Calculate a cubic spline for (given second derivative at (1,4) and (4,5) are zero)

(1,4), (2,3), (4,5)

Solve x" + ? = ???2? , ? (0) = 2, ? ′ (0) = 1
1) manually 2) using Laplace transform. Show step-by-step process

Which of the following sets are subspaces of ℝ³?

A. {(5x,8x,3x) | x arbitrary number }}
B. {(x,0,0) | x arbitrary number }}
C. {(x,y,z) | −8x−3y−7z=−5}
D. {(−6,y,z) | y,z arbitrary numbers }}
E. {(x,y,z) | x<y<z}
F. {(x,y,z) | x+y+z=0}

Answer the following questions. Please show your work or otherwise justify your answers. If you’re asked to create a truth table, show the entire table.

Create the truth table for the statement form: ~(? → ?) ↔ (? ∧ ~?)

Write the two statements in symbolic form and determine if they’re equivalent. Construct a full truth table, and explain how this shows equivalency (or a lack thereof). Statement 1: If you play aggressively and protect your king, then you will win the match. Statement 2: You didn’t play aggressively or you didn’t protect your king or you won the match.

The numbers of polio cases in the world are shown in the table for various years.

Let f(t)f(t) be the number of polio cases in thousands t years since 1980.

Use a graphing calculator to draw a scattergram of the data. Is it better to model the data by using a linear or exponential model? Select an answer Exponential Linear

Find an equation of f. Hint

f(t)=f(t)=    Round the coefficients to 4 decimal places.

The number of polio cases is Select an answer decreasing increasing  by Select an answer 25.55% 368665.63% 3686.6563% 74.45%  per year.

Predict the number of polio cases in 2014.

Hint

Predict in which year there will be 1 case of polio. Hint



Find the approximate half-life of the number of polio cases. Hint

years

1. There are 4 quarters and 3 dimes and 1 nickel in a coin pouch. Two coins are selected at random. If XX assigns the total value of the coins to each outcome, how many distinct values does XX take?

2. Rework problem 5 in section 4.2 of your text, involving the selection of two students from a committee of students. Assume that the committee is made up of 5 males and 6 females. Two are selected at random, and a random variable XX is defined to be the number of males selected.

How many different values are possible for the random variable XX?

3.

Rework problem 9 in section 4.2 of your text, involving a defective vending machine. Assume that the machine yields the item selected 70 percent of the time, and returns nothing 30 percent of the time. Three individuals attempt to use the machine. Let XX be defined as the number of individuals who obtain the item selected.

How many different values are possible for the random variable XX?

2. Let S be the set of strings over the alphabet Σ = {a, b, c} defined recursively by (1) λ ∈ S and a ∈ S; and (2) if x ∈ S, then bxc ∈ S. Recall that λ denotes the empty string which has no letters and has length 0. List all strings of S which are length at most seven.

3. Prove the following theorem by induction: For every integer n ≥ 1,

1 · 2 + 2 · 3 + 3 · 4 + · · · + n(n + 1) = n(n + 1)(n + 2) 3 .

Apply Euler's method twice to approximate the solution to the initial value problem on the interval [0, .5], first with step size h=0.25, then with step size h= 0.1. Compare the three-decimal place values of the two approximations at x=.5 with the value of y(.5) of the actual solution.

y'=y-3x-6, y(0)=8, y(x)=9+3x-e^x

^{a.) The Euler approximation when h=0.25 of y(.5) is:}

^{b.)The Euler approximation when h=0.1 of y(.5) is:}

Given 12 coins, with possibly (but not necessarily) one coin being counterfeit (heavy or light). You have a comparison scale and you want to determine the fairness of all with a minimum number of comparisons.

(a) Give the procedure to identify the false coin or to show all are fair.

Use pigeonhole principle to solve please: will upvote!

Let V = {v1,…,vk} be any set of vectors in R^2 (Real Numbers to the power of 2). Suppose n agents each start at (0,0) and each takes a mV-walk where a mV-walk consists of a sequence of exactly m steps and each step moves the agent along a vector in V. Prove that, if n > (m + k − 1 , k − 1) (these are two separate terms in one parenthesis), then some pair of agents finishes their walk at the same location.

We consider the following homogeneous differential differential equation

y''+ 2ky^'+ ak²y = 0, where k ≥ 0 and a are given parameters in R.

(i) Find the general solution of this ED (there are several cases to be treated).

(ii) Find the solution of PVI for y (0) = 1 and y' (0) = 0, in the case where the roots found in (a) are real and distinct.

Null graph,Nn, n=1,2,3,4...，the graph with n vertices and no edges. (N4=4 vertices with no edges)

4 a) find a graph with 8 vertices with no 3-cycles and no induced sub graph isomorphic to N4

b)prove that every simple graph with 9 vertices with no 3-cycles has an induced sub graph isomorphic to N4

We are given a set of vectors S = {V1, V2, V3} in R 3 where eV1 = [ 2 −1 3 ] , eV2 = [ 5 7 −1 ] , eV3 = [ −4 2 9 ]

Problem 1

• Prove that S is a basis for R^3 .

• Using the above coordinate vectors, find the base transition matrix eTS from the basis S to the standard basis e.

Problem 2 Using your answers in Problem 1

• Compute the base transition matrix STe from the standard basis e to the basis S.

• If eV = [ 5 1 7 ], compute SV (the coordinate vector of V with respect to the basis S). Use this to express V as a linear combination of the vectors in S.

Q5. The laser at the doctor’s office is graduated in 1/20th of an inch intervals. It measures your height. A tape measure instead is graduated in 1/8th of an inch intervals. Which one is more accurate? Which one is more precise? Illustrate your rationale with a few examples of hypothetically measured heights using the two instruments. [20]