Determine the form of a particular solution to the following differential equations (do not evaluate coefficients).

(a)y′′ −4y′ = x+1+ xe^(2x) + e^(4x) + e^(4x)sin4x

Confidence Interval Worksheet with Sample Data MTH 160: Statistics

Read the following scenario and complete each of the problems below:

A flashlight company claims that the new bulb in its heavy-duty flashlight will average 246 hours of light. A statistics student decides that he/she wants to test this claim at a 5% level of significance to determine if there is evidence to support the claim. The student randomly selects and tests 15 flashlight bulbs and records how long the bulb lasts until it burns out. Assume the life of a bulb is normally distributed. T

he data is in the table below

Trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hrs: 246 224 231 242 237 240 243 236 239 255 256 239 247 231 253

A. The standard deviation of the population is 7.4 hours. Construct a 95% confidence interval for this study.

B. Same scenario, but the population’s standard deviation is not known. Construct a 95% confidence interval for this study.

C. Write a statement comparing and contrasting the two confidence intervals and results.

Read the following scenario and complete each of the problems below:

A new car manufacturing company has emerged and has claimed that its new hybrid car, the Pusho, gets a better gas mileage than the highest ranked Toyota Prius. Consumer Reports Magazine decides to test this claim at a 5% level of significance. Consumer Reports randomly selects 10 of each type of car, calculates the miles per gallon for each car in the study, and records the data in the table below. Assume miles per gallon of the cars is normally distributed.

Pusho 54.1 52.4 55.7 49.7 50.6 48.9 51.8 54.5 56.9 49.8

Prius 53.2 54.3 49.8 50.1 50.5 56.1 47.8 53.4 56.8 48.7

A. Construct a 90% confidence interval for the difference in the gas mileage of Pusho and Prius.

B. Construct a 95% confidence interval for the difference in the gas mileage of Pusho and Prius.

C. Write a statement comparing and contrasting the two confidence intervals and the significance of having 0 in both of those intervals.

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

By constructing a suitable bijection, show that the number of subsets of an n-set of odd size is equal to the number of subsets of an n-set of even size.

Let us divide the odd positive integers into two arithmetic progressions; the red numbers are 1, 5, 9, 13, 17, 21, ... The blue numbers are 3, 7, 11, 15, 19, 23,....

(a) Prove that the product of two red numbers is red, and that the product of two blue numbers is red.

(b) Prove that every blue number has a blue prime factor.

(c) Prove that there are infinitely many blue prime numbers. Hint: Follow Euclid’s proof, but multiply a list together, multiply the result by four, then subtract one.

Problem #1 :

Using separation of variables and the formalism demonstrated in class, find the general solution to the Helmholtz equation (also known as the time independent wave equation) :

∇^2 F + k^2 F = 0

Assume that k is a real number. For clarification, the general solution is the solution in the case where boundary conditions are not specified.

Problem #2 :

Express the following functions as an infinite Fourier sine series using sin(nπx/a)

a)f(x) = x

b)f(x) = e^x

c)f(x) = cos(x)

Quiz 4

A manufacturer makes and sales four types of products:  Product X, Product Y, Product Z, and Product W.  

The resources needed to produce one unit of each product and the sales prices are given in the following Table.

• Currently, 4,600 pounds of steel and 5,000 machine hours are available.

• To meet customer demands, exactly 950 total products must be produced.

• Customers also demand that at least 400 units of Product W be produced.

Formulate an LP that can be used to maximize sales revenue for the manufacturer.

LP Formula

Let Pi be the number of product type i produced by the manufacturer, where i = X, Y, X, and W.

MAXIMIZE  4 PX + 6 PY + 7 PZ + 8 PW

Subject To

2 PX + 3 PY + 4 PZ + 7 PW <= 4600   ! Available Steel

3 PX + 4 PY + 5 PZ + 6 PW <= 5000   ! Available Machine Hours

PX + PY + PZ + PW    = 950               ! Total Demand

                            PW >= 400               ! Product W Demand

PX >=0

PY >=0

PZ >=0

PW >=0

Suppose manufacturer raises the price of Product Y by 50¢ per unit. What is the new optimal solution to the LP?

1. How is the ratio of outputs to inputs, y x, different from the slope ratio?
2. When a linear graph passes through the origin, why is the ratio of outputs to inputs, y x the same as the slope?
3. How can you tell from a graph whether a relationship is inverse variation or direct variation? Does this change when we look at direct square variation versus inverse square variation? Why or why not?
4. What is a horizontal asymptote? Do direct or inverse variation graphs have asymptotes? Why?
5. Explain how to tell whether a table varies directly, inversely, or neither. Include what to watch for to determine if a function varies directly or inversely with the square.

** NEED MATLAB**

design a cam with harmonic and cycloidal rise and 3-4-5 polynomial fall

specifications:

*Cycloidal rise (0◦ < θ < 80◦) from 0 mm to 20 mm

*Dwell (80◦ < θ < 100◦)

*Harmonic rise (100◦ < θ < 180◦) from 20 mm to 30 mm

*Dwell (180◦ < θ < 210◦)

*3-4-5 Polynomial fall (210◦ < θ < 300◦) from 30 mm to 0 mm

*Dwell (300◦ < θ < 360◦)

•the radius of the base circle is 40 mm, radius of follower is 5 mm, and cam is driven by a constant speed motor rotating counter-clockwise at 500 rpm with the above specifications

----------------------

1. Plot the displacement, velocity, accleration, and jerk profiles for one revolution

2. Plot the pressure angle as a function of θ (eccentricity is 0)

3. Plot cam contour, pitch curve, prime circle, and base circle (all on same plot)

4. Plot the prime circe and cam contour at various orientations (from θ =0◦ to θ =240◦. AND arrange 9 subfigures as a 3×3 matrix)

How could a function be defined everywhere but continuous at only one point?

The population of mosquitoes in a certain area increases at a rate proportional to the current population, and in the absence of other factors, the population doubles each week. There are 800,000 mosquitoes in the area initially, and predators (birds, bats, and so forth) eat 60,000 mosquitoes/day. Determine the population of mosquitoes in the area at any time. (Note that the variable t represents days.)  

Determine the entropy of the random variable which counts the sum of
three dice.

How to determine whether the following statements about big-O notation are true or false?

(a) Let f(n) = √ n log n − 4, then f(n) = O(n^ 2)

(b) Let f(n) = 4 n + 2 log^ 2 (n), then f(n) = O(log^ 2 (n))

(c) Let f(n) = 5 √ n + 2, then f(n) = Ω(log^ 4 (n))

(d) Let f(n) = 5 n^ 2 + 5 n log n + 4, then f(n) = O(n^3 )

(e) Let f(n) = 2 log^2 (n) + 5√ n + 7, then f(n) = Θ(√ n)

The number of hours worked by 24 employees of a company is given below:
40 43 40 39 36 44 40 39 39 52 27 50
41 47 40 48 38 36 25 41 35 36 16 40
(a) (6 points) Calculate the mean, variance and standard derivation for the given data
(b) (6 points) Calculate the three quartiles (Q1, Q2, and Q3) and the Interquartile range
(IQR)
(c) (4 points) Calculate the values of the lower fence and the upper fence for a boxplot.
(d) (5 points) Construct a box-and-whisker plot. Comment on the shape of the distri-
bution of the data. List any potential outliers, if any