Prove and apply the fundamental theorem of calculus in finding the value of specific Riemann integrals of functions. This is for a class in real analysis. Right now, I just need a basic understanding. Thank you.

Let X = R and A = {disjoint union of the intervals of the form (a, b], (−∞, b] and (a, + ∞)}. Prove that A is an algebra but not a σ-algebra.

1. Prove``The left and right cosets partition G into equal sized chunks." (Cor 5.11 and 5.13 in your book). You have to show the ~ is an equivalence relation, you can't just cite a theorem from the book. Similarly so you have to show phi is 1-1 and onto, you can't just cite a theorem from the book.

(Corollary 5.11. If G is a group and H ≤ G, then the left (respectively, right) cosets of H form a partition of G. Next, we argue that all of the cosets have the same size)

(Corollary 5.13. Let G be a group and let H ≤ G. Then all of the left and right cosets of H are the same size as H. In other words #(aH) = |H| = #(Ha) for all a ∈ G. † The next theorem provides a useful characterization of cosets. Each part can either be proved directly or by appealing to previous results in this section.)

2. Use the above theorem to prove Lagrange's theorem. (Don't use a proof you read online or in the book, your goal is to prove it using what you know about cosets).

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii) maps the x2-axis to itself, and (iii) maps no other line through the origin to itself.

For example, the negating function (n: R2→R2 defined by n(x) =−x) satisfies (i) and (ii), but not (iii).

(b) The function that maps (x1, x2) to the perimeter of a rectangle with side lengths x1 and x2 is not a linear function. Why?

For part (b) I can't come up with any counterexamples that show T(x+y) = T(x) + T(y) or that aT(x) = T(ax) isn't true, and when I tried to use a variables instead of numbers, I ended up showing that it did satisfy both conditions. I'm not sure what I'm missing.

Determine which of the following pairs of functions y1 and y2 form a fundamental set of solutions to the differential equation: x^2*y'' - 4xy' + 6y = 0 on the interval (0, ∞). Mark all correct solutions:

a) y1 = x and y2 = x^2

b) y1 = x^2 and y2 = 4x^2

c) y1 = x^2 and y2 = x^3

d) y1 = 4x^2 and y2 = x^3

e) y1 = [(x^2)+(x^3)] and y2= x^3

Solve Laplace's equation inside a rectangle 0 ≤ x ≤ L, 0 ≤ y ≤ H, with the following boundary conditions [Hint: Separate variables. If there are two homogeneous boundary conditions in y, let u(x,y) = h(x)∅(y), and if there are two homogeneous boundary conditions in x, let u(x,y) = ∅(x)h(y).]:

∂u/∂x(0,y) = 0

∂u/∂x(L,y) = 0

u(x,0) = 0

u(x,H) = f(x)

study of the effect of college student employment on academic performance, the following summary statistics for GPA were reported for a sample of students who worked and for a sample of students who did not work. The samples were selected at random from working and nonworking students at a university. (Use a statistical computer package to calculate the P-value. Use μ_employed − μ_{not employed}. Round your test statistic to two decimal places, your df down to the nearest whole number, and your P-value to three decimal places.)

t= df= P=
Does this information support the hypothesis that for students at this university, those who are not employed have a higher mean GPA than those who are employed? Use a significance level of 0.05.

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

Determine the eigenvalues and eigenfunctions of the following operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your final answer

1-f(x) =1/8(7x-2), x ≤ 3

a-absolute maximum value b-absolute minimum value c-local maximum value(s) d-local minimum value(s)

2-Show that the equation x³ − 16x + c = 0 has at most one root in the interval [−2, 2].

3-If f(1) = 10 and f '(x) ≥ 3 for 1 ≤ x ≤ 4, how small can f(4) possibly be?

Write your own R-function for the linear regression using qr() function. You must not use solve() function. The input of your function would be a design matrix X and a response vector y and the output must contain

- least sqaure estimates and the corresponding stnadard errors.

- residuals and MSE

- fitted values

- ANOVA table

Solve IVP for y(x): dy/dx + (3/x)y = 8y^4, y(1) = 1

Find the values of λ (eigenvalues) for which the given problem has a nontrivial solution. Also determine the corresponding nontrivial solutions​ (eigenfunctions).

y''+2λy=0; 0<x<π, y(0)=0, y'(π)=0