Using the Riemann Summ Estimate ln(4)

a) n = 3

b) n = 4

c) n = 5

Ex 4.

(a) Prove by induction that ∀n∈N,1³+ 2³+ 3³+···+n³=[(n(n+ 1))/2]²

b) Prove by induction that 2ⁿ>2n for every natural number n≥3.

This is a engineering analysis hw question please show your work I will use it to study

PROBLEM 1 (20PT) Perform the following tasks for the function, f(x), shown below.   

f(x) := x*cos(pie*x/2.4)-3*x²+e^{x/1.5 (use this function to answer the following parts a, b, c . there is no figure given) (the work you give me I will use to plug it into Mathcad which is the program we use to show our work) (if you feel like you are unfamiliar with the Taylor series process please pass this question on to someone who does. thanks)}

(a) Use a Taylor series expansion, keeping up to the third derivative term, to approximate the function about the point xo = 2. Plot the function, f(x), and the approximated Taylor Series function, fTay(x), in a single graph. Display x-axis from -2 to 8. Show a marker at x = xo.   

(b) Re-do the Taylor series expansion, fTay2(x), to show an expansion up to the third derivative term about the point xo = 4. Show the function, f(x), and the approximated function, fTay2(x) on a new plot. Display x-axis from -2 to 8. Show a marker at x = xo.   

(c) Do the Maclaurin series expansion, fTay3(x), to show an expansion up to the third derivative term. Show the function, f(x), and the approximated function, fTay3(x) on a new plot. Display xaxis from -2 to 8. Show a marker at x = xo.

Prove the trace of an n x n matrix is an element of the dual space of all n x n matrices.

Determine whether each statement is true or false. If false, give a counterexample.

a. Interchanging 2 rows of a given matrix changes the sign of its determinant.

b. If A is a square matrix, then the cofactor Cij of the entry aij is the determinant of the matrix obtained by deleting the ith row and jth column of A.

c. Every nonsingular matrix can be written as the product of elementary matrices.

d. If A is invertible, the AX = 0 has only the trivial solution.

e. If A and B are invertible matrices of order x, then AB is invertible and (AB)^-1 = (A^-1)(B^-1).

f. If A and B are matrices such that AB is defined, then (AB)^T = (A^T) (B^T).

g. Matrix A is symmetric if A = A^T.

h. Matrix multiplication is associative.

i. Matrix multiplication is commutative.

j. Every matrix has an additive inverse.

k. Every homogeneous system of linear equations is consistent.

l. A system of 2 linear equations in three variables is always consistent.

m. A linear system can have exactly two solutions.

Let G be a connected graph and let e be a cut edge in G. Let K be the subgraph of G defined by:

V(K) = V(G) and

E(K) = E(G) - {e}

Prove that K has exactly two connected components. First prove that e cannot be a loop. Thus the endpoint set of e is of the form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a path in K from v to ṽ, or there is a path in K from w to ṽ

Given a graph G, we obtain the subdivision graph of G, denoted by S(G), by subdividing each edge of G exactly once. Remember to subdivide an edge is to add vertex of degree 2. So if you have an edge (u, v) in G it becomes two edges in S(G). Show that S(G) is bipartite.

Give a direct proof of the following theorem, upon which case you
can use it for future proofs. (Hint: note that we’ve called it a corollary as
in p.81, not just a theorem.)

Corollary 4.12. Every integer is even or odd.

Give an example of a 2x2 matrix of numbers that represent the changes in the size of a predator-prey relationship. Then, conduct an eignvalue/eigenvector analysis of the matrix. Determine the long-term scaling factor for your populations and the long-term distribution of the predators to prey.

Then, compute trajectories for several different starting values of predators and prey. Use eigenvectors to simplify the computation of these trajectories.

There are two lines of students. The first line has students s1, s2, s3, s4 and s5, in that order. The second line has students t1, t2, t3, t4, t5, and t6, in order. Both lines will be merged into one as follows: If both lines still have students in them, one of the two lines is randomly selected and the person in the front of that line will go to the back of the merged line. If only one line remains, then all of those students will join in the back of the merged line, in their original order. For example, one way in which the students could line up is: s1, t1, t2, t3, s2, t4, t5, s3, t6, s4 and s5. Using this procedure, how many possible ways could these 11 students line up?

. Find all x ∈ Z₁₄₃ such that x²Ξ 7 (mod 143).

Please explain all steps and details clearly, it would be much appreciated, thanks!

QUESTION 1. For each of the following, state whether the occurrence of the variable x occurs bound, or free (i.e. unbound), both, or neither.

1. ∃xCube(x)

2. ∀xCube(a) ∧ Cube(x)

3. ∀x((Cube(a) ∧ Tet(b)) → ¬Dodec(x))

4. ∃yBetween(a,x,y)

5. ¬∀x¬(¬Small(d) ∧ ¬LeftOf(c,x))

QUESTION 2. Correctly label each of the following strings of symbols as a sentence, or well-formed formula (but not a sentence), or neither.

1. Fx ∧ Gy

2. ∃bFb

3. ∃z(Fz → Gb)

4. ∀xFc

5. ∀yFy ∨ ¬Fy

6. ¬∃¬xGx

Help me please with these questions thank you

Let p0 = 1+x; p1 = 1+3x+x2; p2 = 2x+x2; p3 = 1+x+x2 2 R[x].

(a) Show that fp0; p1; p2; p3g spans the vector space P2(R).

(b) Reduce the set fp0; p1; p2; p3g to a basis of P2(R).

Is the argument valid? Use rules of inference and laws of logic to prove or disprove (no Truth tables)

1. If John has talent and works very hard, then he will get a job. If he gets a job, then he’ll be happy. Hence if John is not happy, then he either not worked very hard or does not have talent.

2.For spring break Marie will travel to Cancun or Miami. If she goes to Cancun, she will not visit the Miami Zoo. If she does not visit the Miami Zoo she will visit the Cancun Zoo. She did not go to Miami. Therefore she visited the Cancun Zoo.

3. When interest rates go up, then car prices go down. Car prices did not go down. Therefore interest rates went up.

4. Mary plays tennis or Mary plays golf. Therefore, Mary plays golf.

Consider the natural join of the relation R(A,B) and S(A,C) on attribute A. Neither relations have any indexes built on them. Assume that R and S have 80,000 and 20,000 blocks, respectively. The cost of a join is the number of its block I/Os accesses. If the algorithms need to sort the relations, they must use two-pass multi-way merge sort.

QUESTION:

Assume that there are 10 blocks available in the main memory. What is the fastest join algorithm for computing the join of R and S? What is the cost of this algorithm?