For what values of x this set is linearly independent?

{sin(x), cos(x), x}.

Check the true statements below:

A. The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c≠0.

B. If the columns of an m×n matrix A are orthonormal, then the linear mapping x→Ax preserves lengths.

C. If a set S={u1,...,up} has the property that ui⋅uj=0 whenever i≠j, then S is an orthonormal set.

D. Not every orthogonal set in Rn is a linearly independent set.

E. An orthogonal matrix is invertible.

Solve the following problem:

y ''' − 2y '' + y ' = 2 − 24e^ x + 40e ^5x

y(0) = 1/2 , y' (0) = 5/2 , y'''(0) = − 9/2

BOTH LINES ARE PART OF A SYSTEM OF EQUATIONS

Please show the calculation process in Excel, thank you

7. With the gasoline time series data from Table 8.1, show the exponential smoothing forecasts using α=0.1.

   1. Applying the MSE measure of forecast accuracy, would you prefer a smoothing

      constant of α=0.1 or α=0.2 for the gasoline sales time series?

   2. Are the results the same if you apply MAE as the measure of accuracy?

   3. What are the results if MAPE is used?

   4. Week	Sales (1000s of gallons)
      1	17
      2	21
      3	19
      4	23
      5	18
      6	16
      7	20
      8	18
      9	22
      10	20
      11	15
      12	22

Solve the given differential equation by undetermined coefficients. y'' − 6y' + 9y = 21x + 3

Let V = R4 and let U = hu1, u2i, where u1 =    1 2 0 −3    , u2 =     1 −1 1 0    . 1. Determine dimU and dimV/U. 2. Let v1 =    1 0 0 −3    , v2 =     1 2 0 0    , v3 =     1 3 −1 −6    , v4 =     −2 2 0 9    . For any two of the vectors v1,...,v4, determine whether they are in the same coset of U in V or not. 3. Find a basis of V that contains a basis of U. Hence, determine a basis of V/U. 4. Find two (distinct) elements of the coset e1 + U.

a) What is the difference between regression and interpolation?
b) Use least squares regression to fit a straight line to the data given in Table 1 and
calculate the y value corresponding x=3.
c) Find the Lagrange interpolating polynomial using the data given in Table 1 and
calculate the y value corresponding x=3.

Table 1

This question pertains to the probability of rolling a given total (adding the two die face-values) using 2 fair 6-sided dice. The possible results of rolling 2 fair 6-sided dice are the elements of the Cartesian product A × A, where A = {1, 2, 3, 4, 5, 6}.

(a) How many elements does A × A have?

(b) Partition the elements (x, y) of the set A × A according to the sum x + y. For example, (1, 3) goes in the part labeled somehow by ‘4.’

(c) Write down all of the parts of the partition with all of their respective elements.

(d) Define an equivalence relation R on A × A so that the equivalence classes of R equal the parts of the partition in Part (c).

(e) Write down one representative of each equivalence class in the equivalence relation R in Part (d).

(f) The most likely die total is |P|/|A × A|, where P is the largest part of the partition you wrote down in Part (c). What is the most likely die total?

Use the Laplace transform to solve the given initial-value problem.

y'' - 2y'' - 8y = 2sin2t; y(0) = 2, y'(0) = 4

Find the coordinates of the images of A (2,3) and B (-2,3) under the following transformations. Assume that all dilations are centered at the origin. Draw sketches to justify your answer.

a. A dilation with a scale factor of 3 followed by a dilation with a scale factor 2.

b. A dilation with scale factor 2 followed by a translation with a slide arrow from (2,1) to (3,4).

c. A translation with a slide arrow from (2,1) to (3,4) followed by a dilation with scale factor 2.

A small radio transmitter broadcasts in a 48 mile radius. If you drive along a straight line from a city 59 miles north of the transmitter to a second city 56 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Does Elevation Affect Temperature Mid-June?
Suppose that you wanted to determine the effect, if any, that elevation has on temperature. The table below lists the elevations (in feet above sea level) of 24 randomly selected cities in the United States and the low temperatures (in degrees Fahrenheit) of these cities on June 15, 2020.

1. If Westerville is 875 feet above sea level, what is Westerville’s estimated low temperature for June 15, 2020?

2. Before performing any regression analyses, what is the benefit of noting that the lowest and highest values amongst the elevation data are −282 feet and 6910 feet, respectively?

3. What proportion of the variation in low temperatures can be explained by the linear relationship with elevation?

Give one example of a lurking variable that may also influence the response variable in this situation

A dead body was found within a closed room of a house where the temperature was a constant 70° F. At the time of discovery the core temperature of the body was determined to be 80° F. One hour later a second measurement showed that the core temperature of the body was 75° F. Assume that the time of death corresponds to t = 0 and that the core temperature at that time was 98.6° F. Determine how many hours elapsed before the body was found. [Hint: Let  t₁ > 0 denote the time that the body was discovered.] (Round your answer to one decimal place.)

3.11. (a) Let n be any integer such that n is congruent to 0 (mod 7). For any positive integer k, what is the remainder when n^k is divided by 7?

(b) Let n be any integer such that n is congruent to 1 (mod 7). For any positive integer k, what is the remainder when n^k is divided by 7?

(c) Let n be any integer such that n is congruent to 2 (mod 7). For any nonnegative integer k, what is the remainder when (i) n^3k is divided by 7? (ii) n^3k+1 is divided by 7? (iii) when n^3k+2 is divided by 7?

(d) Let n be any integer such that n is congruent to 3 (mod 7). For any nonnegative integer k, what is the remainder when (i) n^3k is divided by 7? (ii) n^3k+1 is divided by 7? (iii) when n^3k+2 is divided by 7?