If f(x)=sign(x−2)+|x+2| for (-4,4] and you extend f(x) to a periodic function on the real line, and F(x) is the Fourier series of f(x). Which of the following options are correct? (Select all that apply.)

I. F(1)=2.

II. F(0)=1.

III. f(x) is continuous on the interval.

IV. F(x) is an odd function.

V. F(x) is continuous on the interval.

Pirates are attacking the coastal town of Townberg. The pirates have

a cannon on their ship which can fire a cannonball at 440 meters per

second.

(a) Suppose the pirates fix the cannon at an angle of π/4. Write a function h(t) which gives the height (in meters) at t seconds for the cannonball (assume that the cannonball has initial height 5 meters) and a function d(t) which gives the horizontal distance of the cannonball (in meters) at t seconds.

(b) How far from Townberg does the pirate ship have to be so that

the cannonball can hit the town (assume Townberg has a height

of 0 meters)?

(c) What is the cannonball’s vertical speed as it hits the ground?

What is the cannonball’s horizontal speed as it hits the ground?

What is the cannonballs total speed as it hits the ground?

ICE13B - “How long will the money last?” You begin on 1 January 2015 by making an initial deposit of $5,000 to open a savings account at Bank of Mason. This account pays 0.135% interest per month. Every even numbered month, you withdraw $500 to pay your tuition installment balance. At the end of December each year you receive a $2,000 end-of-year bonus from your employer which you deposit into your account. Run your simulation until your account balance goes negative. Question 1: During what month does you account balance go negative after making the withdrawal? Question 2: Prepare a plot showing your account for all months A Matlab code is needed

(Advanced Calculus and Real Analysis) - Lebesgue outer measure

* Please prove that the Cantor set C has Lebesgue outer measure zero.

1. Explain the importance of error analysis in numerical methods with suitable example.
2. Out of Bisection method and secant method which one is better and why? Solve one application based problem using that method.
3. Use Newton-Raphson Method to find the root of trigonometric function correct up to seven decimal places. (Trigonometric function should be complex)
4. Solve one problem which is based on the application of Interpolation.
5. Using numerical differentiation solve one application based problem. (Use central difference approximation and problem must include first order as well as second order derivatives)
6. Out of Trapezoidal rule and Simpson’s 1/3^rd rule which one is better explain in detail. Also solve one application based problem using that rule. Compare the exact and approximate result to compute the relative error.

(I saw the same question already uploaded in the chegg but i need other numbers )  

If p,p+2 are twin primes, prove 4((p−1)!+1)+p≡0 modp(p+2)

Convert the base-10 number to Two’s Complement. Show your work.

(a) −12 (b) −32 (c) −57 (d) −112 (e) −24 (f) −85

3. Suppose A and B are non-empty sets of real numbers that are both bounded above.

(a) Prove that, if A ⊆ B, then supA ≤ supB.

(b) Prove that supA∪B = max{supA,supB}.

(c) Prove that, if A∩B 6= ∅, then supA∩B ≤ min{supA,supB}. Give an example to show that equality need not hold.

Use the Intermediate value theorem to find an interval of length one that contains a root of the equation. a) x^3=9 b) 3x^3+x^2=x+5 2. Still consider equations 1a and 1b A)How many iterations are required to find the roots by the Bisection Method within six decimal places B) Use the interval you find in problem 1 that contains a root to compute the first 4 iterations and table the results

We will write Δ(m,n) for the set of order preserving functions with source m and target n.

Let f ∈ ∆(k, n), g ∈ ∆(m, n), and h ∈ ∆(n, p). Suppose im f = im g. Prove that im(h ◦ f) = im(h ◦ g).

Consider all positive integers less than 100. Find the number of integers divisible by 3 or 5?

Consider strings formed from the 26 English letters. How many strings are there of length 5?

How many ways are there to arrange the letters `a',`b', `c', `d', and `e' such that `a' is not immediately followed by`e' (no repeats since it is an arrangement)?

3. Let a and b be positive integers. Is gcd(5a + b, 11a + 2b) = gcd(2a + b, 3a + 2b)?

If yes provide a proof. If not, provide a counterexample.

How to solve it by wolfram mathematica:

The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s.
We shall consider fractions like, 30/50 = 3/5, to be trivial examples.
There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
You have to identify these four fractions.

(This is "almost" Problem 33 from Euler project, with only the requirement modified).

Indicate whether each statement is True or False. Briefly justify your answers ( Please answer C and E both )

(c) The set of all 3 × 3 matrices A with zero trace (T r(A) = 0) is a vector space under the usual matrix operations of addition and scalar multiplication. (

e) If x⃗1 and x⃗2 are in the null space of a square matrix A, then any linear combination of x⃗1 and x⃗2 is also in the null space of A.