Download Review for cumulative test

Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 1
Review for cumulative test
On Friday, January 23, Honors Math 3 will have a course-wide cumulative test covering
Chapters 1-4. You can expect the test to contain 4-to-6 open-response problems of a similar style
and format to those on the chapter tests. This test will count as an ordinary test in your Quarter 3
grade.
Following this topic outline, you’ll find a set of review problems for each chapter, with answers.
Topic outline
Chapter 1: Functions and polynomials





Fitting polynomial function formulas to tables
o Lagrange Interpolation (p. 17)
o “Sasha’s” alternate method (p. 22 exercise 5)
o adding a term to make a formula fit a table at an additional point (p. 25)
Key theorems and their proofs
o Euclidean Property (p. 36)
o Remainder Theorem (p. 37)
o Factor Theorem (p. 41) and its graphical interpretation about x-intercepts
o Corollaries about degree n polynomials (pp. 42-43)
Polynomial graphs (from handouts)
o End behavior based on degree and leading coefficient
o Multiplicity of factors: just touching vs. passing through x-axis
o Fitting formulas to graphs, including finding the leading coefficient
Factoring methods
o “Find a root, find a factor” (p. 59)
o Factoring quadratics (p. 51) including non-monic quadratics (p. 52)
o Difference of squares, sum of cubes, difference of cubes (p. 55)
o Perfect squares (p. 55) and perfect cubes (p. 60 exercise 1)
o Factoring by grouping (p. 57)
o Substitutions for quadratic-like (p. 62) and cubic-like (p. 63) polynomials
o 4th degree with hidden difference-of-squares (p. 64)
o Factoring over R (p. 65)
Calculation skills
o Long division (p. 35) and synthetic division (Challenge Problem Set 1)
o Simplifying, adding, and subtracting rational expressions (p. 69)
Chapter 2: Sequences and series


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
Name
January, 2015





Honors Math 3 review problems
cumulative: Chapters 1-4 page 2
Series concept
o definite and indefinite series
o writing in + notation or in  notation
o ways to visualize sums: staircases, pyramids
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
Limits
o limits of sequences
o limits of series
o limits of geometric series: formula 1ar
o visual examples: “fractals” (pp. 149-151 # 3, 4, 9); Achilles paradox (p. 151 #8)
o repeating decimals using limits of geometric series
Pascal’s Triangle and Binomial Theorem
o patterns, relationships, and symmetry in Pascal’s Triangle
o summation properties: “hockey stick” (pp. 160-161), ///etc. on diagonals (p. 163)
o even/odd and remainder patterns (p. 162)
o Binomial Theorem: use and informal justification
Chapter 3: Statistics




Statistics for a data set
o Find the mean, mean absolute deviation, variance, and standard deviation.
o Prove and use this alternate formula for variance: x 2  (x ) 2 .
o Prove and use that means and variances are additive.
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 For repeated Bernoulli trials, find the probability that the total has a specific value (using the
formula involving a combination number).
o For repeated Bernoulli trials, find the mean, variance, and standard deviation.
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 with 95% or 99+% confidence.
o Understand that correlation does not imply causation.
o Understand the differences between sample surveys, observational studies, and experiments.
Name
January, 2015

Honors Math 3 review problems
cumulative: Chapters 1-4 page 3
Probability distributions
o Make probability histograms.
o For normal distributions, apply the 68/95/99+% rule and use the normalpdf and normalcdf
functions to answer probability questions.
o Apply the Central Limit Theorem to identify normal distributions.
Chapter 4: Trigonometry










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 trigonometric equations by hand, using inverses on the calculator, and graphically on
the calculator
Graph the functions sin x, cos x, and tan x and identify the periods of the graphs
Prove and apply the angle sum identities (formulas for sin( + ) and cos ( + ), see p. 346)
Prove and apply the Law of Sines and Law of Cosines.
Solve triangles (find unknown sides and angles) when given SSS, SAS, ASA, or AAS.
Solve triangles in the potentially ambiguous case SSA.
Review problems
Chapter 1: Functions and polynomials
1. Find a function formula that matches the given table. Do this using two different methods
(Lagrange interpolation 1.02 and “Sasha’s agree-to-disagree method” 1.03).
Input, x
Output, f(x)
1
-6
3
84
5
750
7
2856
9
7650
2. Suppose you have a table that has six entries. What is the greatest possible degree polynomial
that you would need to match the table? What is the least possible degree of polynomial that
you would need?
3. a. Find the quotient and remainder when you divide f (x)  2x 3  4 x 2  x 1 by x  3 .
b. Find f (3) without plugging in.
4. Suppose P(x)  x 3  2x 2  3x  4 . Use the Remainder Theorem to showthat P(x)  P(5) is

divisible by x  5.




Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 4
5. Factor each expression (over Z except where it says otherwise).
4
2
a. 4 x 19x  5
6
b. x  8


c.
x 4 11x 2  36
3
2
d. x  x  8x  6 , factoring over R
 6. Find polynomial function formulas that could fit each of these graphs. Remember that you
will need to determine the value of the leading coefficient.

a.
b.
Chapter 2: Sequences and series
7. Evaluate each sum below using Gauss’ method or Euclid’s method. Show all steps of each
method you use!
a. 3 + 8 + 13 + … + 78
b. 2 + –6 + 18 + … + –486
8. Write each sum below in sigma notation, then evaluate using sum identities.
a. the sum of the even integers from 4 to 68, inclusive
b. the sum of the first 8 terms of the geometric sequence with initial term 12 and common
ratio ½
c. the sum of the first n terms of the arithmetic sequence with first term 9 and fourth term –3
d. the sum of the first n terms of the geometric sequence 36, 12, 4, …
9. Use the given table for f(n) to answer these questions.
a. Fill in the  column of cumulative sums.
n
0
f(n)
–3
b. Write a function formula for f(n).
1
2
2
7
3
12
4
17
c. Express the series associated with f using

notation.
d. Find a closed-form function formula for the series associated with
f. Show how you get it.
Σ
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 5
10. Find a closed-form formula that agrees with this recursively defined function:
ìï
3
if n = 0
f (n) = í
2
ïî f (n -1) + 2n + 7n - 5 if n > 0
11. The Koch snowflake is formed as follows. Begin with an equilateral triangle. On the middle
third of each of the sides attach an equilateral triangle pointing outward, then remove the
original middle third. Repeat this process forever. Stages 0 through 3 are shown below.
a. Find the ratio between the area of an equilateral triangle with side lengths
1
s and the
3
area of an equilateral triangle with side lengths s.
b. If the area of the initial triangle is 1 unit, find the area of the Koch snowflake in stage n.
Then find the area of a true Koch snowflake, formed by infinitely many iterations.
12. a. Draw the first eight rows of Pascal’s triangle.
æ
ö æ
ö æ
ö
ö æ
ö æ
b. Find the sum ç 8 ÷ + ç 4 ÷ + ç 7 ÷ + ç 6 ÷+ ç 5 ÷.
è 3 ø è 2 ø è 2 ø è 4 ø è 3 ø
c. Use the Binomial Theorem to expand and simplify the expression (x – 3y)5.
Chapter 3: Statistics
13. When you subtract a constant c from each element in a data set, what happens to the mean
and what happens to the standard deviation? Prove your claims.
14. Weather data from a recent snowstorm shows that the measured snow accumulation from
many locations in Lexington had approximately a normal distribution with a mean of 8.5"
and a standard deviation of 1.1".
a. You ask two of your friends who live in Lexington about the snow depths in their yards.
They say 7" and 12". Discuss the likelihood of each of these claims. Include
computations of z-scores as part of your analysis.
b. A news story on the storm said “Lexington was blanketed by between 7 and 10 inches of
snow.” What percent of locations received an amount of snow that was in this interval?
c. What is the probability that a location in Lexington got less than 6" of snow?
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 6
15. A researcher investigates whether snow tires are more effective than regular tires at
preventing automobile accidents. Suppose that a researcher conducts a well-designed,
appropriately-randomized statistical study of this question, and tabulates how many cars
had an accident over a period of three winters.
Make two copies of this table then complete it as directed below.
accident
no accident
total
snow tires
1500
regular tires
1500
total
200
2800
3000
a. Complete the table below in such a way that there would be strong evidence that snow
tires helped prevent accidents.
b. Complete the table below in such a way that it would be difficult to say whether or not
the snow tires helped prevent accidents (that is, such that there’s possible evidence but
not strong evidence).
16. A snow tubing area has 400 snow tubes that are used each day. Each tube has as 1% chance
of needing repair after each day of snow tubing.
a. Calculate the probability that there will not be any snow tubes needing repair after a day.
b. Calculate the mean and standard deviation of the number of snow tubes needing repair on
each day.
c. Calculate the probability that the number of snow tubes needing repair will be 10 or
more.
17. The snow tubing area surveys randomly-selected customers on a particular day about how
many times they have visited so far this season.
Number of snow tubing visits How many customers
1
133
2
43
3
17
4
2
5
5
>5
0
a. Calculate the mean, variance, and standard deviation of number of visits per customer.
b. Does this data have a normal distribution? Explain.
Chapter 4: Trigonometry
18. Solve the following equations over the given domain. Find exact answers where possible.
If you use your calculator, solve for degree values to the nearest tenth.
a. cos x  2  3 cos x
for general solution(s) where x is in degrees
b. sin 2 x  sin x  cos 2 x
for 0° ≤ x ≤ 360°
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 7
c. 2 cos 2 x   3 cos x
for 0° ≤ x ≤ 360°
d. 4 cos 2 x  1
for –180° ≤ x ≤ 180°
e. sin 2 cos   cos 2 sin  
2
for general solution(s) where x is in degrees
2
19. Find the exact value for cos 15 two ways by applying two different identities.
20. Find the area of quadrilateral ABCD.
D
55°
A
8
120°
B
C
10
21. Solve ABC given a = 10, b = 12.93, and C = 20.5°
22. A hot air balloon is seen over Boston simultaneously by two observers at points A and B that
are 1.75 miles apart. Their angles of elevation are 33 and 37  respectively. How high above
the ground is the balloon?
Hot Air Balloon
37°
33°
A
B
1.75 miles
23. Use two different methods (unit circle and angle sum identities) to prove that
cos180      cos 
24. a. Sketch a graph of y  tan x 0  x  360
b. Use your graph to determine the number of solutions to the equation tan x 
1
2
1
c. Use x  tan 1   to find all values of x for 0  x  360 to the nearest degree.
2
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 8
Review problem answers
(first draft; if any answers don’t look right, check with teacher)
Chapter 1: Functions and polynomials
f (x)   641 (x  3)(x  5)(x  7)(x  9)  78 (x 1)(x  5)(x  7)(x  9)
119
 375
32 (x 1)(x  3)(x  7)(x  9)  4 (x 1)(x  3)(x  5)(x  9)
1. Lagrange:
 1275
64 (x 1)(x  3)(x  5)(x  7)
“Sasha”:
f (x)  6  45(x 1)  72(x 1)(x  3) 18(x 1)(x  3)(x  5)
 (x 1)(x  3)( x  5)(x  7)

Both equivalent to f (x)  x 4  2x 3  4 x 2  5x .
2. Greatest degree 5; least degree 0.

3. a. quotient 2x 2  2x  7 ; remainder -22
b. –22

4. P(x)  P(5)  x 3  2x 2  3x  90. When substituting 5 for x, the result is 0. So, P(x) – P(5) is
divisible by x – 5.

5. a. (4 x 2 1)(x 2  5)
b. (x 2  2)(x 4  2x 2  4)

c. (x 2 - x + 6)(x 2 + x + 6)
d. (x  3)(x 1  3)( x 1 3)
It’s OK if your leading coefficients are slightly off.
 6. Here are possible answers. 
a. f (x)  14 (x  2) 3 (x 1) 2using (0, -2)
b.
f (x)   12 x(x  2) 2 (x  3) 3 using (-2, 16)
 Chapter 2: Sequences and series
7. a. See the MIA on p. 91 for the steps required for Gauss’ method. S = 648.

b. See the Example on p. 92 for the steps required for Euclid’s method. S = –364.
8. a.
32
32
32
32
32
k=0
k=0
k=0
k=0
k=0
å(4 + 2k) = å 4 + å2k = 4å1 + 2åk = 4(33) +
2(32)(33)
=1188
2
æ æ 1 ö8 ö
ç ç ÷ -1 ÷
7
7
æ1ö
æ1ö
è2ø
÷ , which simplifies to 23 29
b. å12 ç ÷ = 12åç ÷ = 12 ç
ç æ1ö ÷
è2ø
è ø
32
k=0
k=0 2
ç çè 2 ÷ø -1 ÷
è
ø
k
k
c. Let d be the common difference. Then 9 + 3d = -3 , so d = –4.
n-1
n-1
n-1
n-1
n-1
k=0
k=0
k=0
k=0
k=0
å(9 - 4k) = å9 + å-4k = 9å1 - 4åk = 9n -
4(n -1)(n)
= -2n 2 +11n
2
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 9
d. Series is geometric with a = 36, r = 1/3.
æ æ 1 ön ö
ç ç ÷ -1 ÷
k
k
n-1
n-1
ææ 1 ön ö
æ1ö
æ1ö
è 3ø
ç
÷
ççç ÷ -1÷÷
36
=
36
=
36
=
-54
å çè 3 ÷ø
åçè 3 ÷ø
ç
1
÷
èè 3 ø
ø
k=0
k=0
ç 3 -1 ÷
è
ø
9. a.
b.
c.
d.
n
0
f(n)
–3
Σ
–3
1
2
–1
2
7
6
3
12
18
4
17
35
f (n) = 5n - 3
n
n
k 0
k=0
 f (k ) or å(5k - 3)
n
n
n
n
n
k=0
k=0
k=0
k=0
k=0
å(5k - 3) = å5k + å-3 = 5åk - 3å1 =
5(n)(n +1)
5
1
- 3(n +1) = n 2 - n - 3
2
2
2
10. See the “unstacking” method on p. 119, and the For Discussion prompt on p. 120.
n
n
n
k=1
k=1
3+ 2å k + 7å k - 5å1 = 3+ 2
2
k=1
(n)(n +1)(2n +1)
n(n +1)
2
9
7
+7
- 5n = n3 + n 2 - n + 3
6
2
3
2
6
11. a. The area of the equilateral triangle with sides
1
1
s is the area of the triangle with sides s.
3
9
b. area = initial area + area added stage 1 + area added stage 2 + … + area added stage n
æ1ö
æ1ö
æ1ö
area = 1+ 3ç ÷ +12 ç ÷ + 48ç ÷ +... + area added stage n
è9ø
è9ø
è9ø
2
3
With the exception of the first 1, the rest is a geometric series with a = 1/3 and r = 4/9.
æ æ 4 ön ö
ç ç ÷ -1 ÷
n
n-1
ö
1æ 4ö
1 n-1 æ 4 ö
1çè 9 ø
3 ææ 4 ö
÷
So total area at stage n: 1+ å ç ÷ = 1+ åç ÷ = 1+
= 1- ççç ÷ -1÷÷ .
è ø
3 k=0 è 9 ø
3 ç 4 -1 ÷
5 èè 9 ø
ø
k=0 3 9
ç 9
÷
è
ø
k
k
As n approaches infinity, area at stage n approaches 1-
3
8
( 0 -1) , which simplifies to .
5
5
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 10
12. a.
æ
ö
b. Use Pascal’s triangle, where we know ç n ÷ is the entry in row n, entry k. Remember we
è k ø
start by labeling the first row n = 0, and the first entry in a row k = 0.
æ 8 ö æ 4 ö æ 7 ö æ 6 ö æ 5 ö
ç
÷+ç
÷+ ç
÷ = 56 + 6 + 21 + 15 + 10 = 108.
÷+ç
÷+ç
è 3 ø è 2 ø è 2 ø è 4 ø è 3 ø
c. By Binomial Theorem,
æ 5 ö 5
æ 5 ö 4
æ 5 ö 3
æ 5 ö 2
æ 5 ö 1
æ 5 ö 0
(x - 3y)5 = ç
÷ x (-3y)0 + ç
÷ x (-3y)1 + ç
÷ x (-3y)2 + ç
÷ x (-3y)3 + ç
÷ x (-3y)4 + ç
÷ x (-3y)5
0
1
2
3
4
5
è
ø
è
ø
è
ø
è
ø
è
ø
è
ø
æ
ö
Use the entries in row n = 5 of Pascal’s triangle to find values of ç 5 ÷ , then simplify.
è k ø
Final expansion: (x - 3y)5 = x 5 -15x 4 y + 90x 3 y2 - 270x 2 y3 + 405x y 4 - 243y 5
Chapter 3: Statistics
13. The mean decreases by c. The variance stays the same.
Proof (mean): Suppose x1, …, xn have a mean of x . Then x1–c, …, xn–c have a mean of
( x1  c)    ( xn  c) x1    xn nc


 x c.
n
n
n
Proof (std. dev.): Since each data value decreases by c and the mean also decreases by c,
all of the deviations are unchanged, so the squares of the deviations are unchanged, so the
variance and the standard deviation are unchanged.
14. a. The 7" amount is between 1 and 2 standard deviations below the mean (z ≈ –1.36), so
lower than average but well within the interval [6.3, 10.7] where 95% of the data would
lie. However, the 12" amount is more than 3 standard deviations above the mean
(z ≈ 3.18) where far less than 1% of the data would fall. So the 7" amount was in the
typical range but the 12" was extraordinary.
b. normalcdf(7,10,8.5,1.1) ≈ 82.73%
c. normalcdf(0,6,8.5,1.1) ≈ 1.15%
Name
January, 2015
Honors Math 3 review problems
cumulative: Chapters 1-4 page 11
15. If there were no association between the variables, the number of accidents with snow tires
would have mean 100 and a standard deviation of 1500  151  14
15 ≈ 9.66, or about 10.
a. To show strong evidence, the number of accidents with snow tires would need to be
significantly lower than 100. An appropriate choice would be 2 or more standard
deviations below the mean, i.e., below 80.
a. To show possible evidence, the number of accidents with snow tires would need to be a
bit lower than 100, but not enough to show strong evidence. Around 90 would be a
reasonable choice.
16. a. (0.99)400 ≈ 0.018 or 1.8%.
b. mean = 4, std. dev. =
400  0.01 0.99 ≈ 1.990.
c. normalcdf(10,400,4,1.990) ≈ 0.0013 or 0.13%.
17. a. mean = 1.515, variance ≈ 0.780, std. dev. ≈ 0.883.
b. No. The graph is not symmetrical about the mean. Also, 7 out of 200 customers fall about
3-to-4 standard deviations above the mean, where in a normal distribution it would be
uncommon to have even 1 customer that high.
Chapter 4: Trigonometry
18. a. x = 360° n
b. x = 90°, 210°, 330°
c. x = 90°, 150°, 210°, 270°
d. x = ±60°, ±120°
e. x = 45° + 360° n, 135° + 360° n
19. cos 2  15  2 cos 2 15  1 
32
4


or cos 15  cos 45  30 
6 2
4
20. 108.32 square units
21. c  5 A  44.5 B  115
22. height = 0.6 miles
23. Use cos(180   )  cos 180 cos   sin 180 sin  or
use the diagram shown at the right.
(cosθ,sinθ)
24. a. Use your calculator to check your graph.
θ
180+θ
b. two
c. x = 27°, 207°
(-cosθ,sinθ)