Write a MATLAB code for discrete least squares trigonometric polynomial S3(x), using m = 4 for f(x) = e^x * cos(2x) on the interval [-pi, pi]. Compute the error E(S3).

1. Consider the following second-order differential equation. d^2x/dt^2 + 3 dx/dt + 2x − x^2 = 0 (a) Convert the equation into a first-order system in terms of x and v, where v = dx/dt. (b) Find all of the equilibrium points of the first-order system. (c) Make an accurate sketch of the direction field of the first-order system. (d) Make an accurate sketch of the phase portrait of the first-order system. (e) Briefly describe the behavior of the first-order system

Cadbury is preparing a special edition of its famous egg-fondant. The new design of the chocolate shell of the small egg is created from the following two paraboloids: z = 2−2x²−2y² and z = x² + y²−1. The x, y and z coordinates are given in centimeters. Answer the following questions using triple integrals and show steps.

a) Determine the volume of the egg using the rectangular coordinates

b) Knowing that the density of the fondant inside the egg (in g / cm3) at a point (x, y, z) is 3 times the distance from this point to the Oz axis, determine the mass of the egg using the cylindrical coordinates. (We neglect here the mass of the chocolate shell.)

c) You share this egg with your teacher by cutting it with the horizontal plane z = c. Determine the value of the constant c to separate your egg into two pieces of equal volumes.Use rectangular coordinates

d) Determine the mass of each of the two pieces found in (c) using the cylindrical coordinates. Which piece should you keep to have the one with the most fondant?

*** fondant is the chocolate white milk inside the egg ****

Please explain in the most simple way possible what Bipolar coordinates are and what each coordinate represent in them compared to cartesian coordinates. A reference book would be great too.

1. ¬B∨(G↔J), H→(B&C) ∴(H&J)→G

2. A∨B, C↔¬(B∨D) ∴C→A

3. (A&B) ↔ (F→G), (A&F) & B∴(G→R)→R

4. T→¬B, T→¬D ∴ T→¬(B∨D)

5. ¬(M∨¬S), S→(R→M) ∴A → (¬R∨T)

6. (F&G) → I, (I∨J) → K ∴F→(G→K)

7. ¬U, O→G, ¬(O∨G) →U ∴G

Prove that the arguments are valid by constructing a dedication using the rules MP, MT, DN, Conj, Simp, CS, Disj, DS, DM, CP, HS, BE, and DL. Use CP if needed.

3. True or false? All vectors below are in Rⁿ for some n. If your answer is “True”, explain why. If your answer is “False”, give a counterexample. Please specify for all if the given statement is true or false, thank you!

(1) If two vectors are linearly independent, then they are orthogonal.

(2) If x is orthogonal to both u and v, then x must be orthogonal to u − v.

(3) If W is a subspace of Rⁿ, then W and W^⊥(W transpose) have no vectors in common.

(4) If z is orthogonal to v₁ and v₂ and if W =span(v₁,v₂), the z must be in W^⊥​​​​​​​(W transpose) .

(5) ∥cv∥ = c∥v∥ for all scalars c and vectors v in Rⁿ .

1. For each of the following statements indicate if it is true or false. If the answer is
false, briefly explain why.

(a) (2 points) Let V be a vector space and consider the subspace W = Span{v1, v2, v3, v4}.
If v1 = 2v2 + v3, then {v2, v3, v4} is a basis for W.

(b) (2 points) If A and B are invertible n × n matrices, then A is row equivalent to B.

(c) (2 points) Let P3 be the vector space of all polynomials of degree less than or equal
to 3. If {p1, p2, p3} are linearly independent in P3, then they are a basis of P3.

(d) (2 points) If A is an invertible n × n matrix, then det(A3) > 0.

Find the number of r-permutations of the multiset {∞?1, ∞?2, … , ∞??} such that in
every such permutation each type of an element of the multiset appears at least
once. (You do not need to provide a short answer. Assume r ≥ n.)

(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear transformation such that • (0, 0, 1, 0) and (0, 0, 0, 1) lie in the kernel of T, and • all vectors of the form (x1, x2, 0, 0) are reflected about the line 2x1 − x2 = 0.

(a) Compute all the eigenvalues of A and a basis of each eigenspace.

(b) Is A invertible? Explain.

(c) Is A diagonalizable? If yes, write down its diagonalization (you can leave it as a product of matrices). If no, why not?

Linear Algebra: Explain what a vector space is and offer an example that contains at least five (5) of the ten (10) axioms for vector spaces.

a. Consider a non-equilateral triangle. Try to create a tessellation around a point as you did before. Be sure you have no gaps or overlaps. Do you think any triangle with tessellate? Why or why not? Defend your reasoning.

Decide, with justification, on the truth of the following propositions, both when the Universe of discourse is the set of all positive integers, and when the Universe of discourse is the set of all real numbers.

1.18. ∃x∀y,x≤y.
1.19. ∀y∃x,x≤y.
1.20. ∃x∀y,x<y.
1.21. ∀y∃x,x<y.
1.22. ∃x ∀y, y ≤ x.
1.23. ∀y ∃x, y ≤ x.
1.24. ∃x ∀y, y < x.
1.25. ∀y ∃x, y < x.
1.26. ∃x∀y,(x < y ⇒ x2 < y2). 1.27. ∀y∃x,(x<y⇒x2 <y2).