Co-planar test is an important step in continuous collision detection. Let p0= [0,0,0], p1= [1,0,1], p2= [0,1,1], p3= [1,1,0] be four points and v0= [0,0,0], v1= [1,0,0], v2= [0,1,0], v3= [0,1,1] be their velocities. Please derive the formula to compute the time when these four points are co-planar. (Hint: You may use Unity or calculator to compute dot and cross products. You don’t need to give the time solution.)

4.

(a) Derive an integral expression for the probability of a gas molecule of mass m, at temperature T is moving faster than a certain speed v_min.

(b) A particle in the atmosphere near the earth’s surface traveling faster than 11 km/s has enough kinetic energy to escape from the earth’s gravitational pull. Therefore, molecules in the upper atmosphere will escape if they do not have collisions on the way out. The temperature of the upper atmosphere is about 1000K. Using the result for (a) and numerical computation to determine the probability that a N₂ molecule at this temperature will escape from the upper atmosphere? Repeat the calculation for a He atom. The escape velocity for the moon is about 2.4 km/s. Explain why the moon has no atmosphere

Name and sketch two ways in which an open method for root finding can fail to locate a root even when a root exists.

In this problem we prove that the Strong Induction Principle and Induction Principle are essentially equiv-
alent via Well-Ordering Principle.
(a) Assume that (i) there is no positive integer less than 1, (ii) if n is a positive integer, there is no
positive integer between n and n+1, and (iii) the Principle of Mathematical Induction is true. Prove
the Well-Ordering Principle: If X is a nonempty set of positive integers, X contains a least element.
(b) Assume the Well-Ordering Principle and prove the Strong Mathematical Induction Principle.

Find the general solution to the ODE y'' − y'/x − 4x^2y = x^2 sinh(x^2)

(a) Use a direct proof to show that the product of two odd numbers is odd.

(b) Prove that there are no solutions in integers x and y to the equation 2x2 + 5y2 = 14.

(c) Prove that the square of an even number is an even number using (a) direct proof, (b) an indirect proof, and (c) a proof by contradiction.

Q. 2. Maximum score = 25 (parts (a) 9 points, part (b-i) and (b-ii) 8 points)

(a) Show that 13 + 23 + …. +n3 = [n(n+1)/2]2 whenever n is a positive integer.

(b) Use induction to prove the following for all natural numbers n.

i) -1 + 2 + 5 + 8 +...+ (3n – 4) = (n/2) (3n-5)

ii) ½ + ¼ + 1/8 + … + 1/2n = (2n – 1)/ 2n

Q.3. Maximum score = 25 Prove that 1 + (1/4) + (1/9) +……(1/n2) < 2 - (1/n) for n, a positive integer >1.

Q. 4 Maximum score 25 Determine the formular for an given by the recurrence relation an = an-1 + 6an -2 ; a0 = 1, a1 = 8.

Find the general solution to the differential equation below. y′′ − 6y′ + 9y = 24t−5e3

Calculate the inverse Laplace transform of ((3s-2) e^(-5s))/(s^2+4s+53)

Calculate the Laplace transform of y = cosh(at) using the integral definition of the Laplace transform. Be sure to note any restrictionson the domain of s. Recall that cosh(t) =(e^t+e^(-t))/2

Consider the m by n grid graph: n vertices in each of m rows, and m vertices in each of n columns arranged as a grid, and edges between neighboring vertices on rows and columns (excluding the wrap-around edges in the toric mesh). There are m n vertices in total.

a)What is the diameter of this graph?

b) From the top left vertex to the bottom right vertex, how many shortest paths are there? Please explain.

Consider the following all-integer linear program:

Max 5x1 +8x2

s.t.  

6x1 + 5x2 <= 30

9x1 + 4x2 <= 36

1x1 + 2x2 <=10

x1, x2 $ 0 and integer

a. Graph the constraints for this problem. Use dots to indicate all feasible integer solutions.

b. Find the optimal solution to the LP Relaxation. Round down to find a feasible integer solution.

c. Find the optimal integer solution. Is it the same as the solution obtained in part (b) by rounding down?

I NEED EXCEL FILE THAT HAS EXCEL SPREAD SHEET AND SENSITIVITY REPORT TO SOLVE THIS PROBLEM (ALSO NEED STEP TO CREATE THEM)

1) There is a continuous function from [1, 4] to R that is not uniformly continuous. True or False and justify your answer.

2) Suppose f : D : →R be a function that satisfies the following condition: There exists a real number C ≥ 0 such that |f(u) − f(v)| ≤ C|u − v| for all u, v ∈ D. Prove that f is uniformly continuous on D

Definition of uniformly continuous: A function f: D→R is called uniformly continuous iff for all sequences {a_n} and {b_n} in D if (a_n) - (b_n)→ 0 then f(a_n) - f(b_n)→ 0

A children's fairy tale tells of a clever elf who extracted from a king the promise to give him one grain of wheat on a chess board square today, two grains on an adjacent square tomorrow, four grains on an adjacent square the next day (and so on), doubling the number of grains each day until all 64 squares on the chess board were used.

(a) How many grains of wheat did the hapless king contract to place on the 64th square?


(b) There are about 1.1 million grains of wheat in a bushel. Assume that a bushel of wheat sells for $4.40. What was the value of the wheat on the 64th square?

Consider the sequence: x₀=1/6 and x_n+1 = 2x_n- 3x_n² | for all natural numbers n.

Show:

a) x_n< 1/3 for all n.

b) x_n>0 for all n.

Hint. Use induction.

c) show x_n isincreasing.

d) calculate the limit.

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a.

(b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a.

(c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be an even number).

Explain how technology can be used to address one specific misconception in secondary-level geometry. (When thinking of a misconception, you want to mention something specific within a geometry course where a student may make an error).