Name Honors Math 3 Review Problems

Name Honors Math 3 review problems

January, 2015cumulative: 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

• Lagrange Interpolation (p. 17)
• “Sasha’s” alternate method (p. 22 exercise 5)
• adding a term to make a formula fit a table at an additional point (p. 25)

• Key theorems and their proofs

• Euclidean Property (p. 36)
• Remainder Theorem (p. 37)
• Factor Theorem (p. 41) and its graphical interpretation about x-intercepts
• Corollaries about degree n polynomials (pp. 42-43)

• Polynomial graphs (from handouts)

• End behavior based on degree and leading coefficient
• Multiplicity of factors: just touching vs. passing through x-axis
• Fitting formulas to graphs, including finding the leading coefficient

• Factoring methods

• “Find a root, find a factor” (p. 59)
• Factoring quadratics (p. 51) including non-monic quadratics (p. 52)
• Difference of squares, sum of cubes, difference of cubes (p. 55)
• Perfect squares (p. 55) and perfect cubes (p. 60 exercise 1)
• Factoring by grouping (p. 57)
• Substitutions for quadratic-like (p. 62) and cubic-like (p. 63) polynomials
• 4th degree with hidden difference-of-squares (p. 64)
• Factoring over R (p. 65)

• Calculation skills

• Long division (p. 35) and synthetic division (Challenge Problem Set 1)
• Simplifying, adding, and subtracting rational expressions (p. 69)

Chapter 2: Sequences and series

• Working with tables of numbers

•  columns for differences (sometimes repeated: 2, 3, etc.)
•  columns showing cumulative sums (sometimes repeated: , , etc.)

• Sequence concept

• a sequence as a list of numbers
• a sequence as a function with domain of nonnegative integers (or all integers from some starting value)
• visual examples: figurate numbers

• Series concept

• definite and indefinite series
• writing in + notation or in  notation
• ways to visualize sums: staircases, pyramids

• Formulas for sequences and series

• closed-form formulas for the kth term of a sequence or for the sum of an indefinite series
• recursive formulas for the kth term of a sequence or for the sum of an indefinite series

• Evaluating definite and indefinite series

• arithmetic series using Gauss’s Method or its result
• geometric series using Euclid’s Method or its result
• k2, k3, k4, k5 using Bernoulli Formulas (these formulas will be given if needed)
• use of  identities to break series into simpler pieces that can be evaluated

• Limits

• limits of sequences
• limits of series
• limits of geometric series: formula
• visual examples: “fractals” (pp. 149-151 # 3, 4, 9); Achilles paradox (p. 151 #8)
• repeating decimals using limits of geometric series

• Pascal’s Triangle and Binomial Theorem

• patterns, relationships, and symmetry in Pascal’s Triangle
• summation properties: “hockey stick” (pp. 160-161), ///etc. on diagonals (p. 163)
• even/odd and remainder patterns (p. 162)
• Binomial Theorem: use and informal justification

Chapter 3: Statistics

• Statistics for a data set

• Find the mean, mean absolute deviation, variance, and standard deviation.
• Prove and use this alternate formula for variance: .
• Prove and use that means and variances are additive.

• Repeated experiments

• Given the statistics (mean, variance, and/or standard deviation) for a single experiment, find the statistics for the experiment repeated n times.
• For repeated Bernoulli trials, find the probability that the total has a specific value (using the formulainvolving a combination number).
• For repeated Bernoulli trials, find the mean, variance, and standard deviation.

• Assessing the effectiveness of a treatment

• Identify appropriate randomization methods for selecting the treatment and control groups.
• Recognize outcomes that show strong evidence, possible evidence, or little or no evidence that a treatment is effective.

• Sample surveys

• Identify appropriate methods for random sampling.
• Given a sample proportion, use simulation results to assess what values are plausible for the population parameter.
• Estimate margins of error with 95% or 99+% confidence.
• Understand that correlation does not imply causation.
• Understand the differences between sample surveys, observational studies, and experiments.

• Probability distributions

• Make probability histograms.
• For normal distributions, apply the 68/95/99+% rule and use the normalpdf and normalcdf functions to answer probability questions.
• 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 by .

b. Find without plugging in.

4.Suppose . Use the Remainder Theorem to show that is divisible by .

5.Factor each expression (over Z except where it says otherwise).

a.

b.

c.

d., 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.

n / f(n) / Σ
0 / –3
1 / 2
2 / 7
3 / 12
4 / 17

a.Fill in the  column of cumulative sums.

b.Write a function formula for f(n).

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.

10.Find a closed-form formula that agrees with this recursively defined function:

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 s and the 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.

b. Find the sum .

c. Use the Binomial Theorem to expand and simplify the expression (x – 3y)5.

Chapter 3: Statistics

13.When you subtract a constant cfrom each element in a data set, what happens to the mean and whathappens 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?

15. A researcher investigates whether snow tires are more effective than regular tires at preventing automobile accidents.Suppose that a researcher conducts a welldesigned, appropriately-randomized statistical study of this question, and tabulates how many cars hadan 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 thesnow tires helped prevent accidents (that is, such that there’s possibleevidence 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. Ifyou use your calculator, solve for degree values to the nearest tenth.

a. for general solution(s) where x is in degrees

b.for 0° ≤ x ≤ 360°

c.for 0° ≤ x ≤ 360°

d. for –180° ≤ x ≤ 180°

e. for general solution(s) where x is in degrees

19.Find the exact value for two ways by applying two different identities.

20.Find the area of quadrilateral ABCD.

21.Solve 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 and respectively. How high abovethe ground is the balloon?

23.Use two different methods (unit circle and angle sum identities) to prove that

24.a.Sketch a graph of

b.Use your graph to determine the number of solutions to the equation

c.Use to find all values of x for to the nearest degree.

Review problem answers(first draft; if any answers don’t look right, check with teacher)

Chapter 1: Functions and polynomials

1.Lagrange:

“Sasha”:

Both equivalent to .

2.Greatest degree 5; least degree 0.

3.a. quotient ; remainder -22b. –22

4.. When substituting 5 for x, the result is 0. So, P(x) – P(5) is divisible by x – 5.

5.a. b.

c.d.

6.Here are possible answers. It’s OK if your leading coefficients are slightly off.

a. using (0, -2)

b. 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.

b., which simplifies to

c. Let dbe the common difference. Then , so d = –4.

d. Series is geometric with a = 36, r = 1/3.

9. a.

n / f(n) / Σ
0 / –3 / –3
1 / 2 / –1
2 / 7 / 6
3 / 12 / 18
4 / 17 / 35

b.

c. or

d.

10.See the “unstacking” method on p. 119, and the For Discussion prompt on p. 120.

11.a. The area of the equilateral triangle with sides s is the area of the triangle with sides s.

b. area = initial area + area added stage 1 + area added stage 2 + … + area added stage n

area = area added stage n

With the exception of the first 1, the rest is a geometric series with a = 1/3 and r = 4/9.

So total area at stage n: .

As n approaches infinity, area at stage n approaches , which simplifies to .

12.a.

b. Use Pascal’s triangle, where we know is the entry in row n, entry k. Remember we start by labeling the first row n = 0, and the first entry in a row k = 0.

= 56 + 6 + 21 + 15 + 10 = 108.

c.By Binomial Theorem,

Use the entries in row n = 5 of Pascal’s triangle to find values of , then simplify.

Final expansion:

Chapter 3: Statistics

13.The mean decreases by c. The variance stays the same.

Proof (mean): Suppose x1, …, xn have a mean of . Then x1–c, …, xn–c have a mean of
.

Proof (std. dev.): Since each data value decreases by c and the mean also decreases by c, allof 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%

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 ≈ 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. = ≈ 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.or

20.108.32 square units

21.

22.height = 0.6 miles

23.Use or
use the diagram shown at the right.

24.a.Use your calculator to check your graph.

b.two

c.x = 27°, 207°