, Limits and Convergence

Please see the attached file for the fully formatted problems. Question 1 Differentiate the function f(x) = (a) xlnx - x (b) x5lnx (c) (lnx)2 (d) 1-x ________________________________________lnx Question 2 Figure 2.1 ?(x) = ln ^/¯ (9-x2) ________________________________________(4+x2)

There are twelve problems involving functions, continuity, finding slope using predictor formula, tangent line to a curve, trajectory of a projectile, finding limits, finding limits using squeeze law and continuity of functions.

Please refer to the attached file to view the complete questions. ======================================== Question 1 Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2 . Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. Then write the equation of the line tangent to the graph of f at

Real Analysis : Differentiable Functions

Does there exist a differentiable function f: R-->R such that f'(0) < 0 for all x &#8804; 0 and f'(x) > 0 for all x > 0? Give an example of such function or prove that it does not exist.

Series Convergence and Divergence

Please see attached file for full problem description. 1) Consider the series where . Show that and for . 2) Use the result of the previous problem to find . 3) The series converges. Find its sum. 4) Determine whether the series converges or diverges. Fully justify your answer. 5) Determine wheth

Trigonometric Limits

Use one-sided limits to find the limit or determine that the limit does not exist. 16-x^2 /4-x lim x => 4 Find the trigonometric limit: sin3x/2x limx => 0 Please show work.

Antiderivatives

? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of Please see the attached file for the fully formatted problems.

Finding Limits

Please see the attached file for the fully formatted problems.

Real Analysis : Limits Proof

Prove sqrt(x+1) - sqrt(x) goes to 0 as x --> infinity Please see the attached file for the fully formatted problems.

Dominating Random Variables : Measure Theory and Dominated Convergence Theorem

Please see the attached file for the fully formatted problems. I have provided a solution to the attached problem. I do not understand or like the solution - I was hoping you could provide an alternate solution or expand upon the solution I have provided in more detail. Exercise (moment-generating function). ? Let X be

Limits

Problem #30, 32, 34, 38 and 40 in both files.

Finding Limits from a Table of Values

Estimate the value of the limit by filling out a table and then making an educated guess about the tendency of the numbers. Then graph the lim of tanx///x as x approaches 0 to verify own conclusion lim of tanx///x as x approaches o table should be 2 colums with the headings |x|tanx/x|

Limits

The function has a limit as f(x) = (1/x) + 3 has a limit of L=3 as x approaches x. This means that if x is sufficiently large (that is if x > N for some number N), the values of f(x) are closer to L=3 than a number epsilon > 0. a) Sketch the graph y=(1/x) +3 and a horizontal strip of points (x,y) such that (if y