Which one of the improper integrals below converges or diverges?
[int _a ^b] means integral from a to b, we use inf to indicate infinity.

a) [int _0 ^1] 1/x dx
b) [int _0 ^1] 1/x^(1/2) dx
c) [int _0 ^1] 1/x^2 dx

d) [int _1 ^inf] 1/x dx
e) [int _1 ^inf] 1/x^(1/2) dx
f) [int _1 ^inf] 1/x^2 dx

g) [int _1 ^inf] lnx / x^2 dx
h) [int _1 ^inf] lnx / x dx

i) [int _(-inf) ^inf] 1/(1+x^2) dx
j) [int _1 ^inf] exp(-x^2) dx
k) [int _1 ^inf] (sinx / x )^2 dx
l) [int _2 ^inf] 1 / sqrt(x^2 - 1) dx
m) [int _0 ^Pi/2] tanx dx

Complete the proof for the claim that any open ball B(x₀,r) in Euclidean space Rⁿ is homeomorphic to Rⁿ.

proof is given below the theorem. Show that suggested map g is in fact homeomorphism.

Theorem: Let X₀, X₁, and X₂ be topological spaces and let f: X₀ -> X₁ and g : X₁ -> X₂ be continuous functions. Then g∘f : X₀ -> X₂ is continuous.

proof : Suppose that V is open in X₂. Since g is continuous, g^-1(V) is open in X₁. Since f is continuous, f^-1(g^-1(V)) = (g∘f)^-1(V) is open in X₀. It follows that g∘f is continuous.

Prove: There are infinitely many primes congruent to 3 modulo 8. Hint: Consider N = (p₁p₂···p_r)² + 2.

A small business owner contributes $2,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly. (a) How long will it be until the account is worth at least $150,000? (Round your answer UP to the nearest quarter.) 43 quarters (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $4,000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.) $

Use Gauss’s Lemma to find the Legendre symbol values (8/11), (5/19), and (6/31).

Before we begin graphing systems of equations, a good starting point is to review our knowledge of 2-D graphs. These graphs are known as 2-D because they have two axes. Find an online image of a graph to use as the foundation of your discussion. (This is easily accomplished by searching within Google Images.) Using your graph as the example:

1.) Select any two points on the graph and apply the slope formula, interpreting the result as a rate of change (units of measurement required).

2.) Use rate of change (slope) to explain why your graph is linear (constant slope) or not linear (changing slopes).

Embed the graph into the post by copying and pasting into the discussion. You must cite the source of the image. Also be sure to show the computations used to determine slope.

prove or disprove using logical equivalences

(a) p ∧ (q → r) ⇐⇒ (p → q) → r

(b) x ∧ (¬y ↔ z) ⇐⇒ ((x → y) ∨ ¬z) → (x ∧ ¬(y → z))

(c) (x ∨ y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ⇐⇒ ¬y → (x ↔ z)

Find Taylor series expansion of log(1+z) and show radius of convergence

a) Let S ⊂ R, assuming that f : S → R is a continuous function, if the
image set {f(x); x ∈ S} is unbounded prove that S is unbounded.

b) Let f : [0, 100] → R be a continuous function such that f(0) = f(2),
f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to zero in at most
two points of the interval [0, 100].

Prove that (f(50) − f(49))(f(25) − f(24)) > 0.

Consider a Math class with 15 female students and 14 male students.

a) How many different 5 people committees with exactly 3 females and 2 males are possible? Justify your answer

b) How many different 5 people committees with representation of both genders are there? Justify your answer

c) Suppose that two of the students refuse to work together. How many different 5 people committees are possible? Justify your answer

d) How many different ways to arrange them in a row with no two males together? Justify your answer

e) Show that there are at least 3 students with the same gender whose were born on the same day of the week.

Find a characterization of primes p such that 3 is a square mod p and then prove it.

incorrect,7.3.13 Twelve different video games showing substance use were observed and the duration of times of game play​ (in seconds) are listed below. The design of the study justifies the assumption that the sample can be treated as a simple random sample. Use the sample data to construct an 80​% confidence interval estimate of sigma​, the standard deviation of the duration times of game play. Assume that this sample was obtained from a population with a normal distribution. 4 .607 3 . 970 3 . 926 4 . 652 3 . 915    4 . 400 4 . 065 4 .264 4 . 237 4 . 102 4 . 687 3 . 804 LOADING... Click the icon to view the table of​ Chi-Square critical values. The confidence interval estimate is nothing secless thansigmaless than nothing sec. ​(Round to one decimal place as​ needed.)

What is the benefit of using Fourier transforms in complex formulation.

1. Implement the Explicit Euler Scheme for Initial Value Problems of the form:
        y'(t) = F(t, y(t)) ,  t0 ≤ t ≤ tend
        y(t0) = y0

   The function F(t,y) should be coded in a function subprogram FCN(...).
   Input data: t0, y0, tend, Nsteps.  Thus the time-step will be h=(tend-t0)/Nsteps.
   Your code should print out the input data and then the pairs:
                tn          Yn
   At the end, it should print out the final n, tn, Yn
   (appropriately labelled, of cource).

2. Solve, on paper, the simple (integration) problem:
                y' = 2t ,   0 ≤ t ≤ 1
                y(0) = −1

3. To debug your code, run it on the problem above. 
   Compare the numerical solution Yn with the exact solution yEXACT(tn),
   i.e. modify your output to print out:
        tn      Yn      yEXACTn         ERRn
   where ERRn = |Yn - yEXACT(tn)|, and keep track of the maximum error.
   At the end of the run, print out the above values (at time tend) 
   and the maximum overall error ERRmax.  Test with N=10 and N=100.
   Turn off printing of tn  Yn ... and test with N = 1000, 10000 and larger.

   Once the code is debugged, only FCN(...) and input data need to be changed 
   to solve other IVPs.

4. Now solve the IVP:
        y' = −t/y ,   0 ≤ t ≤ 1
        y(0) = 1

   Find the exact solution at t = 1 (by hand),
   and compare the numerical and exact values at t = 1.  
   Try small (N=10) and larger (N=100, 1000, 10000, ... ) number of time-steps.
   At which time does the worst error occur in this problem ?
   Plot the exact solution.  Do you see why it occurs there?