1. Suppose A is (10, 2, 5, 9, 1, 8, 2, 4). Consider the function:

int BBOX(int n, int k)

            if (n <= 0) return 0;

            else if (A[n] < k) return (1+ 2*BBOX(n-1,k+1));

            else return BBOX(n-1,k-2);

            Find BBOX(8, 5)

Use Newton's method to find a solution for the equation in the given interval. Round your answer to the nearest thousandths. ? 3 ? −? = −? + 4; [2, 3] [5 marks] Answer 2.680

Q6. Use the Taylor Polynomial of degree 4 for ln(1 − 4?)to approximate the value of ln(2). Answer: −4? − 8?2 − 64 3 ? 3 − [6 marks]

Q7. Consider the curve defined by the equation 2(x2 + y2 ) 2 = 25(x2 − y2 ). Find the equation of the line tangent to the curve at the point(3, 1). [5 marks]

1. Find the solution to the given linear systems by Jacobi and Gauss Seidel iteration methods.

          2x + 5y = 16                      20x + y – 2z = 17            5x – y +2z = 12

          3x + y = 11                        3x +20y – z = -18           3x +8y -2z = -25

                                                      2x – 3y +20z = 25            x + y +4z = 6

2. Solve the equation Ax = b by using the LU decomposition method given the following linear systems of equations:

a. 3x – 7y -2z = -7

-3x +5y + z = 5

6x – 4y      = 2

b. 2x – y +2z = 1

-6x     -2z = 0

8x – y+5z = 4

  

Consider the given matrix.

3		0		0
0		2		0
16		0		1

Find the eigenvalues. (Enter your answers as a comma-separated list.)

λ = 1,2,3

Find the eigenvectors. (Enter your answers in order of the corresponding eigenvalues, from smallest eigenvalue to largest.)

Let Q1, Q2, Q3 be constants so that (Q1, Q2) is the critical point of the function f(x, y) = (90)x 2 + (0)xy + (90)y 2 + (−72)x + (96)y + (40), and Q3 = 1 if f has a local minimum at (Q1, Q2), Q3 = 2 if f has a local maximum at (Q1, Q2), Q3 = 3 if f has a saddle point at (Q1, Q2), and Q3 = 4 otherwise. Let Q = ln(3 + |Q1| + 2|Q2| + 3|Q3|). Then T = 5 sin2 (100Q)

satisfies:— (A) 0 ≤ T < 1. — (B) 1 ≤ T < 2. — (C) 2 ≤ T < 3. — (D) 3 ≤ T < 4. — (E) 4 ≤ T ≤ 5

General Directions: Show all necessary supporting work and box in each answer. If a blank is provided, state the answer in the blank.

Use dimensional analysis to convert. If unable to state an exact answer, round to the nearest tenth.

1. 0.35 mi. =                           yd.                                          2. 182 ft =                              in.

3.143 ¾ in (fraction 3 over 4) in. =             cm                                          4. 5,237,833 in = _____ mi

Consider the “metric staircase” to convert to the specified unit. If unable to state an exact answer, round to the nearest tenth.

5. 24 hm =                              dam                                        6. 18,426 mm =                                  m

7. 650,000 mm =                                km                               8. 50,000 cm =                                     km

Theorem 3.4. Let a and b be integers, not both zero, and suppose that b = aq + r

for some integers q and r. Then gcd(b, a) = gcd(a, r).

a) Suppose that for some integer k > d, k | a and k | r. Show that k | b also. Deduce that k is a common divisor of b and a.

b) Explain how part (a) contradicts the assumption that d = gcd(b, a).

Consider the two savings plans below. Compare the balances in each plan after

1111

years. Which person deposited more money in the​ plan? Which of the two investment strategies is​ better?

YolandaYolanda

deposits

​$450450

per month in an account with an APR of

55​%,

while

ZachZach

deposits

$ 5000$5000

at the end of each year in an account with an APR of

5.55.5​%.

The balance in

YolandaYolanda​'s

saving plan after

1111

years was ​$

.

​(Round the final answer to the nearest cent as needed. Round all intermediate values to seven decimal places as​ needed.)

The following kets name vectors in the Euclidean plane:

|a>, |b>, |c>.

Some inner products: <a|a> = 1, <a|b> = −1, <a|c> = 0, <b|c> = 1, <c|c> = 1

(a) Which of the kets are normalized?

(b) Which of these are an orthonormal basis?

(c) Write the other ket as a superposition of the two basis kets. What is the norm |h·|·i| of this ket (i.e., the length of the vector)? What is the angle between this ket and the two basis kets?

(d) In the same basis, write as a superposition a ket that has the same direction but is normalized.

^{The number of new businesses established in the US since 1990 can be modeled by the function Nx=110.8x^3-5305.5x^2+76,701x+332,892 where x = 0 represents 1990 and the domain is [0, 25].}      

1) What was the average rate of change in the number of new businesses established between 2000 and 2010? Don’t forget to label and interpret the answers.  

Construct a linear equation for this model

What is the value of the y-intercept? What does it mean?

What is the value of the slope? What does it mean?

If you use 10 megabytes of data how much would you expect to pay for your bill?

Where else might you use this type of model? Be specific.

What are all the values of k for which the series [(k^3+2)*(e^-k)]^n converges from n=0 to n=infinity?

Suppose that the coefficient matrix of a homogeneous system of equations has a column of zeros. Prove that the system has infinitely many solutions. What are the possibilities for the number of solutions to a linear system of equations? Can you definitively rule out any of these?

( X, τ ) is normal if and only if for each closed subset C of X and each open
set U such that C ⊆ U, there exists an open set V satisfies C ⊆ V ⊆clu( V) ⊆ U

Q1- Out of Trapezoidal rule and Simpson’s 1/3rd 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 errore.