Let (x_n) be a sequence with positive terms. (a) Prove the following: lim inf x_n+1/ x_n ≤ lim inf ⁿ√ x_n ≤ lim sup ⁿ√x_n ≤ lim sup x_n+1/ x_n .

(b) Give example of (x_n) where all above inequalities are strict. Hint; you may consider the following sequence x_n = 2ⁿ if n even and x_n = 1 if n odd.

We define SO(3) to be the group of 3 × 3 orthogonal matrices whose determinant is 1. This is the group of rotations in three-space, and you can visualize each element as a rotation about some axis by some angle.

(1) Check that SO(3) satisfies the three axioms of a group. You may take for granted that matrix multiplication is associative, as well as any standard properties of transposes and determinants.

(2) Prove that any reflection about a two-plane (or rather the matrix representation of such a linear transformation) is not included in SO(3). Hint: what is the determinant of such a matrix? Note that after a change of basis you can take the plane to be the xy-plane.

(3) Show that SO(3) is nonabelian.

(4) Consider the cube in R 3 whose set of eight vertices is {(i, j, k) : i, j, k ∈ {1, −1}}. Let H ⊂ SO(3) be the subgroup consisting of those rotations which map this cube to itself setwise.3 What is the order of H? You should provide some justification for your answer.

(5) Observe that each element of H determines a permutation of the set of 8 vertices. Give an example of such a permutation which does not arise from an element of H. Note: if you wish to identify such a permutation with an element of S8, you will first need to choose an ordering of the vertices.

(6) Similarly, each element of H determines a permutation of the set of 6 faces. Give an example of such a permutation which does not arise from an element of H.

A car moving at a constant speed of 40m/s overtakes a stationary police car which immediately starts to chase the speeding car at a constant acceleration.

What is the speed of the police car when it overtakes the speeding car?

(Note: No other knowns except speed of speeding car and that the police car has constant acceleration).

1. 110 ice creams of different flavors are purchased weekly in a student residence: vanilla, passion fruit, and coconut. The budget for this purchase is 540 euros and the price of each ice cream is 4 euros for vanilla, 5 euros for passion fruit and 6 euros for coconut. Once the students' tastes are known, it is known that between 20% passion fruit and coconut ice cream, vanilla must be purchased. to. A. Solve using a system of linear equations to calculate how many ice creams of each flavor are bought per week. b. Solve, using the Gauss Jordan Method

Exercise

Minimize            Z = X₁ - 2X₂

Subject to            X₁ - 2X₂ ≥ 4

                            X₁ + X₂ ≤ 8

                           X1, X2 ≥ 0

Solve the following linear programs graphically.

Minimize            Z = 4X₁ - X₂

Subject to            X₁ + X₂ ≤ 6

                            X₁ - X₂ ≥ 3

                           -X₁ + 2X₂ ≥ 2

                           X₁, X₂ ≥ 0

Solve the initial value problem. y'=(y^2)+(2xy)+(x^2)-(1), y(0)=1

Show that drastic sum and drastic product satisfy the law of excluded middle and the law of contradiction. [Hint: substitute the s-norm and t-norm operation for intersection and union in the two laws, respectively]

What are the square roots of 3+4i?

Find the inverse of the following 4x4 matrix:

1-j j 1+j 2

-j    4    2-j    3

1-j 2+j j 3-j

2 3 3+j 1

1. If f(x) = ln(x/4)
-(a) Compute Taylor series for f at c = 4
-(b) Use Taylor series truncated after n-th term to compute f(8/3) for n = 1,.....5
-(c) Compare the values from above with the values of f(8/3) and plot the errors as a function of n
-(d) Show that Taylor series for f(x) = ln(x/4) at c = 4 represents the function f for x element [4,5]

Exercise

Solve the following linear programs graphically.

Maximize            Z = X1 + 2X2

Subject to            2X1 + X2 ≥ 12

                            X1 + X2 ≥ 5

                           -X1 + 3X2 ≤ 3

                           6X1 – X2 ≥ 12

                           X1, X2 ≥ 0

1a. Proof by induction: For every positive integer n,
1•3•5...(2n-1)=(2n)!/(2n•n!). Please explain what the exclamation mark means. Thank you for your help!

1b. Proof by induction: For each integer n>=8, there are nonnegative integers a and b such that n=3a+5b

A tree is a circuit-free connected graph. A leaf is a vertex of degree 1 in a tree. Show that every tree T = (V, E) has the following properties: (a) There is a unique path between every pair of vertices. (b) Adding any edge creates a cycle. (c) Removing any edge disconnects the graph. (d) Every tree with at least two vertices has at least two leaves. (e) | V |=| E | +1.