Category: Mathematics

The integral of a function

Posted on | by Brian Leakey

Part I. Define Integration. What does it mean to take the integral of a function? How can
integration be used in real life applications? How is integration related to differentiation?
Part II. Go to
http://webspace.ship.edu/msrenault/geogebracalculus/geogebracalculusapplets.html This will
take you to a page “Calculus Applets using GeoGebra”. Scroll down to “The Integral” section,
numbers 25. – 30. Pick FOUR of the following:

  1. Introduction to Integration – The Exercise Bicycle Problem Part 1
  2. Introduction to Integration – The Exercise Bicycle Problem Part 2
  3. Introduction to Integration – Gaining Geometric Intuition
  4. The Riemann Sum
    Answer questions: What is the Riemann Sum? Why is it important in mathematics?
  5. The Area Function
  6. The Fundamental Theorem of Calculus, Part I (Theoretical Part)
  7. The Fundamental Theorem of Calculus, Part II (Practical Part)
    As you complete the above applets, make sure you clearly label which exercise you are
    completing and answer all EXPLORE questions associated with the applet on your paper to
    turn in. Also, make sure to screen shot the applet and inset picture(s) where appropriate.

http://webspace.ship.edu/msrenault/geogebracalculus/geogebracalculusapplets.htm

The integral of a function

Posted on | by Brian Leakey

Part I. Define Integration. What does it mean to take the integral of a function? How can
integration be used in real life applications? How is integration related to differentiation?
Part II. Go to
http://webspace.ship.edu/msrenault/geogebracalculus/geogebracalculusapplets.html This will
take you to a page “Calculus Applets using GeoGebra”. Scroll down to “The Integral” section,
numbers 25. – 30. Pick FOUR of the following:

  1. Introduction to Integration – The Exercise Bicycle Problem Part 1
  2. Introduction to Integration – The Exercise Bicycle Problem Part 2
  3. Introduction to Integration – Gaining Geometric Intuition
  4. The Riemann Sum
    Answer questions: What is the Riemann Sum? Why is it important in mathematics?
  5. The Area Function
  6. The Fundamental Theorem of Calculus, Part I (Theoretical Part)
  7. The Fundamental Theorem of Calculus, Part II (Practical Part)
    As you complete the above applets, make sure you clearly label which exercise you are
    completing and answer all EXPLORE questions associated with the applet on your paper to
    turn in. Also, make sure to screen shot the applet and inset picture(s) where appropriate.

http://webspace.ship.edu/msrenault/geogebracalculus/geogebracalculusapplets.htm

The statistical analyses

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The statistical analyses

 

Complete the statistical analyses that follow for the sample data below. If you do the work for this project by hand, you must show the work you do to arrive at your results. If you use technology (Excel, graphing calculator, etc.) to obtain the results, you must state the technology you used to obtain the results.
Project Data: Lengths of 30 random rivers in the world measured in miles
431 800 485
500 800 309
926 790 618
383 375 605
425 380 531
538 434 300
540 800 420
301 659 865
360 512 424
338 430 652
1. (10 points) Construct a grouped frequency distribution for the sample data. Use 6 classes. Use the minimum data value as the lower limit of the first class.
Class
Limits Class
Boundaries Frequency Relative
Frequency Cumulative Frequency
2. (3 points) Draw a histogram for the data set. Title the graph and label the axes appropriately.
3. (2 points) Does the distribution appear to be normal (yes or no)? Explain.
4. (10 points) Find the following descriptive statistics for the sample data.
Mean
Median
Mode(s)
Range
Standard Deviation
5. (3 points) Find the quartiles for the data.
Q1 = _______________
Q2 = _______________
Q3 = _______________
6. (2 points) Sketch a box plot. Do your best to draw it to scale.
7. (2 point) What is the IQR for this data set? ____________
8. (3 points) Outlier identification
Any value in this data set that is
less than ________ or greater than _______ is to be considered an outlier.
Therefore, the following values in this data set are outliers (write “none” if there are no outliers):
9. (3 points) Construct a 95% confidence interval for the population’s mean.
____________ < μ < ____________
10. (2 points) If someone were to ask you what this particular confidence interval means, what would you say?
11. (10 points) Test the hypothesis that the population’s mean is different from the sample’s median value. Use the traditional method. Use α = 0.05.
AlternatZve hypothesis (H1): ________________
Null hypothesis (H0): _____________________
Critical value: ____________
Test statistic: ____________
Conclusion:
Do you reject the null hypothesis (yes or no)? _________
If someone were to ask what the conclusion means in this particular study, what would you say?

 

 

Personality data

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please label your answers with a 1,2, or 3.
1= Pick one type of Personality data (S, I, L, or B). a. Describe/Define this type of data and b. tell me the pros and cons of using this type of data

2= Are some kinds of data “privileged” for some kinds of questions? For example, if a person says he is happy (S data), but his acquaintances say he is unhappy (I data), is it possible that the I data could be more valid than the S data? Would it be meaningful to say something like, “He’s not as happy as he thinks he is”
3=Define triangulation. Then, tell me why you think it is useful in assessing personality.

 

Using and Interpreting R^2.

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You have been introduced to the notion that R^2 can capture “the % of variation in Y explained by the regression model.” This interpretation holds for simple and multivariate regression (as you have seen: as long as the model contains an intercept.) It is important to note, however, that R^2 cannot be used formally to hypothesis test whether a model is “good” or “not.” Therefore, the magnitude is often open to interpretation. I have noticed that there is often a desire to identify a “global value for R^2” that suggests the model is “good.” (Global in the sense that the value could be used across all models, 0.70? 0.80? 0.90?) Often I am asked, “what is a good value for R^2?”

 

 

 

ALGEBRA AND LOGIC GATES

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Section 1[ALGEBRA AND LOGIC GATES]: Show all necessary calculations.

1. Consider the following list of numbers. Your job is to erase as few of these numbers as possible such that the remaining numbers appear in increasing order. For example, erasing everything except the first two numbers laves an increasing sequence; erasing everything except for the first, third, sixth and eighth numbers does the same. [2]
9 44 32 12 7 42 34 92 35 37 41 8 20 27 83 64 61 28 39 93 29 17 13 14 55 21 66 72 23 73 99 1 2 88 77 5 65 83 84 62 5 11 74 68 76 78 67 75 69 70 22 71 24 25 26

2. Solve the above exercise [1], such that the remaining numbers are in decreasing order [2]

3. Solve the following simultaneous equations for x_1,x_2,x_3,and x_4
E_1: x_1+x_2+0 +3x_4=4
E_2: x_1+x_2-x_3 +x_4=1
〖 E〗_3: 3x_1-x_2-x_3 +〖2x〗_4=-3
〖 E〗_4: – x_1+〖2x〗_2+3x_3-x_4=4
[4]

4. A foundation student bought a mobile phone and smart watch for a total of £1,250, excluding VAT (value added tax), paying 3 times as much for the mobile phone
as for the smart watch.
Write an expression to show the total cost in terms of the cost of the smart watch. [2]
What was the cost of each item? [2]
Calculate the VAT on the mobile phone, assuming a VAT rate of 14% [2]

5. Find the value of k to satisfy the equation referring to the discharge of water from a tank
t_1-t_2=2A/a √((1+k)/2g) (√(h_1 )-√(h_2 ))
given that t_2=0.45,t_1=2.6,A=15.6,a=12.57/144 ,g=32.2,h_1=36 and h_2=25 [2]
6. Construct a chart giving values of u and v to satisfy the equation 1.2v+0.64u=0.85 the range of u being 5 to 100.(Use Excel) [2]

7. Construct a chart giving values of u and v to satisfy the equation 2u+7v=52 the range of v being 2 to 50. (Use Excel) [2]
8. Consider the following:
An online trading company wants to offer discounts to new customers. The company has recently emailed discount code to the customers. New customer must have a discount code to be eligible but returning customer are not eligible for a discount.
Let
A = Returning Customer
B = Discount Code
(a) Draw a truth table to represent when a customer has a discount. [2]
(b) From your truth table in (a) produce a Boolean expression to represent discounted customer (Q) for the trading company. [2]

Section 2[Statistics]
9. The data below shows the monthly rent for 24 students: 1500, 1350, 350, 1200, 850, 900, 1500, 1150, 1500, 900, 1400, 1100, 1250, 600,610, 960, 890, 1325, 900, 800, 2550, 495, 1200, 690.
(a) Create a relative frequency distribution table using 7 classes
(b) Use your table in (a) to produce a histogram to represent your data [3]

10. The following is a frequency table showing the ages of members of a symphony orchestra for young adults.
Age Frequency
15 2
16 5
17 11
18 9
19 14
20 13
21 10
22 13
Find the sample mean and median of the ages of the 77 members of the symphony. [3]

Section 3 [Flowchart and Algorithm Development]

11. Given the following list of steps in the admission process:
Search for a school
Prepare for the admission test and write test
Did you pass the exam?
Submit necessary documents and get admission
End
Are seats available?
Start
Is there an admission test?
Re-arrange the above steps in the correct order and use MS Visio to draw a flowchart for the admission process. [5]

12. Construct an algorithm that has as input an integer n≥1,n+1 points and x_0,x_1,⋯x_n and a point x and which produces as output the product
P=(x-x_0 )(x-x_1 )⋯(x-x_n ) [5]

 

 

A topic in probability that interests you

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Research a topic in probability that interests you. Explain the importance of this data and what you find interesting about the results. Possible topics include polygraphs and telling the truth, finding a blood type for a transfusion or kidney for a transplant, right-handedness or left-handedness in athletics, types and use of social media in different countries, or the chance of getting a disease overseas. Describe the example, the probabilities, and what you can determine given your research. Use a website reference for your data and cite your source.