Write a program (fortran 90) that calls a subroutine to approximate the derivative of y=sin(x)+2x^2 using a one-sided difference approach fx = (fi-fi-1)/deltaX and a centered difference approach fx = (fi+1-fi-1)/deltaX. The value of the function f and its derivative fx should be evaluated at x=3.75. Your code should print both values tot he screen when it runs.

Express sin 6θ as a polynomial in sin θ and cos θ.

Find the general solution of the differential equation y′′+9y=13sec²(3t), 0<t<π/6.

Use C1, C2,... for the constants of integration. Enter an exact answer. Enter ln|a| as ln(|a|), and do not simplify.

y(t)=

Use the method of variation of parameters to find a particular solution of the differential equation

4 y′′−4 y′+y=32et²

Find the inverse Laplace transform for

1 / ((s^2+1)(s+1))

How would you go about finding which of the following PDEs can be separated into ODEs by assuming a product of unknown functions x,y, and z?

1) aUxy+bU=0

2) x^2Uxx+yUyy=0

3) Uxx+3Uxy+7Uy=0

4)Uxx+Uyy+Ux+Uy=0

Where y=y(x) solves...

(5.9y-(23.6)x)dx+(5.9x-1)dy=0   y(0.27)=2.15

(5xy-(-1.1)ye^x -1)dx+(2.5x^2-(-1.1)e^x)dy=0     y(1.2)=2.3

The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars 21 26 28 35 43 Number of Bids 1 3 5 6 9 Table Step 6 of 6 : Find the value of the coefficient of determination. Round your answer to three decimal places.

4. The check digit for ISBNs is one of the numbers 0, 1, 2, . . . , 9, or the letter X. One of the your fellow students comments ”Gee, it sure is a pain to have to use that X all time. Why don’t they just compute the check digit sum modulo 10 instead of modulo 11, so that we can get rid of the X?” Would this plan work? Prove your answer.

Assume that all ISBNs are 10 digits with the tenth digit the check digit. I know that using modulo 10 to compute the check digit will catch single digit errors, but it will not always catch errors regarding transposition of numbers in the ISBN. I am having a difficult time proving that the transposition errors will not always be caught.

Let (X, d_X) and (Y, d_Y) be metric spaces.Define the function

d : (X × Y ) × (X × Y ) → R

by

d ((x₁, y₁), (x₂,y₂)) = d_x(x₁,x₂)+d_y(y₁,y₂)

Prove that d is a metric on X × Y .

Consider the differential equation y' = -18 + 11x - x².

(a) Find the equilibria (i.e. constant solutions).

(b) By analyzing the sign of dy/dt around the equilibria, determine whether each equilibrium solution is stable, unstable, or neither.

(c) Find an explicit general solution to the equation.

(d) Graph the equilibrium solutions, as well as some non-constant solution curves, verifying visually the stability properties determined in (b).

Consider the following relation

AIRLINE TABLE

Some of the requirements that this table is based on are as follows:

1. The AIRLINE is flying number of flights on various routes daily.
2. A flight with a particular Flight Id always flies on the same route (same Origin and same Destination).
3. Every captain has a unique Captain Id, and a non-unique Captain Name.
4. For each instance of a flight (a particular flight on a particular Flight Date) we keep track of who was the captain of that instance of a flight and how many passengers were on board (Number of Passengers on the Flight).
5. A captain can fly multiple flights during the same day.

Normalize the relation to the second normal form (2NF).

Normalize the relation to the third normal form (3NF).

Exercise 1.8

1. Your long distance phone service has a base monthly charge and a per-minute charge. When you used 350 minutes in a month the total cost was $32.50. When you used 400 minutes in a month the total cost was $36.50. You want an equation that will allow you to calculate your phone bill.

Please provide:

a. the definition of x, including the units

b. the definition of y, including the units

c. the equation in the form y = mx + b

d. the units of the slope

e. the units of b and an interpretation of b.

f. How much will your phone bill be if you talked for 711 minutes?

2. The dosage for a medicine is linear with the weight of the patient. There is a minimum dosage onto which is added a per pound dosage. You find that your dosage, at 110 pounds is 48 milliliters (ml). Your brother’s dosage, at 170 pounds, is 66 ml. You would like an equation that will relate the dosage to the weight of the patient. Please provide:

a. the definition of x, including the units

b. the definition of y, including the units

c. the equation in the form y = mx + b

d. the units of the slope

e. the units of b and an interpretation of b.

f. what is the dosage for a 165 lb patient?

For each of the problems,6 –10, cost information at a certain level of production for a manufacturing process is given. The revenue per unit is also given. Assume a linear relationship between the number of units produced and cost. For each problem please find:a. The cost,revenue and profit functions and the units of the slope and of b.b. The variable cost per unit, also known as the marginal cost, and the fixed cost. c. The cost, revenue and profit when z units are produced. (z will be specified in each problem)d. The break-even point.e. The average cost per unit of producing w units and the equation of the average cost per unit function. (w will be specified in each problem)

6. The cost of producing 1,250 units is $45,000. The cost of producing 1,500 units is $50,000. Each unit produced can be sold for $30. z = 3,000; w = 2,500.
7. The cost of producing 23,000 units is $225,000. The cost of producing 30,000 units is $260,000. Each unit produced can be sold for $12. z = 45,000; w = 68,000.

3.5. Each of the following measurements is a rounded value. We have no way of knowing the exact value that was rounded to obtain these rounded values. For each, i) state the range of possible exact values; ii) stating the absolute value of the maximum possible measurement rounding error that may have resulted from the rounding; and iii) state the minimum and maximum possible relative measurement error as a percent to two significant digits.

a. 0.02 ft b. 0.07 ft c. 2.634E+02 km d. 9.167E+02 km

a 600gal tank initially containing 75lb of salt. Brine containing 1lb of salt per gallon enters the tank at a rate of 5gal/s, and the well mixed brine in the tank flows out at a rate of 3gal/s. How much salt will the tank contain when it is full of brine?