Download Week 1 Lecture Notes - NIU Math

1. Math 101
Core Competency in Mathematics
Dept. of Mathematical Sciences
Northern Illinois University
http://www.math.niu.edu
Professor Richard Blecksmith
[email protected]
http://www.math.niu.edu/∼richard/math101
2. Core Competency
The mathematics core competency requirement can be satisfied
by
• passing Math 101 (with a D or higher)
• obtaining a grade of C or better in
– Math 155 (trig)
– Math 201 (math for elementary education)
– Math 206 (discrete mathematics)
– Math 210 (finite mathematics)
– Math 211 (business calculus)
– Math 229 (calculus)
• obtaining credit for one of the courses listed above, except
Math 101, through credit by examination (Advanced Placement)
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3. Core Competency Cont’d
• obtaining a grade of C or better in Stat 208, Stat 301, Stat
350, Ieng 335, or Ubus 223; and obtaining
– a grade of C or better in Math 110
– an ACT mathematics score of at least 24
– an SAT mathematics score of at least 560
– an A- or B-level placement on the mathematics placement exam
• obtaining equivalent transfer credit
• passing the Mathematics Core Competency Exam
4. Text
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Text: Mathematical Thinking and Quantitative Reasoning
By: Linda Sons, Peter Nicholls, and Joseph Stephen
third edition
Kendall/Hunt Publishers
Plus Additional Handouts
Available on the Math 101 Webpage
5. Instructor
Prof. Richard Blecksmith
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Office: Watson Hall 344
Phone: (815)753-1835
Office Hours: M W 3–4, W 1–2
and by appointment
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6. Recitation
• Frank Gambino
– B1 Tues 2:30–3:20 DU 340
– B3 Tues 3:30–4:20 DU 418
– B5 Thurs 2:30–3:20 FW 201
– B7 Thurs 3:30–4:20 DU 348
• Kenny Albright
– B2 Tues 2:30–3:20 WZ 103B
– B4 Tues 3:30–4:20 DU 268
– B6 Thurs 2:30–3:20 DU 340
– B8 Thurs 3:30–4:20 DU 318
7. Important Dates
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Test 1: Wed Sept 27, 2006
Last Day to Withdraw: Fri Oct 20, 2006
Test 2: Fri Oct 27, 2006
Test 3: Mon Nov 20, 2006 Thanksgiving week
All makeups will resolved at the end of the semester
Final Exam: Wed Dec 13, 8-9:50 P.M.
8. Tests and Quizzes
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3 tests worth 100 points each
Final exam worth 200 points
Homework (9/11) for 90 points
Recitation Quizzes (9/11) for 90 points
In-class MiniQuizzes (10/11) for 30 points
Projects (2) for 20 points each
Total: 750 points
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9. Grading Scale
The scale will be at least as generous as
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A: 637 points (85%)
B: 562 points (75%)
C: 450 points (60%)
D: 375 points (50%)
Note: passing the final exam (with a score of at least 100
points out of 200) automatically guarantees that you will
pass the course.
10. How to Pass the Course
To get a C in this course, turn in the all the homework and
study for the quizzes and miniquizzes
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Homework: 85 pts
Recitation Quizzes: 80 pts
Miniquizzes: 25 pts
Projects: 40 pts
Total: 230 points
11. How to Pass the Course
Now suppose you know how to solve 30% of the test questions:
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30% of 500 = 150 points
Guess on the remaining 350 points
The probability of guessing correctly is 1/5
1/5 of 350 = 70 points
Total: 220 points
230 + 220 = 450, the guaranteed cutoff for a C.
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12. Observations
• This strategy requires that you:
– work very hard on the non test part of the course
– know something (30 percent of the material)
– are an average guesser
• The secret is to
– get as many easy points as you can
– come to class and recitation regularly
– work consistently
13. Calculators
• You will need a calculator for this course. It needs to have
– an exponentiation button (ˆ or xy )
– parentheses buttons ( and )
– at least one memory
• There are two types of calculators which you may use:
– scientific or business
– graphing calculator such as the TI83
14. Some calculating problems
• Compute 2 × 3 + 5
– Pressing 2 × 3 + 5 = on your calculator gives the answer:
11,
– as expected.
• Compute 2 + 3 × 5
– Pressing 2 + 3 × 5 = on your calculator gives the answer:
17,
6
– which may be a surprise if you expected your calculator
to add 2 + 3 to get 5 and then multiply by a seond five
to get 25 for an answer.
15. What happened?
• When evaluating an expression involving
+,
−,
×, and ÷,
without parentheses, your calculator will
compute the multiplication and divisions first
• Your calculator computes 2 + 3 × 5 = as
2 + (3 × 5) = 2 + 15 = 17.
16. Another Example
2+3
• Suppose you want to calculate
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• If you press the buttons 2 + 3/5 = on your calculator, the
answer will be 2.6
• Without parentheses, your calculator is thinking
2 + 3/5 = 2 + 0.6 = 2.6
• To make your calculator add the 2 and 3 first,
• you must use parentheses:
(2 + 3)/5 =
• which gives the correct answer 1
17. Order of Operations
• 1. Parenetheses
• 2. Functions (such as square root)
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3. Exponentiation
4. Multiplication and Division
5. Addition and Subtraction
.05 240
Example: To calculate 1000 1 +
12
• Enter: 1000 × (1 + .05/12)ˆ240 = to get the answer
• 2712.64
18. Find the Maximum
• Which is larger? 624 or 2510 ?
• My calculator says
– 624 = 4.73838133832E18
– 520 = 9.53674316406E13
• The second number is NOT larger than the first
• even though 9.5 > 4.7.
• Why not?
19. Scientific Notation
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These two numbers are written in scientific notation
624 = 4.73838133832 × 1018
520 = 9.53674316406 × 1013
624 is a 19 digit number
520 is only a 14 digit number
So 624 is the larger number
Can you think of a way to tell which number is bigger without using a calculator?
20. Precision
• How many digits should you write down?
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– As many as on the calculator display?
– Just 2 or 3?
• The correct answer is: as many as you need to solve the
problem.
• Common sense should be your guide: For example, when
working with money
– An exta value meals costs $3.55
– My new car lists for 16, 549
– Bill Gates is worth 55 billion dollars
21. Precision Cont’d
• As a rule of thumb, your answer should be to an many
significant digits as the data you used in the calculations.
• When giving a purely mathematical solution, however, you
should avoid saying something like:
• The square root of 2 is 1
arguing that your answer is accurate to one digit, because
2 is just a one digit number.
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• Instead, use at least four digits: 2 = 1.414
22. The Quadratic Formula
The equation ax2 + bx + c = 0 has the solution
√
−b ± b2 − 4ac
x=
2a
• Find the roots of the equation 2x2 + 5x − 11 = 0
• Plug a = 2, b = 5, and c = −11 into the formula
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52 − 4 · 2 · (−11)
√ 2·2
−5 ± 52 + 4 · 2 · 11
=
2·2
x=
−5 ±
23. Quadratic Formula Cont’d
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On your
compute these two answers separately
√ calculate,
2
(−5 +
(5 + 4 × 2 × 11))/(2 × 2) =
Answers:
√ 1.4075
(−5 −
(52 + 4 × 2 × 11))/(2 × 2) =
Answers: −3.9075
24. Using the Memory
There is a short cut in computing the previous problem.
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The expression 52 + 4 × 2 × 11 was computed twice.
To save time, you could compute it once and save the result in
memory.
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On
√ a 2graphing calculator, compute
(5 + 4 × 5 × 11) → X
Then the two roots are calculated
(−5 + X)/4 = and (−5 − X)/4 =
On a scientific calculator, use the Store Memory and Recall
Memory keys.
25. Size comparisons
• Which is larger: a liter or pepsi or a quart of coca-cola?
What is the difference in ounces?
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• In American track, the quarter mile race has been replaced
by the 400 meter race. Which is larger: a quarter of a mile
or 400 meters? What is the difference in feet?
• Which is larger: a football field (between the sidelines and
from endzone to endzone) or one acre? What is the difference in square feet?
• Are you over one billion seconds old? Are your parents?
Exactly how many years old is this?
26. Conversions
Metric to U.S.
1 millimeter [mm]
0.03937 in
1 centimeter [cm] 10 mm 0.3937 in
1 meter [m]
100 cm 1.0936 yd
1 kilometer [km] 1000 m 0.6214 mile
U.S. to metric
1 inch [in]
1 foot [ft]
12 in
1 yard [yd]
3 ft
1 mile
1760 yd
1 int nautical mile 2025.4 yd
2.54 cm
0.3048 m
0.9144 m
1.6093 km
1.853 km
27. How Big is an Acre?
Which is bigger: an acre or a football field?
• Webster Unabridged Dictionary: An acre is a U. S. and
English measurement, meant to represent the amount of
land a yoke of oxen could plow in one day.
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• One acre equals 160 square rods
• What’s a rod?
• By Webster again, one rod equals 5.5 yards
28. An Acre Cont’d
• 1 square rod = 5.52 square yards = 30.25 square yards
• 1 acre = 160 × 30.25 square yards = 4840 square yards
• Now a collegiate football field measures 100 by 53 13 yards.
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a football field = 5333 square yards
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• So a football field is about 10% bigger than an acre.
29. A Football Field
1 acre
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10
20
30
40
50
40
Go Huskies!
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20
10
0
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30. Area Conversions
Metric to U.S.
1 sq cm [cm2 ] 100 mm2
0.1550 in2
1 sq m [m2 ] 10,000 cm2 1.1960 yd2
1 hectare [ha] 10,000 m2 2.4711 acres
1 sq km [km2 ]
100 ha
0.3861 mile2
U.S. to metric
1 sq inch [in2 ]
1 sq foot [ft2 ]
144 in2
1 sq yd [yd2 ]
9 ft2
1 acre
4840 yd2
1 sq mile [mile2 ] 640 acres
6.4516 cm2
0.0929 m2
0.8361 m2
4046.9 m2
2.59 km2
31. Fluids and Weights
Metric to U.S.
1 gram [g]
1,000 mg
1 kilogram [kg] 1,000 g
1 tonne [t]
1,000 kg
1 liter [`]
0.0353 oz
2.2046 lb
1.1023 ton
1.0569 qt
U.S. to metric
1 fluid ounce
29.574 ml
1 pint
16 fl oz 0.4731 `
1 gallon
4 qt
3.7854 `
1 pound [lb] 16 oz 0.4536 kg
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32. The Price of Gasoline
A few years ago, I put the following problem on an exam:
In Canada gasoline costs 74.9 Canadian cents per liter. The
money exchange rate is
1.2374 Canadian dollars = 1 U. S. dollar.
The metric conversion is
1 liter = .26417 gallons.
What is the price in U.S. currency for one gallon of gasoline in
Canada?
Criticize the following actual student solutions:
$16,540
$0.023
33. What’s the correct answer?
dollar
S. dollar
liter
0.749 Canadian
× 1.23741 U.
liter
Canadian dollar × .26417 gallons
=
0.749
U. S. dollar
1.2374×.26417 gallon
S. dollar
= 2.29 U. gallon
Two years ago this seemed expensive
Now it seems cheap!
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34. Killer Summation
• The computation S = 1 + 2 + 3 + 4 + 5 + · · · + 999 + 1000
could be done on a hand calculator by entering each number,
starting with 1 and ending with 1000.
• The chances for making an error, by pressing a wrong key
while entering a digit, is very great.
• Besides, it takes a long time to enter a thousand numbers!
35. A better way:
• Write the list once, and directly below it, write the same
list with the numbers in reverse order:
S =1 + 2 + 3 + 4 + 5 + · · · + 999 + 1000
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S =1000 + 999 + · · · + 5 + 4 + 3 + 2 + 1.
• Now add each entry in the top row with the entry in the
row directly below it: 1 + 1000 = 1001, 2 + 999 = 1001,
3 + 998 = 1001, etc.
• You get 1000 sums of 1001.
• So 2S = 1000 × 1001 = 1001000.
• Dividing by two gives us the value
S = 1001000/2 = 500500.
36. Exercises:
Try the “backwards” method on:
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1. 1 + 3 + 5 + 7 + 9 + · · · + 95 + 97 + 99.
2. 11 + 12 + 13 + · · · + 98 + 99 + 100.
3. 5 + 10 + 15 + 20 + · · · + 190 + 195 + 200.
4. 2 + 5 + 8 + 11 + 14 + 17 + · · · + 992 + 995 + 998.
Sequences like these are called arithmetic progressions.
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• Note that we obtain each term by adding the same number
to the previous term.
37. Another Sequence
• A second type of sum we will meet in this class is obtained
by multiplying the previous term by a common factor.
• These are called geometric progressions.
• If the first term is 1, then the sum of a geometric progression
is S = 1 + x + x2 + x3 + · · · + xn−1 + xn .
• The trick for adding a geometric progression is to multiply
S by x on a line just above S:
38. Geom. Sequences Continued
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xS =x + x2 + x3 + · · · + xn−1 + xn + xn+1
S =1 + x + x2 + x3 + · · · + xn−1 + xn .
• Now subtact the second line from the first and notice that
almost every term in the first row cancels a term in the
seond row.
• The only uncancelled terms are the xn+1 in line 1 and the
1 in line 2.
• Hence xS − S = xn+1 − 1
39. Geom. Sequences Completed
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Start with: xS − S = xn+1 − 1
Pull out S: (x − 1)S = xn+1 − 1
Divide by x − 1 to get a formula for S:
n+1
S = x x−1−1
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xn+1 − 1
• 1 + x + x + ···x =
x−1
2
n
40. Example
Compute the sum 1 + 2 + 4 + 8 + 16 + 32
• 1 + 2 + 4 + 8 + 16 + 32 = 1 + 21 + 22 + 23 + 24 + 25
• In the formula
xn+1 − 1
2
n
• 1 + x + x + ···x =
x−1
• x = 2 and n = 5
• Plugging these values into the formula gives
6
−1
• 1 + 2 + 4 + 8 + 16 + 32 = 22−1
• Since 26 = 64
63
• 1 + 2 + 4 + 8 + 16 + 32 = 64−1
2−1 = 1 = 63