Download Review for Cumulative Test

Name_____________________________________
Chapters 1 – 4A
Math 3
February, 2017
Review for Cumulative Test
On Wednesday 2/8 or Thursday 2/9, CP1 Math 3 will have a course-wide cumulative test
covering Chapter 1 (Functions and Polynomials), Chapter 2 (Sequences and Series), Chapter 3
(Statistics) and Chapter 4A (Trigonometry). You can expect the test to contain about 4 openresponse problems of a similar style and format to those you have had on other tests and quizzes.
This test will count as 5% of your year grade.
This review problem set does not include Chapter 4A (Sections 4.0-4.04) review problems since
we have completed these sections this past week.
Topic outline
Chapter 1: Functions and polynomials
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Fitting polynomial function formulas to tables
o Lagrange Interpolation
Key theorems and their proofs
o Euclidean Property
o Remainder Theorem
o Factor Theorem
o Corollaries about degree n polynomials
Factoring methods
o “Find a root, find a factor”
o Factoring quadratics including non-monic quadratics
o Difference of squares, sum of cubes, difference of cubes
o Perfect squares
o Factoring by grouping
o Substitutions for quadratic-like polynomials
Calculation skills
o Long division
o Simplifying, adding, and subtracting rational expressions
Chapter 2: Sequences and series
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Working with tables of numbers
o  columns for differences (sometimes repeated: 2, 3, etc.)
o  columns showing cumulative sums (sometimes repeated: , , etc.)
Sequence concept
o a sequence as a list of numbers
o a sequence as a function with domain of nonnegative integers (or all integers from some
starting value)
o visual examples: figurate numbers
Series concept
o definite and indefinite series
o writing in + notation or in  notation
o ways to visualize sums: staircases
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Formulas for sequences and series
o closed-form formulas for the kth term of a sequence or for the sum of an indefinite series
o recursive formulas for the kth term of a sequence or for the sum of an indefinite series
Evaluating definite and indefinite series
o arithmetic series using Gauss’s Method or its result
o geometric series using Euclid’s Method or its result
o k2, k3, k4, k5 using Bernoulli Formulas (these formulas will be given if needed)
o use of  identities to break series into simpler pieces that can be evaluated
Infinite Sequences and Series
o The real number a sequence approaches, if it exists
o The real number a series approaches, if it exists
Pascal’s Triangle and Binomial Theorem
o patterns, relationships, and symmetry in Pascal’s Triangle
o Binomial Theorem
Chapter 3: Statistics
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Mean, variance, and standard deviation
o Given a data set, find the mean, variance, and standard deviation.
o Given a random variable (data values with probabilities), find the mean (a.k.a. expected
value), variance, and standard deviation.
o Apply the fact that means and variances are additive.
o Calculate and interpret z-scores.
Repeated experiments
o Given the statistics (mean, variance, and/or standard deviation) for a single experiment, find
the statistics for the experiment repeated n times.
o Find probabilities using polynomial expansions (found by hand or using WolframAlpha)
or using Pascal’s Triangle numbers.
o For repeated Bernoulli trials, find the mean, variance, and standard deviation (using the
formulas on page 249).
Assessing the effectiveness of a treatment
o Identify appropriate randomization methods for selecting the treatment and control groups.
o Recognize outcomes that show strong evidence, possible evidence, or little or no
evidence that a treatment is effective.
Sample surveys
o Identify appropriate methods for random sampling.
o Given a sample proportion, use simulation results to assess what values are plausible for the
population parameter.
o Estimate margins of error.
o Understand that correlation does not imply causation.
o Understand the differences between sample surveys, observational studies, and experiments.
Probability distributions
o Make and interpret probability histograms.
o For normal distributions: apply the 68/95/99.7% rule and use the normalpdf and normalcdf
functions to answer probability questions.
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Chapter 4: Trigonometry (Sections 4.0-4.04)
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Understand and apply the unit circle definitions of the trigonometric functions
Find trigonometric function values using the unit circle, using special triangles (for 30°, 45°,
60°, and angles related to these), and in general using a calculator
Prove and apply the Pythagorean identity (sin2  + cos2 )
Find trigonometric function values when given other values using quadrant relationships, the
Pythagorean identity, and other identities
Solve simple trigonometric equations, using the inverse trig. buttons on the calculator.
Review Problems
Chapter 1: Functions and Polynomials
1. An input-output table for the polynomial function g(x) is given.
a. Identify a known factor of g(x).
b. If g(x) is divided by (x  2), what will the remainder be?
x
1
2
5
g(x)
3
4
0
c. Identify a division problem involving g(x) that would have a remainder of 3.
2. Consider the polynomial function p(x) = 2x 3 + x 2 -13x + 6 .
a. Calculate p(2) and use the result to identify a factor of p(x).
b. Divide p(x) by the factor found in part a to get a quadratic factor of p(x).
c. Factor the quadratic from part b into two linear factors.
d. List the zeros of p(x).
3. Perform the following calculation, expressing your final answer as a simplified rational
expression in factored form.
25x 2 - 9
x 2 + 2x - 8
5x - 3
x 2 + 4x
3
Chapter 2: Sequences and Series
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4. Tony showed the following work when evaluating the sum
å(3k - 4) . Find any mistakes
k=0
Tony made and explain why they are mistakes.
æ 12 öæ 12 ö 12
36(12∗13)
å(3k - 4) = çå3÷çåk ÷ + å-4 = 2 − 48
è k=0 øè k=0 ø k=0
k=0
12
5. Sasha reads about a man who was wearing a hat he bought for $4. When the man tried to
remove his hat, there was another, larger hat underneath it. When he tried to take that hat off,
there was yet another hat underneath it, too! Each hat was one and a half times as nice (and
as expensive) as the previous hat. Sasha wanted to describe the situation mathematically.
Would this be an arithmetic or geometric situation? Write a closed form and recursive
function for the situation. How would you find the sum of the values for the first 18 hats?
6. Evaluate each sum using an appropriate summation method:
a. 0.25 + 0.5 + 1 + … + 16
b. 17 + 13 + 9 + … - 51
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c.
å(3
k
k=2
+ 3k )
Chapter 3: Statistics
7. Suppose you are flipping a bent coin that lands heads 80% of the time and tails 20% of the
time. You flip the coin 9 times.
a. Describe the different ways that you can find the probability of getting exactly 6 heads.
Then find the probability using each of your methods.
b. Find the probability of getting at least 6 heads. Find the exact probability using one your
methods from part a and an estimate from the normal curve.
c. List the values for the number of heads that are within one standard deviation of the
mean.
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8. The probability that 0, 1, 2, 3, or 4 people will be placed on hold when they call a radio talk
show is shown in the table below.
𝑥
0
1
2
3
4
𝑃(𝑥)
0.18
0.34
0.23
0.21
0.04
Find the mean, variance and standard deviation for the data.
9. The following questions involve:
 a number cube (not the standard 1-to-6 number cube) whose values on a single roll have a
mean of 4 and a standard deviation of 1.5
 a spinner whose values on a single spin have a mean of 10 and a standard deviation of
2.5.
a. Suppose you roll the number cube and spin the spinner and add the results. What is the
mean and standard deviation for the sum? Show your work or explain how got your
answer.
b. Suppose you repeated the experiment described in part a 100 times, and the total sum
from all the experiments was 1302. Would this be a fairly typical result, an exceptionally
high result, or an exceptionally low result? Use statistics to justify your answer.
Answers
1. a. (x  5) is a factor of g(x) because x = 5 is a zero of g(x).
g(x)
b.
has a remainder of 4 because g(2) = 4.
x-2
g(x)
c.
has a remainder of 3 because g(1) = 3.
x +1
2. a. p(2) = 0 so (x  2) is a factor of p(x).
2x 3 + x 2 -13x + 6
b.
= 2x 2 + 5x - 3
x-2
c. (2x -1)(x + 3)
d. Zeros of p(x) are x = 2, x = ½, x = 3
5
3.
25x 2 - 9
2
2
x 2 + 2x - 8 = 25x - 9 × x + 4x = (5x + 3)(5x - 3) × x(x + 4) = x(5x + 3)
5x - 3
x 2 + 2x - 8 5x - 3
(x + 4)(x - 2) 5x - 3
x-2
x 2 + 4x
4. There are two major mistakes. First, the sum of a product does not equal the product of the
sums. Tony could have just done 3 times the sum of k. Second, there are thirteen 4s to be
added, making it 52 instead of 48.
12
 12  12
Here is a corrected calculation: å(3k - 4) = 3  k     4 = 3 12213  52 = 182.
 k 0  k 0
k=0
5. This is a geometric situation since each term is being multiplied by a common ratio. The
value of the nth hat can be represented by the equation 𝑣(𝑛) = 4(1.5)𝑛−1 or recursively by
4
if n  1

v ( n)  
v(n  1)  1.5 if n  1
You could use Euclid’s method or a sigma identity to evaluate the sum of the first 18 hats.
b. 306 c. 265911
æ9ö
7. a. Combinatorics: ç ÷ (0.8)6 (0.2)3 = 0.176 =17.6%
è6ø
Polynomial expansion: (0.8ℎ + 0.2𝑡)9 and identify the coefficient of ℎ6 𝑡 3 which is
6. a. 31.75
0.176 =17.6%
NormalPDF(6, 7.2, 1.2) = 0.201, which is an approximation and differs slightly from the
above answers because the coin is flipped a small number of times. This answer would
be closer to the above two answers if the coin was flipped many, many times.
æ9ö
æ9ö
æ9ö
æ9ö
b. ç ÷ (0.8)6 (0.2)3 + ç ÷ (0.8)7 (0.2)2 + ç ÷ (0.8)8 (0.2)1 + ç ÷ (0.8)9 = 0.914 or
è6ø
è 7ø
è8 ø
è9ø
Alternatively you could use a normal distribution, but again it’s not a very good
approximation. NormalCDF (6, 100, 7.2, 1.2) = 0.841, or you could estimate 84%
from the fact that 6 heads is one standard deviation (1.2) below the mean (7.2).
c. 6, 7, 8 (these need to be integer values because we are looking for # of heads)
8. 𝑥̅ = 1.59
Variance = 1.262
Standard Deviation = √1.262 = 1.123
9. a. The means are additive: mean of sum = 4 +10 = 14
Standard deviation is not additive, so you must find the variance first, which is additive:
s 2 = 2.25+ 6.25 = 8.5 . Then s = 8.5 = 2.915 .
b. x =100 ×14 =1400
s = 850 = 29.155
s 2 =100 ×8.5 = 850
Typical results will fall within 2 standard deviations of the mean: 1400 ± 2(29.155) , so the
range would be 1341.69 to 1458.31.
Since 1302 is more than 2 standard deviations below the mean, this is an exceptionally
low result.
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