Given the parametrized curve r(u) = a cos u(1 − cos u)ˆi + a sin u(1 − cos u)ˆj, u ∈ [0, 2π [ , (with a being a constant)

i) Sketch the curve (e.g. by constructing a table of values or some other method)

ii) Find the tangent vector r 0 (u). What is the tangent vector at u = 0? And at u = 2π? Explain your result.

iii) Is the curve regularly parametrized? Motivate your answer using the definition.

iv) Compute the length of the arc corresponding to the interval [0, π/2].

A person borrows money at i^(12) = .12 from Bank A, requiring level payments starting one month later and continuing for a total of 15 years (180 payments). She is allowed to repay the entire balance outstanding at any time provided she also pays a penalty of k% of the outstanding balance at the time of repayment. At the end of 5 years (just after the 60th payment) the borrower decides to repay the remaining balance, and finances the repayment plus the penalty with a loan at i^(12) = .09 from Bank B. The loan from Bank B requires 10 years of level monthly payments beginning one month later. Find the largest value of k that makes her decision to refinance correct.

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

The linear transformation is such that for any v in R², T(v) = Av.

a) Use this relation to find the image of the vectors v₁ = [-3,2]^T and v₂ = [2,3]^T. For the following transformations take k = 0.5 first then k = 3,

T₁(x,y) = (kx,y)

T₂(x,y) = (x,ky)

T₃(x,y) = (x+ky,y)

T₄(x,y) = (x,kx+y)

For T₅ take theta = (pi/4) and then theta = (pi/2)

T₅(x,y) = (cos(theta)x - sin(theta)y, sin(theta)x + cos(theta)y)

b) Plot v₁ and v₂ and their images under the transformations. Write a short description saying what the transformations is doing to the vectors.

Use Newton-Raphson to find the real root to five significant figures 64x^3+6x^2+12-1=0. First graph this equation to estimate. Use the estimate for Newton-Raphson

“A real number is rational if and only if it has a periodic decimal expansion”

Define the present usage of the word periodic and prove the statement

1) Determine whether these statements are true or false. Please explain why so I can understand where the answers came from

a) ∅ ∈ {∅}

b) ∅ ∈ {∅, {∅}}

c) {∅} ∈ {∅}

d) {∅} ∈ {{∅}}

e) {∅} ⊂ {∅, {∅}}

f ) {{∅}} ⊂ {∅, {∅}}

g) {{∅}} ⊂ {{∅}, {∅}}

2) Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find

a) A x B x C

d) B x B x B

3) Show that if A and B are sets, then

b) (A ⊕ B) ⊕ B = A. (prove every step! you may use the fact that symmetric difference of sets is associative)

Examine the propagation of roundoff through division.

Determine whether the statements in (a) and (b) are logically equivalent.

1. Bob is both a math and computer science major and Ann is a math major, but Ann is not both a math and computer science major.

2. It is not the case that both Bob and Ann are both math and computer science majors, but it is the case that Ann is a math major and Bob is both a math and computer science major.

In a (2, 5) Shamir secret sharing scheme with modulus 19, two of the shares are (0,11) and (1,9). Another share is (3,k), but the value of k is unreadable. Find the correct value of k.

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

5. Calculate the conditional probability distribution, ?(?|?)P(A|B).

6. Calculate the conditional probability distribution, ?(?|?)P(B|A).

7. Does ?(?|?)=?(?|?)P(A|B)=P(B|A)? What do we call the belief that these are always equal?

8. Does ?(?)=?(?|?)P(A)=P(A|B)? What does that mean about the independence of ? and B?

Using traditional methods it takes 8.2 hours to receive a basic flying license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique on 26 students and observed that they had a mean of 8.0 hours with a variance of 2.89 . Is there evidence at the 0.1 level that the technique reduces the training time? Assume the population distribution is approximately normal. Step 1 of 5: State the null and alternative hypotheses.

Discribe in 300. words systems like RSA; authentication in public-key systems uses digital signature it was invented in 1977 by Ron Rivest, Adi Shamir, and Leonard. please type

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and K = h(2; 1; 1)i of G. (a) Find all cosets (along with the elements they contain) to H and K, respectively. (b) Write down Cayley tables for the factor groups G=H and G=K, and classify them according to the Fundamental Theorem of Finite Abelian Groups.