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ACM 3 / Math 4
Final Exam Study Guide
Unit 1 Data Analysis
I.
Multiple Choice. Choose the one best answer.
1. In a large population of adults, the mean IQ is 115 with a standard deviation of
15. Suppose 100 adults are randomly selected for a market research
campaign. The distribution of the sample mean IQ is
A)
exactly normal, mean 115, standard deviation 20.
B)
approximately normal, mean 115, standard deviation 0.1.
C)
approximately normal, mean 115, standard deviation 1.5.
D)
approximately normal, mean 115, standard deviation 20.
E)
exactly normal, mean 115, standard deviation 1.5.
2. The central limit theorem is important because it allows us to use the Normal distribution to
make inferences about the population mean
A) if the sample size is reasonably large
B) if the population is Normally distributed and the sample size is reasonably large
C) if the population is Normally distributed
D) if the population is Normally distributed and the population variance is known
E) if the population size is reasonably large
3. A 90% confidence interval for the mean math scores for a population of fifth-graders is
(43.6, 58.9). If you computed a 95% confidence interval using the same information, which
of the following statements is correct?
A) The intervals have the same width.
B) The 95% confidence interval is wider.
C) The 95% confidence interval is shorter.
D) The margin of error can’t be found.
E) The answer can’t be determined from the information given.
4. The weights of 12 men have mean 180 pounds and standard deviation 15 pounds. What is
the standard error of the mean?
A) 3.098
B) 4.330
C) 4.648
D) 5.196
E) Cannot be determined with information given
5. If all other parts of problem stay the same, which of the following would reduce the width of
a confidence interval for a given population.
1
I.
II.
III.
Increase the sample size.
Have a smaller sample standard deviation.
Increase the confidence level.
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
II.
Free Response questions. Answer completely, but be precise.
6. Consider the sampling distribution of sample means obtained by random sampling from an
infinite population. This population has a distribution that is highly skewed toward the larger
values.
a. How is the mean of the sampling distribution related to the mean of the population?
b. How is the standard deviation of the sampling distribution related to the standard deviation
of the population?
c. How is the shape of the sampling distribution affected by the sample size?
7. A random sample of 100 households in a certain affluent community yields a
mean weekly food budget of $100 and a standard deviation of $10. Find the 95%
confidence interval for weekly food budgets of households in this community.
8. It is believed that the average amount of money spent per US household per
week on food is about $98. Is the community in problem number 9 representative
of the US household? Justify your answer.
2
Unit 2 Sequences and Series
1. Show that you know the difference between a sequence and a series by writing the first three
terms of a geometric sequence and a geometric series, each of which has a first term 10 and
common ratio 3.
Sequence:
Series:
2. Show that the sequence 2, 6, 24, … is neither arithmetic nor geometric.
3. For the following sequence: 6, -18, 54, …
a. Find a possible formula for tn
b. Find t21
4. Determine the sum of the series: 81 + 77 + 73 + … + 5
5. What is the 11th term of the sequence an = 2n – 6?
6. What is the sum of the first five terms of a if an = 4n2?
7. What is the sum of the first six terms of the sequence an = 2n + 6?
8. What is the sum of the first 45 terms in the arithmetic sequence a1 = 3; an = an-1 + 5
6
9. Calculate:
 (4i  5)
i1
15
10. Calculate:
3  2
(n1)
n1
11. What is the limit of the infinite series: 2  1 1  1 ... ?
2
12. An arithmetic series has t1 = 19 and t2 = 32.
a. Find the common difference
b. Find t300.
3
4
c. Find S300.
d. Find n if tn = 7377.
e. Find n if Sn = 1086.
13. A geometric series has t1 = 100 and t2 = 90.
a. Find the common ratio.
b. Find t30.
c. Find n if tn = 28.2429… (Hint: solve the exponential equation using logarithms)
d. Find S3.
e. Find the limit of the infinite series Sn.
14. Assume that this year, n = 1, the average annual salary for women in the United States is
$30,000 and it is increasing by 4% per year, meaning that each year the salary is 1.04 times what
it was the previous year. Assume that the average annual salary for men is $35,000 and that it is
increasing by $1,600 per year.
a. Write the first three terms of each sequence of salaries. What kind of sequence is each?
b. In the first three years, whose salaries are increasing faster, women’s or men’s?
c. Assuming that the same pattern of salaries continues, who will be making more, women
or men, in year n = 25?
15. The student council is going to sell tickets for a candy-give-away for $5. All the tickets will be
drawn, with the first person receiving 1 piece of candy, the second person receiving 2 pieces,
the third 4 pieces, the fourth 8 pieces and so on. The student council expects to sell 25 tickets.
a. How many pieces of candy will the 25th person receive?
b. How many pieces of candy will the need for all 25 tickets?
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c. If the cost for the candy is $1 per 10,000 pieces, what will be the profit/loss for the
fundraiser?
Unit 3 Rational Functions
1. Give the degree, the number of real zeros, the number of (nonreal) complex zeros, for this polynomial function.
• Degree: _________________________________
• Real zeros: ______________________________
• Complex zeros: __________________________
2. Sketch the graph of y  1x .
3. The function y 
1
x 2
is what transformation of y 
1
x
in Problem 2?
4. What kind of discontinuity does the graph in Problem 3 have at x = –2?
5. Find the limit for lim
f ( x) 
x3  5 x 2  8 x  6
x 3
As x approaches 3
5
6.Simplify and then sketch the graph of
y
( x  3)( x  2)
( x  3)
What kind of discontinuity will the graph have at
x = 3?
7. This cubic function has three zeros, although only two of them seem to show up on the graph. Write the three
zeros.
Rational Function Problem: The figure shows the graph of the rational function
y
5x  13
( x  1)( x  5)
10. Explain why the discontinuities at x = 1 and x = 5 are vertical asymptotes rather than removable discontinuities.
11. . Find the limit for lim
g ( x) 
x3  5 x 2  8 x  5
x 3
As x approaches 3
12. The instantaneous rate of change of a function is also called as the _____________________ of a function.
6
Unit 4 : Trigonometric Functions
Change between Radians and Degrees
1. 35o
2.
7
13
3. 315o
4.
11
12
Tell the quadrant the terminal side of the given angle would be.
5.
13
2
6. 225o
7. 
5
3
8. -567o
Find one positive and one negative co-terminal angle with the given angle.
9. 550
10.
11. If sin  
17
4
5
what would tan be?
9
12-14. Find the missing sides and angles of the given triangle.
A
A = ____
12
C
a = _____
56o
B = 56o
b = 12
C = 90o
c = _____
Draw the following angles in standard position, find the reference angles and mark them.
16. 2580 12
15. 156.52’
Use the given point on the terminal side of an angle x in standard position. Evaluate the six
trigonometric functions of x.
17. (-12, 5)
Sin x =
Csc x =
cos x =
Sec x =
Tan x =
7
Cot x =
Given a trig ratio of an angle x and the quadrant the terminal side of the angle is in, evaluate the
remaining 5 trigonometric functions of x.
18. Sin x = 
Sin x = 
3
and cos   0 .
7
3
7
Csc x =
Cos x =
Sec x =
Tan x =
Cot x =
19. In the figure to the right, what is the value of y?

A) r•cos 
B) r•sin 
D) r•cos 
E) r•sin 
C) r•tan 
r
20. If cot θ = 1.9312, find θ in degrees.
A) 62.62
B) 27.38
C) .5178
D) .0337
Unit 5: Graphs of Trigonometric Functions
Graph each function.
1.
y = 2 sin x + 1
2.
y = -3cos (2x - )
3.
y = tan 

1
x 
2
2
8
E) .0090
Write the equation of the following graphs.
4.
5.
Write the equation of a cosine function with the following conditions.


, phase shift: 
4
2
6.
amplitude: 2, period:
7.
amplitude: 3, period: 2, phase shift: -1, vertical shift: -4
Graph each function.
2 
x + 2
3 
8.
y = 2cot 
9.
y = 3sec 
10.
y = -3sin  x 
1 
x
2 




4
Find the following.
___________ 11.
__________ 13.
 1

 2
sin-1  
tan-1 (-1)
__________ 12.
__________ 14.
9
arccos (-1)
 2

 2 


cos-1 
__________ 15.

 3 

tan  arccos
 2 





__________ 16. cos tan 1
 3  + arccos (-1))
Complete the chart with exact values.
17.
Period
Amplitude
Phase Shift
Vertical Shift
Period
Amplitude
Phase Shift
Vertical Shift
Period
Amplitude
Phase Shift
Vertical Shift
y = -8cos(3x - ) + 2
y = 5sin(4x + ) - 3
18.
y = 3tan(2x) + 1
y = 2cot(3x) + 2
19.
y = 2sec(3x) + 1
1 
x + 2
2 
y = -3csc 
Unit 6: Identities
Find the following using trigonometric identities. Assume  is in
quadrant I.
1.
If sec  = 3, find tan .
2.
10
1
If cot  = , find sin .
5
Simplify.
3.
cos2x  tan x  csc x
tan x  csc x
1  tan 2 x
4.
Prove the following identities
5. cos2β∙tan2β + cos2β = 1
7.
6.
sec 
 csc 
tan 
1  tan 2 x
 tan 2 x
1  cot 2 x
8.
Solve for values of  such that 0   < 360.
9.
4cos2 - 1 = 0
10.
11.
2sin2θ = sin  + 1
12.
cos A
 sin A  tan A
cot 2 A
3 sec θ + 2 = 0
2cos2 = cos 
Find the exact value using sum/difference or half-angle identities.
13. sin 195
14. cos 165
11
If α and β are measures of two first quadrant angles, find the exact value
of each function.
15.
If sin α =
12
5
and cos β =
, find sin (α + β).
20
13
16.
If cos α =
15
7
and sin β =
, find cos (α – β).
17
25
Unit 7: Law of Sines / Law of Cosines / Heron’s Formula
Solve the triangle.
1. C = 125.7o
a = 6.25
b = 2.15
2. C = 145o
b=4
c= 14
3. A = 110.25o
a = 48
b = 16
4. C = 110o
b = 100
c = 125
Use Heron’s formula to find the area of the triangle.
5. a = 5
b=7
c = 10
12
6. C = 120o
a=4
b=6
Solve each problem. Draw any diagrams necessary and show all work.
7. Because of the prevailing winds, a tree grew so that it was leaning 6o from the vertical. At a point
100 feet away from the tree, the angle of elevation to the top of the tree is 22.83o. Find the height
of the tree.
8. The angles of elevation to an airplane from two points A and B on level ground are 51o and 68o
respectively. The points A and B are 6 miles apart, and the airplane is between those positions. Find
the altitude of the plane.
9. Find the area of triangular parcel of land if its sides are 400 ft, 500 ft and 700 ft.
Vectors
1, Find the component vector and determine the magnitude of the vector if the initial point is (2, 4) and
the terminal point is (5, 9 )
2. Find the unit vector and show that its magnitude is 1 for the vector v = < 3, 5 >
3. If v = < 2, - 4 > and u = < 3, 4 >, find a) v+u b) 2v – 3u c) 2v + 3u
4. If v + u = 2i + 3j and v – u = 6i – 5j find the value of vector v and also u.
13
Key For Data Analysis
1.
2.
3.
4.
5.
6.
C
A
B
B
A
a) Mean of sample = mean of population
b) sample standard deviation = population standard deviation divided by square root of
sample size
c) increasing sample size makes sample distribution more normal with less variation
7. ($98.02, $101.98)
8. This community is NOT representative of the US because $98 is not in the
confidence interval obtained from the data.
Key for Sequences and Series
1. Sequence: 10, 30, 90
Series: 10 + 30 + 90
2. 6 – 2 = 4, 24 – 6 = 18; no common difference therefore not arithmetic
6/2 = 3, 24/6 = 4; no common ratio therefore not geometric
3. a. tn = (-1)n-1(2)(3)n
b. t21 = 20,920,706,406
4. 860
5. 16
6. 220
7. 78
8. 5085
9. 54
10. 98,301
11. 4
12. a. 13
14
b. 3, 906
c. 588,750
d. 567
e. 12
13. a. .9
b. 4.7101…
c. 13
d. 1,000
14. a. women: $30,000, $31,200, $32,448; geometric
men: $35,000, $36,600, $38,200; arithmetic
b. men’s
c. women approximately $76,899
15. a. 16,777,216 pieces
b. 33,554,431 pieces
c. loss of $3,231
15