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Real Analysis

Real numbers analysis

A. let f(x):= 1/(x^2), x does not equal zero, x elements of Reals determine the direct image f(E) where E= {x elements Reals: 1<=x<=2} determine the inverse image f^-1(G) where G= { x element of Reals: 1<=x<=4} b. show that if a<b then a< 1/2(a+b)<b c. for a,b,c elements of reals with a<b, find an explicit bijection of

Mathematics of Flight Time

Two planes leave O'Hare Airport in Chicago at the same time.One flies directly east at 540 miles per hour.The other flies directly west at 720 miles per hour.How long will it take them to be 1890 miles apart.

Power series solution

Compute the power series solution for the ODE y''+x^2y=0 and determine the radii of convergence for the two series solutions.

Effect of Changing a Company's Credit Terms

The Dunstill company has obtained the following information: ? Annual Credit Sales= 30 million ? Collections period is= 60 days ? Credit terms= Net 30 days ? Interest rate = 12% The company is considering changing its credit terms to 4/10 net 30. The company anticipates 50% of the customers will take advantage of this poli

Cluster point proof

Prove that a set S included R has no cluster points if and only if S intersection [-n,n] is a finite set for each n in N.

Convergent Sequences and Limits

Suppose {an} is a convergent sequence and {bn} is a sequence with the property that bn < an if n is even and bn > an if n is odd. Does bn converge? Determine the limit of the given sequence (2 problems). (3n^3+2n+1)/(5n^2-2*n+3) 1/[1+(n^3 / 3^n)] Please see the attached file for the fully formatted problems.

Statistics - 2.1 Calculate the most appropriate measures of central tendency for each variable listed. Show all work. Explain your choices in each case. Discuss what each result means in real terms. ... [See the attached questions file.]

2.1 Calculate the most appropriate measures of central tendency for each variable listed. Show all work. Explain your choices in each case. Discuss what each result means in real terms. A. GNDR B. TRAK C. ADV D. GPA 2.2 Calculate the most appropriate measures of dispersion for each variable listed. Show all work. Explai

Series and Binomial Theorem

See the attached. Q2 a) If the x3, the 3rd term of a series is 3, and x17, the 17th term is -27, state the series. Find if the series has a common difference. b) Coolant hose and nozzle design that to be used on CNC machines, has to ensure that adequate coolant is delivered to the cutter/workpiece interface

Limits and sequences

See attachment a. Use the definition of limits to show that b. Compute the following limits (justify your answers). 1. ( ) 2. c. Let ( be a sequence. Show that ( is convergent if and only if the sequence ( and ( are convergent to the same limit.

Definition of Maclaurin Series

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. (Assume that f has a power series expansion. Do not show that Rn(x) --> 0) Also find the associated radius of convergence. f(x)= ln(1+x) Please show steps.

Differentiability and Limits

Decide whether each of the following statements is true or false. If true, explain why. If false, give a counter-example and explain why the counter-example contradicts the statement. Suppose F(x) is differentiable at ALL x in R. Suppose lim x->0 f '(x) = L, does it follow that f '(0) = L?

Singularities, Convergence of Taylor Series, Residues and Contour Integral

Consider the complex function: f(z) = 1/(z^2 + z + 1)(z + 5i) a) Find and classify the singularities of f(z); b) Without finding the series explicitly, determine the region of uniform convergence of the Taylor series taken about the origin; c) Find the residues of f(z) at each of the singular points; d) Find the val

Integrals and Convergence Evaluated

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

The Sum of Converging Series

Consider the following series. SUM (n =0, infinity) of (x + 7)^n / 4^n a) Find the values of x for which the series converges. (Enter the smaller number first.) b) Find the sum of the series for those values of x.

Sequence Convergence and Limit Proof

Suppose {p_n} converges to p. Prove that there is at most one alpha for which the limit as n goes to infinity of |p_n+1-p|/(|p_n-p|^alpha) is a positive finite number. (See attachment for mathematical notation)

Multiplicity of a Root and Taylor's Theorem

Let f(x) be defined as f(x) = tanx/x if x =/ 0 = 1 if x = 0 and let g(x) = f(x) - 1. Then g(x) is continuous at x = 0 and, in fact, g(x) has derivatives of all orders at x = 0. Determine the multiplicity of the root g(x) has at x = 0. Hint: Apply Taylor's Theorem.

Real Analysis Proofs : Set Operations

Provide counterexamples for each of the following.... From Set Theory, Set Operation. College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.