S_3 is the vector space of polynomials degree <= 3. V is a subspace of poly's s(t) so that s(0) = s(1) = 0. The inner product for two poly. s(t) and f(t) is def.: (s,f) = ([integral from 0 to 1] s(t)f(t)dt). I would like guidance finding (1) an orthogonal basis for V and (2) the projection for s(t) = 1 - t + 2t^2. Thank you!

Think of a password 8 character long which uses at last one lower-case letter, one capital letter, and one number, but uses none more than once.
What's your password?

How many possible permutations are there for an 8 character password which meets those criteria?

How many fewer combinations without regard to order are there for an 8 character password which meets those requirements.

How would you calculate how many permutations there are for an 8 to 12 character password which meets those criteria?

1. When the process of solving a linear inequality results in the variable being eliminated with a false statement remaining, what does this mean about the solution set of the inequality?

2. When the process of solving a linear inequality results in the variable being eliminated with a true statement remaining, what does this mean about the solution set of the inequality?

3. How is the process of solving linear inequalities similar and/or different than the process of solving a linear equation? How do the solution sets of a linear inequality and a linear equation differ?

Un cuerpo que pesa 3 libras fuerza cuelga de un resorte y lo estira 6 pulgadas. Determine la amplitud de movimiento armónico si el cuerpo se libera 4 pulgadas por encima de la posición de equilibrio con una velocidad inicial de 2 pulgadas por segundo. Considere magnitudes hacia abajo negativas y hacia arriba positivas.

Let f: [0 1] → R be a function of the class c ^ 2 that satisfies the differential equation f '' (x) = e^xf(x) for all x in (0,1). Show that if x0 is in (0,1) then f can not have a positive local maximum at x0 and can not have a negative local minimum at x0. If f (0) = f (1) = 0, prove that f = 0

How many distinct arrangements of the letters in MANANGATANG have no A’s in the first six places?

NOTE: MANANGATANG has 11 letters, so the last five places must have four A's and one other letter.

PLEASE SHOW FULL LEGIBLE WORKING.

Question 2. Consider a simple economy with two types of workers. There are capable workers (type A), constituting 70% of the population, with remaining workers (type B) being of average ability. There are two types of job opportunities as well, labeled Good and Bad. In a Bad job, either type of worker produces 20 units of output. In a Good job, Type A worker produces 80 units, and the Type B worker produces 0. The economy is doing well so there is enough demand for workers. This means that for each type of job companies must pay what they expect the employee to produce. Companies must hire each worker without observing his type and pay him before knowing his actual output. But Type A workers can signal their qualification by getting educated. For a Type A worker, the cost of getting educated to level n is 0.4n2, whereas for a Type B worker, it is n2. These costs are measured in the same units as output, and n must be an integer.

(a) What is the minimum level of n that will achieve separation?

(b) Now suppose the signal is made unavailable. Which kind of jobs will be filled by which kinds of workers and at what wages? Who will gain and who will lose from this change?

1. Find the Taylor polynomial of degree ?=3 for ?(?)=?−?22 expanded about ?0=0.
2. Find the error the upper bound of the error term ?₅(?) for the polynomial in part (1).

Describe briefly the following method of Out of Sample forecast criterion
1... Root Mean Square error
2...Mean absolute error
3...Mean absolute percentage error
4...Maximum Absolute Error
5...Maximum Absolute Percentage error.

Write down their formulae and discuss briefly how you interpret them

Using Burnside's Lemma, determine a formula for the number of orbits under the action of D8 on the set of colourings.

(We colour each side of a square with one of k ≥ 1 colours.)

a) Write an function that finds the independent variable value at which a mathematical function is maximized over a specified interval. The function must accept a handle to a function, and evaluate that function at values ranging from x1 to x2 with an increment of dx. The value returned is the value of x at which the maximum value of f(x) occurs.

Function syntax: xm = xmax(f,x1,x2,dx);

As was the case in the previous problem, this function does not find the true value of x that maximizes f(x). It only identifies the value of x that yields the maximum value of f(x) from among those values of x at which the function is evaluated. How closely this approximates the true maximum point of f(x) will depend on the step size, dx.
Your function should use only basic arithmetic operations and loop structures. You may not use any built-in MATLAB functions (e.g., mean.m, sum.m, max.m, min.m, etc.).
Next, you will investigate how the estimated maximum point varies as a function of dx.

b) Write an m-file, ENGR112_HW6_2.m, in which you define the following as an anonymous function.

?(?)=−?^2+15??+21.843

Define a logarithmically-spaced vector, dx, of 2000 step sizes, from 10-4… 100, at which xmax.m will be used to approximate the maximizing value of f(x) over the interval of 0 ≤ x ≤ 50. Call xmax.m at each value of dx and build a vector of maximizing independent variable values, xm. In

c) Plot the vector of approximate maximizing values as a function of step size using a logarithmic axis for the step size. You should see that the approximation converges as step size gets small enough.

"Suppose we have a simple random sample comprised of the following data: 12, 46, 22, 21, 7. What is the point estimate of the population standard deviation?"

Lots of things impact sampling error. Which one of the following does not?

"Given the information provided, what is the probability of obtaining a sample mean within +/- 2 of the population mean? Population information: mean = 425, and standard deviation = 25. Sample size is 210."

A sample size of 95 is selected from a population with the proportion equal to 0.41. The sample proportion has an expected value of __ and standard deviation of __.

The CEO of Snapchat thinks that 65% of users are under the age of 25. A simple random sample of 70 users will be used to estimate the proportion of those under the age of 25. What is the probability the sample proportion will be between 0.6286 and 0.6714?

1.

A) How many three-digit numbers are there for which the sum of the digits is at least 25?

B) How many three-digit numbers can be formed if only odd numbers are allowed to be re-used

Please combinatorics principles where applicable.

1. Space travel is expensive! For their trip to the Moon, the Apollo astronauts' living quarters were only 213 cubic feet (that's smaller than a typical small bathroom in a house). How many dollar bills could fit in there?

The room in the top half of the image is the laboratory. At the lower left is the sleeping area, lower center is bathroom, and lower right is the living area. (Image from NASA).

2.

Would a stack of dollar bills in the amount of the U.S. national debt reach from the Earth to the Moon?

3. Pluto has been hard to measure from Earth because of its atmosphere. In 2007 Young, Young, and Buie measured Pluto as having a diameter of 2322 km. In 2015 the New Horizons probe reached Pluto and measured it up close and we now know the actual diameter is 2372 km. What was the percent error of the 2007 measurement?

Enter your answer as a percent, with a negative value if the 2007 measurement was too small and a positive value if it was too large.