Find the root of the function f(x) = 8 - 4.5 ( x - sin x ) in the interval [2,3]. Exhibit a numerical solution using Newton method.

Consider a two-dimensional ideal flow in the x-y plane (with radial coordinate r2 = x2 + y2). Given the velocity potentials of 1) a uniform flow, 2) a source φ = (q/2π) ln r, and 3) a dipole φ = −d · r/(2πr2):

a) Using the principle of superposition, construct a linear combination of the ingredients above that gives the flow past an infinite cylinder. [10 points]

b) Sketch the streamlines of the flow everywhere in space. [10 points]

3) Prove that the cardinality of the open unit interval, (0,1), is equal to the cardinality of the open unit cube:

{(x,y,z) E R^3|0<x<1, 0<y<1, 0<Z<1}.

[Hint: Model your argument on Cantor's proof for the interval and the open square. Consider the decimal expansion of the fraction 12/999. It may prove handdy]

Let X and Y be T2-space. Prove that X*Y is also T2

solve using powers series method, full procedure please:

y´(t) = -y(t) + t

Solve the nonhomogeneous heat equation:

ut-kuxx=sinx, 0<x<pi, t>0

u(0,t)=u(pi,t)=0, t>0

u(x,0)=0, 0<x<pi

Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W).

Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E B(V, W), V E V, W E W.

We denote by N^∞= {(a_1, a_2, a_3, . . .) : a_j ∈ N for all j ∈ Z^+}.   Prove that N^∞is uncountable.

Please answer questions in clear hand-writing and show me the full process, thank you (Sometimes I get the answer which was difficult to read).

Construct a TM that computes the function: f(x) = 2x. (first give a brief outline, then show how it works for x=11 (in unary system, not eleven) using instantaneous description.

Problem 3. A standard deck of cards contains 52 cards:

13 Ranks: A,2,3,4,5,6,7,8,9,10,J,Q,K
4 Suits: Clubs, Diamonds, Hearts, Spades

Suppose you draw a hand of 5 cards from the deck. Consider the following possible hands of cards. For each part, explain how you derive your solutions.

3(a) How many different hands of 5 cards exist?

3(b) How many ways can you draw a single pair? A single pair means that there are exactly two cards with the same rank (no additional pairs and no 3 of a kind, etc.).

3(c) How many ways can you draw 4 of a kind? (4 cards of the same rank)

3(d) How many ways can you draw a full house? A full house is a pair of one rank and three cards of another rank.

3(e) How many ways can you draw a straight? A straight is 5 consecutive ranks. The Ace (A) can either go before 2 or after K.
E.g., A,2,3,4,5 and 10,J,Q,K,A are both valid, as are 4,5,6,7,8 and 7,8,9,10,J etc.

Note: if you were to divide the number of hands of a particular type by the total number of 5 card hands, you would achieve the probability (odds) of randomly drawing a hand of that type.

Block copy, and paste, the argument into the window below, and do a proof to prove that the argument is valid.
1. (p   ⊃   x) • (x ⊃ ~q)
2. p •   n
3.   t v q     : .     t

Block copy, and paste, the argument into the window below, and do a proof to prove that the argument is valid.

1. (p   • q)   •   (q ⊃   x)
2. ~x v ~s
3. n   v   s     : .    t ⊃ n

  A company makes four types of gourmet chocolate bars. Chocolate Bar I contains 5 grams of nuts, 5 grams of biscuit, 10 grams of caramel, and 100 grams of chocolate, and sells for $5.40. Chocolate Bar II contains 10 grams of nuts, 10 grams of biscuit, 10 grams of caramel, and 90 grams of chocolate and sells for $6.25. Chocolate Bar III contains no nuts, 10 grams of biscuit, 10 grams of caramel, and 100 grams of chocolate and sells for $5.25. Chocolate Bar IV contains 20 grams of nuts, no biscuit, 10 grams of caramel, and 90 grams of chocolate and sells for $7.00. The ingredient costs are $0.15 per gram of nuts, $0.025 per gram of biscuit, $0.02 per gram of caramel, and $0.015 per gram of chocolate. The company has supplier agreements that force them to order at least 50,000 grams of each ingredient per week (nuts, biscuit, caramel, and chocolate). However, due to the space in their warehouse and the expense of holding too much inventory, the company cannot hold more than 100,000 grams of nuts; 75,000 grams of biscuits; 85,000 grams of caramel; and 500,000 grams of chocolate per week. How many chocolate bars of each type should the company produce each week in order to maximize its weekly profit?   

A metric space X is said to be locally path-connected if for every x ∈ X and every open neighborhood V of x in X, there exists a path-connected open neighborhood U of x in X with x ∈ U ⊂ V.

(a) Show that connectedness + local path-connectedness ⇒ path-connectedness

(b) Determine whether path-connectedness ⇒ local path-connectedness.

Find the eigenvalues and eigenvectors of the following matrix. Justify if its diagonaliazble or not.

1 2 0

-3 2 3

-1 2 2