General Mathematics - Evaluating Functions
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tutorial5.ppt
This document provides an overview of combinatorics and number theory concepts including basic counting techniques, recurrence relations, binomial coefficients, prime numbers, congruences, and proofs by induction. It discusses topics such as permutations, subsets, Pascal's triangle for calculating binomial coefficients efficiently, and using recurrence relations to solve problems like calculating the Fibonacci sequence or the number of ways to reach the last stage in a multi-stage process.
Pre-Calculus Quarter 4 Exam 1 Name ___________.docx
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
Pre-Calculus Quarter 4 ExamName _________________________ S.docx
Pre-Calculus Quarter 4 Exam
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
2.
Locate the foci of the ellipse. Show your work.
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3, f(x)
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8.
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 12 + 42 + 72 + . . . + (3n - 2)2 =
9.
A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?
12
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 2 + 5 + .
Multiple Choice Type your answer choice in the blank next to each.docx
Multiple Choice:
Type your answer choice in the blank next to each question number.
_____1.
Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
_____2.
Graph the ellipse and locate the foci.
A.
foci at (0,
6) and (0, -6)
C.
foci at (
, 0) and (-
, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0,
5) and (0, -5)
_____3.
Solve the system by the substitution method.
2y - x = 5
x2 + y2 - 25 = 0
A.
B.
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
_____4.
Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x - 3,
f(x)
A. 5
B. 25
C. 2
D. 22
_____5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C.
∅
D. {(2, -7, -1)}
_____6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z
= -5
x - y + 3z
= -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
_____7.
A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Graph an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the graph to quadrant I only.
A.
C.
B.
D.
Short Answer Questions:
Type your answer below each question. Show your work.
8
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
9
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month a.
Name ____________________________Student Number ________________.docx
Name: ____________________________
Student Number: ___________________
Short Answer:
Type your answer below each question. Show your work.
1
Verify the identity.
Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
Show your work.
(c) Find the y-intercept.
Show your work.
f(x) = x
2
(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan
2
u)(1 - sin
2
u) = 1
7
Verify the identity
. Show your work.
cot
2
x + csc
2
x = 2csc
2
x - 1
8
Verify the identity. Show your work.
1 + sec
2
xsin
2
x = sec
2
x
9
Verify the identity.
Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
The formula A = 118e
0.024t
models the popula.
Name _________________________ Score ______ ______1..docx
The document contains a series of math word problems and questions. It asks the reader to:
1) Solve various math equations and systems of equations, showing the work.
2) Write mathematical statements and prove they are true for different values of n.
3) Graph functions and find limits.
4) Solve optimization problems to maximize profits or minimize costs given certain constraints.
ARITHMETIC SEQUENCE, MEAN AND SERIES WEEK 2 QUARTER 1
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
Pre-Calculus Midterm ExamScore ______ ______Name _______.docx
Pre-Calculus Midterm Exam
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1
Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
7
Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8
Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
9
Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
T.
Recommended
tutorial5.ppt
This document provides an overview of combinatorics and number theory concepts including basic counting techniques, recurrence relations, binomial coefficients, prime numbers, congruences, and proofs by induction. It discusses topics such as permutations, subsets, Pascal's triangle for calculating binomial coefficients efficiently, and using recurrence relations to solve problems like calculating the Fibonacci sequence or the number of ways to reach the last stage in a multi-stage process.
Pre-Calculus Quarter 4 Exam 1 Name ___________.docx
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
Pre-Calculus Quarter 4 ExamName _________________________ S.docx
Pre-Calculus Quarter 4 Exam
Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
2.
Locate the foci of the ellipse. Show your work.
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4.
Graph the function. Then use your graph to find the indicated limit. You do not have to provide the graph
f(x) = 5x - 3, f(x)
5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Write an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8.
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 12 + 42 + 72 + . . . + (3n - 2)2 =
9.
A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10.
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?
12
A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 2 + 5 + .
Multiple Choice Type your answer choice in the blank next to each.docx
Multiple Choice:
Type your answer choice in the blank next to each question number.
_____1.
Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
_____2.
Graph the ellipse and locate the foci.
A.
foci at (0,
6) and (0, -6)
C.
foci at (
, 0) and (-
, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0,
5) and (0, -5)
_____3.
Solve the system by the substitution method.
2y - x = 5
x2 + y2 - 25 = 0
A.
B.
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
_____4.
Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x - 3,
f(x)
A. 5
B. 25
C. 2
D. 22
_____5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C.
∅
D. {(2, -7, -1)}
_____6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z
= -5
x - y + 3z
= -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
_____7.
A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Graph an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the graph to quadrant I only.
A.
C.
B.
D.
Short Answer Questions:
Type your answer below each question. Show your work.
8
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
9
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month a.
Name ____________________________Student Number ________________.docx
Name: ____________________________
Student Number: ___________________
Short Answer:
Type your answer below each question. Show your work.
1
Verify the identity.
Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
Show your work.
(c) Find the y-intercept.
Show your work.
f(x) = x
2
(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan
2
u)(1 - sin
2
u) = 1
7
Verify the identity
. Show your work.
cot
2
x + csc
2
x = 2csc
2
x - 1
8
Verify the identity. Show your work.
1 + sec
2
xsin
2
x = sec
2
x
9
Verify the identity.
Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
The formula A = 118e
0.024t
models the popula.
Name _________________________ Score ______ ______1..docx
The document contains a series of math word problems and questions. It asks the reader to:
1) Solve various math equations and systems of equations, showing the work.
2) Write mathematical statements and prove they are true for different values of n.
3) Graph functions and find limits.
4) Solve optimization problems to maximize profits or minimize costs given certain constraints.
ARITHMETIC SEQUENCE, MEAN AND SERIES WEEK 2 QUARTER 1
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
Pre-Calculus Midterm ExamScore ______ ______Name _______.docx
Pre-Calculus Midterm Exam
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1
Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
7
Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8
Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
9
Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
T.
Chapter 5
The document discusses exponential and logarithmic functions. It defines logarithms as exponents and explains that logarithms were once used to simplify calculations before calculators. It then covers several topics related to exponential functions including:
- Basic laws of exponents using integral exponents
- Examples of applying the order of operations to exponents
- Extending the rules of exponents to include rational exponents
- Exponential growth and decay models and examples
- Graphing and properties of exponential functions
- The number e and the natural exponential function ex
- Compound interest formulas including continuous compounding
Simultaneous Equations Practical Construction
The document discusses solving simultaneous equations using algebraic methods and graphing. It provides examples of setting up and solving systems of two equations with two unknowns to find the values of the unknowns. Various word problems are presented and worked through step-by-step to show how to set up the appropriate equations to find the unknown values being asked about, such as costs, numbers of items, etc. Strategies for setting up simultaneous equations from word problems are emphasized.
Stochastic Processes Homework Help
I am Geoffrey J. I am a Stochastic Processes Homework Expert at excelhomeworkhelp.com. I hold a Ph.D. in Statistics, from Edinburgh, UK. I have been helping students with their homework for the past 6 years. I solve homework related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Homework.
Mit6 006 f11_quiz1
The document provides instructions for Quiz 1 of the MIT course 6.006 Introduction to Algorithms. It states that the quiz has 120 minutes and 120 total points. It is closed book except for one crib sheet. Students are to write their solutions in the provided space and show their work for partial credit. The quiz contains 7 problems worth various point values testing topics like asymptotics, recurrences, sorting algorithms, and graph algorithms.
Problem descriptionThe Jim Thornton Coffee House chain is .docx
Problem description
The Jim Thornton Coffee House chain is planning expansion into Calgary. It has selected many
possible sites for new coffee houses. The possible sites are joined by roads that form a spanning tree.
To eliminate competition with itself, the company has determined that it should not choose two
sites that are adjacent in this tree. From its market evaluation, the company has also determined
the expected profit per year for each site. Your job is to determine what sites Jim Thornton should
choose for his coffee houses.
a. Define this problem precisely, by defining the required Input and the required Output. Use
mathematical notation.
b. Write a recursive equation for the function that maximizes the profit per year. Include a
short argument that your equation is correct.
c. Write an efficient algorithm that computes the maximum profit per year, and also computes
the sites that should be chosen. Your algorithm should run in time that is polynomial in the
input size.
d. What is the asymptotic running time of your algorithm? Defend your answer.
Your algorithm and proofs should be precise and concise (as well, of course, as correct.) Elegance
of your solution counts.
2
CPSC 413 — Winter, 2013
Home Work Exercise #8
March 16, 2013
1. Exercise 1 Chapter 6, page 312 of the textbook.
Answer:
(a) A counterexample is given in Figure 1. The given algorithm finds an independent set of
weight 8. However, the maximum total weight is 10 by adding the nodes at two ends to
the independent set.
5 58
Figure 1: A counterexample to the algorithm of 1(a)
(b) A counterexample is given in Figure 2. The given algorithm finds an independent set of
weight 11. However, the maximum total weight is 19 by adding the nodes at two ends to
the independent set.
10 12 9
Figure 2: A counterexample to the algorithm of 1(b)
(c) Input: An array A = (a1, . . . , an) of n integers. (I use “array” rather than “set” here
because the elements in a set do not have the notion of order.)
Output: A set S = {aα1, . . . , aαm} ⊆ A, which satisfies
(1)
m∑
i=1
aαi is maximum;
(2) ∀i ∈ {1, . . . , m}, ̸ ∃j ∈ {1, . . . , m} s.t. |αi − αj| = 1.
Optimization function. Indset[i] denotes the maximum independent set of an array with
i positive integers (a1, . . . , ai). Let OPT [i] denote the total weight of Indset[i]. The
optimization function is
OPT [0] = 0, OPT [1] = max{0, a1},
OPT [i] =
{
OPT [i − 1] if OPT [i − 1] ≥ OPT [i − 2] + ai
OPT [i − 2] + ai otherwise
Correctness of the optimization function. It is trivial if n = 0 or 1. When n ≥ 2, there
are two cases to be considered. Case 1: Indset[n] includes an. Then Indset[n] must not
include an−1. And Indset[n]−{an} must be a maximum independent set of (a1, . . . , an−2).
This can be easily verified by a replacement argument. If the statement is not true, then
the total weight of Indset[n] − {an} is less than OPT [n − 2]. Then Indset[n − 2] ∪ {an}
has total weight larger than Indset[n] does, and this co.
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This module discusses quadratic functions and their application to solving word problems. Students will learn to 1) recall steps to solve word problems, 2) translate problems to symbolic expressions, and 3) apply quadratic equations. Examples provided cover number, geometry, and motion problems. Students must first translate word problems into equations before solving. Key steps include identifying unknowns, writing the equation, solving, and checking solutions. Practice problems are provided to help students apply the concepts.
Design and analysis of algorithm ppt ppt
The document discusses asymptotic analysis and algorithmic complexity. It introduces asymptotic notations like Big O, Omega, and Theta that are used to analyze how an algorithm's running time grows as the input size increases. These notations allow algorithms to be categorized based on their worst-case upper and lower time bounds. Common time complexities include constant, logarithmic, linear, quadratic, and exponential time. The document provides examples of problems that fall into each category and discusses how asymptotic notations are used to prove upper and lower bounds for functions.
Fundamental Problems in Number Theory.pptx
I am Marvin Jones, a Number Theory Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from Columbia University, and have been assisting students with their homework for the past six years. I specialize in number theory assignments.
For any number theory assignment solution or homework help, visit mathsassignmenthelp.com, email info@mathsassignmenthelp.com, or call +1 678 648 4277. This sample assignment solution is a prove of our work.
Carmen sanchez
En esta ocasión te presentamos esta información sobre expresiones algebraicas, además de ello ejercicios variados que te pueden ser mucha utilidad
Pre-Calculus Midterm Exam 1 Score ______ ____.docx
Pre-Calculus Midterm Exam
1
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1 Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2 A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
Pre-Calculus Midterm Exam
2
3 The wind chill factor represents the equivalent air temperature at a standard wind speed that would
produce the same heat loss as the given temperature and wind speed. One formula for computing
the equivalent temperature is
W(t) = {
𝑡
33 −
(10.45+10√𝑣−𝑣)(33−𝑡)
2204
33 − 1.5958(33 − 𝑡)
if 0 ≤ v < 1.79
if 1.79 ≤ v < 20
if v ≥ 20
where v represents the wind speed (in meters per second) and t represents the air temperature .
Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second.
(Round the answer to one decimal place.) Show your work.
4 Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis
and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
Pre-Calculus Midterm Exam
3
5 For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is
the best choice for modeling the data.
Number of Homes Built in a Town by Year
6 Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Pre-Calculus Midterm Exam
4
7 Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8 Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Pre-Calculus Midterm Exam
5
9 Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10 The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You
ppt math6 qt3-wk3.pptx
The document provides instructions on subtracting integers. It explains:
1) To subtract integers, transform the subtraction into addition by keeping the first number and changing the second number's sign.
2) Examples are provided of subtracting integers with different signs and the same sign.
3) A multi-step word problem is worked out as an example of subtracting integers.
Guia 3 abril calculo
This document provides information and instructions for students regarding a math guide assignment. It includes the names and contact information for 8 teachers, as well as the topic of equations and inequalities. It gives details about the assignment such as it should take 4 weeks to complete, and provides criteria for how it will be evaluated including justifying work shown and submitting a PDF file of photos of work in their notebook. The document then provides content on the topic including definitions and examples of equations, how to solve various types of equations, and example problems to work through.
Ib math slsummerassignment2014
This document provides instructions for a mathematics summer packet for students entering IB Mathematics SL. It explains that the purpose of the packet is to review key algebra, problem solving, and math concepts. It provides expectations for completing the packet neatly and with work shown. It informs students that the first day of school will involve reviewing the content and answers to the packet. A quiz will be given during the first week to assess the skills and knowledge from the packet. Students are instructed to circle any problems they have trouble with for additional review.
Ppt2 u4 funcion_lineal
- The document discusses linear functions and how to determine the equation of a line.
- A linear function can be expressed as f(x) = mx + c, where m is the slope and c is the y-intercept.
- The slope formula is given as m = (y2 - y1) / (x2 - x1) using two points on the line (x1, y1) and (x2, y2).
- The point-slope formula for the equation of a line is given as y - y1 = m(x - x1), which allows determining the equation of a line given the slope and a point.
Semana 12 ecuaciones polinomiales i álgebra-uni ccesa007
The document is about algebra and solving equations. It discusses the history and importance of equations, defines key terms like solutions and sets of solutions. It also provides examples of solving different types of equations step-by-step, including linear equations, quadratic equations through factoring, completing the square, and the quadratic formula. The document emphasizes that solving equations involves finding the value(s) of the variable that satisfy the equality.
Direct proportion
This document discusses direct proportion and methods for solving direct proportion problems. Direct proportion exists when two quantities change at a constant rate with respect to each other. The cross-multiplication method can be used to solve direct proportion problems by setting up a proportion between the known quantities and cross-multiplying to solve for the unknown quantity. Graphs of direct proportion relationships will always produce a straight line passing through the origin.
M112rev
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
November 16, 2015
The document is a class notes document that contains:
1) Information about upcoming tests, Khan Academy topics, and the day's lessons which include introducing literal equations and working on class work problems.
2) Examples of solving literal equations by isolating the variable being solved for and working through examples without and with variables.
3) Tips for solving literal and other equations by distributing, clearing fractions, and isolating variables.
4) A few practice problems for solving literal equations and calculating rates.
Arithmetic Sequence Presentation- Mathematics in the modern worls
This presentation discusses on the ways to find the arithmetic sequence of the given series of numbers
Terminado matematica ii
This document contains a mathematics exam with 11 questions covering topics like quadratic curves, linear systems of equations, matrices, determinants, and matrix operations. The exam asks the student to graph functions, solve systems of equations, perform matrix operations, calculate determinants, and model word problems using matrices. It provides multiple parts to each question to comprehensively assess the student's understanding of core mathematics concepts.
Educational Technology in the Health Sciences
Plenary presentation at the NTTC Inter-university Workshop, 18 June 2024, Manila Prince Hotel.
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The document provides instructions for Quiz 1 of the MIT course 6.006 Introduction to Algorithms. It states that the quiz has 120 minutes and 120 total points. It is closed book except for one crib sheet. Students are to write their solutions in the provided space and show their work for partial credit. The quiz contains 7 problems worth various point values testing topics like asymptotics, recurrences, sorting algorithms, and graph algorithms.
Problem descriptionThe Jim Thornton Coffee House chain is .docx
Problem description
The Jim Thornton Coffee House chain is planning expansion into Calgary. It has selected many
possible sites for new coffee houses. The possible sites are joined by roads that form a spanning tree.
To eliminate competition with itself, the company has determined that it should not choose two
sites that are adjacent in this tree. From its market evaluation, the company has also determined
the expected profit per year for each site. Your job is to determine what sites Jim Thornton should
choose for his coffee houses.
a. Define this problem precisely, by defining the required Input and the required Output. Use
mathematical notation.
b. Write a recursive equation for the function that maximizes the profit per year. Include a
short argument that your equation is correct.
c. Write an efficient algorithm that computes the maximum profit per year, and also computes
the sites that should be chosen. Your algorithm should run in time that is polynomial in the
input size.
d. What is the asymptotic running time of your algorithm? Defend your answer.
Your algorithm and proofs should be precise and concise (as well, of course, as correct.) Elegance
of your solution counts.
2
CPSC 413 — Winter, 2013
Home Work Exercise #8
March 16, 2013
1. Exercise 1 Chapter 6, page 312 of the textbook.
Answer:
(a) A counterexample is given in Figure 1. The given algorithm finds an independent set of
weight 8. However, the maximum total weight is 10 by adding the nodes at two ends to
the independent set.
5 58
Figure 1: A counterexample to the algorithm of 1(a)
(b) A counterexample is given in Figure 2. The given algorithm finds an independent set of
weight 11. However, the maximum total weight is 19 by adding the nodes at two ends to
the independent set.
10 12 9
Figure 2: A counterexample to the algorithm of 1(b)
(c) Input: An array A = (a1, . . . , an) of n integers. (I use “array” rather than “set” here
because the elements in a set do not have the notion of order.)
Output: A set S = {aα1, . . . , aαm} ⊆ A, which satisfies
(1)
m∑
i=1
aαi is maximum;
(2) ∀i ∈ {1, . . . , m}, ̸ ∃j ∈ {1, . . . , m} s.t. |αi − αj| = 1.
Optimization function. Indset[i] denotes the maximum independent set of an array with
i positive integers (a1, . . . , ai). Let OPT [i] denote the total weight of Indset[i]. The
optimization function is
OPT [0] = 0, OPT [1] = max{0, a1},
OPT [i] =
{
OPT [i − 1] if OPT [i − 1] ≥ OPT [i − 2] + ai
OPT [i − 2] + ai otherwise
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are two cases to be considered. Case 1: Indset[n] includes an. Then Indset[n] must not
include an−1. And Indset[n]−{an} must be a maximum independent set of (a1, . . . , an−2).
This can be easily verified by a replacement argument. If the statement is not true, then
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Design and analysis of algorithm ppt ppt
The document discusses asymptotic analysis and algorithmic complexity. It introduces asymptotic notations like Big O, Omega, and Theta that are used to analyze how an algorithm's running time grows as the input size increases. These notations allow algorithms to be categorized based on their worst-case upper and lower time bounds. Common time complexities include constant, logarithmic, linear, quadratic, and exponential time. The document provides examples of problems that fall into each category and discusses how asymptotic notations are used to prove upper and lower bounds for functions.
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I am Marvin Jones, a Number Theory Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from Columbia University, and have been assisting students with their homework for the past six years. I specialize in number theory assignments.
For any number theory assignment solution or homework help, visit mathsassignmenthelp.com, email info@mathsassignmenthelp.com, or call +1 678 648 4277. This sample assignment solution is a prove of our work.
Carmen sanchez
En esta ocasión te presentamos esta información sobre expresiones algebraicas, además de ello ejercicios variados que te pueden ser mucha utilidad
Pre-Calculus Midterm Exam 1 Score ______ ____.docx
Pre-Calculus Midterm Exam
1
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1 Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2 A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
Pre-Calculus Midterm Exam
2
3 The wind chill factor represents the equivalent air temperature at a standard wind speed that would
produce the same heat loss as the given temperature and wind speed. One formula for computing
the equivalent temperature is
W(t) = {
𝑡
33 −
(10.45+10√𝑣−𝑣)(33−𝑡)
2204
33 − 1.5958(33 − 𝑡)
if 0 ≤ v < 1.79
if 1.79 ≤ v < 20
if v ≥ 20
where v represents the wind speed (in meters per second) and t represents the air temperature .
Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second.
(Round the answer to one decimal place.) Show your work.
4 Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis
and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
Pre-Calculus Midterm Exam
3
5 For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is
the best choice for modeling the data.
Number of Homes Built in a Town by Year
6 Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Pre-Calculus Midterm Exam
4
7 Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8 Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Pre-Calculus Midterm Exam
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9 Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
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Ppt2 u4 funcion_lineal
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- A linear function can be expressed as f(x) = mx + c, where m is the slope and c is the y-intercept.
- The slope formula is given as m = (y2 - y1) / (x2 - x1) using two points on the line (x1, y1) and (x2, y2).
- The point-slope formula for the equation of a line is given as y - y1 = m(x - x1), which allows determining the equation of a line given the slope and a point.
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Direct proportion
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General Mathematics - Evaluating Functions
- 2. EXERCISE
- 3. N ans ans ans ans ans ans ans 2n - 1 𝑛2 𝑛 5 + 20 (n- 5)2 𝑛 3 𝑛 + 7 11 (30 – 4n) Let n = 12
- 4. N ans ans ans ans ans ans ans 2n - 1 𝑛2 𝑛 5 + 20 (n- 5)2 𝑛 3 𝑛 + 7 11 (30 – 4n) Let n = 15
- 6. ACTIVITY Meet best friends Emily and Anton. They have formed a partnership. Anton, being creative one, makes costumes jewelry. Emily, being business- minded, markets the jewelry Anton makes. In a recent month, they spent P1,000 on raw materials to make 50 pieces of jewelry and sold each for P25.00. Assuming that they are not required to pay a sales tax, their net profit depends on the number of jewelry sold.
- 7. 1. The problem includes three constants: the fixed cost of the raw materials (P1,000), the selling price for each piece (P25.00), and the total number of jewelry pieces made (50)
- 8. 2. Let us use n to represents the number of pieces of jewelry sold and P for the net profit in pesos. a. What should be greatest value of n? the greatest of value of P? b. What should be the least value of n? the least value of P? (Hint: refer to the table in the next page)
- 9. 3. Some of the data for selling the costume jewelry are shown in the table below. Number of Pieces of Costume Jewelry Sold (n) Net Profit in Pesos (P) 0 -1,000 10 -750 20 -500 25 -375 30 -250 45 125
- 10. 4. A function machine that clearly shows the input, the process and the output after selling the costume jewelry is given as follows.
- 11. 5. Use the function machine to write an equation that shows algebraically how to compute the net profit given the number of pieces of jewelry sold. 𝑃 𝑛 = __________________
- 12. 6. If there are 40 pieces of jewelry sold, how much should be the profit?
- 13. Questions: 1. In the previous activity we could also replace P (n) with y or any variable you like, what does the notation y = P (n) means? 2. How do you call possible values of n that would satisfy the equation y the function P (n)? How about the resulting values of y or P (n)? 3. How do you solve the values of P (n) given n? (Hint: refer to the previous activity)
- 14. Example 1. If 𝑓(𝑥) = 𝑥 + 8, evaluate each. a. 𝑓(4) b. 𝑓(−2) c. 𝑓(𝑥 + 3) d. 𝑓(−𝑥) Solution: a. 𝑓(𝑥) = 𝑥 + 8 𝑓(4) = 4 + 8 Substitution Property of Equality. 𝑓(4) = 4 + 8 Simplify 𝑓(4) = 12
- 15. Example 2. The price function 𝑝(𝑥) = 640 − 0.2𝑥 represent the price for which you can sell x printed T-shirts. What must be the price of the shirt for the first 3 entries in the table? Target Number of Shirts (x) 500 900 1300 Price per T-shirt P(x) P540 P460 P380 1 700 P300
- 16. ASSESSMENT A. Directions. Evaluate each function at the indicated values of the independent variable and simplify the result. 1. 𝑓 𝑥 = 9 − 6𝑥 a. 𝑓 −1 b. 𝑓 1 c. 𝑓 −3 + 𝑥 2. 𝑔 𝑥 = 𝑥2 − 4𝑥 a. 𝑔 2 b. 𝑔 𝑎 + 𝑏 c. 𝑔 2 − 𝑥
- 17. B. Answer the problems that follow. 1. The function A described by 𝐴 𝑠 = 𝑠2 3 4 gives the area of equilateral triangle with side s. a. Find the area when a side measures 8 inches. b. Find the area when a side measures 16 cm. 2. The function C described by 𝐶 𝐹 = 5 9 𝐹 − 32 gives the Celsius temperature corresponding to the Fahrenheit temperature F. a. Find the Celsius temperature equivalent to 14℉. b. Find the Celsius temperature equivalent to 68℉.
- 18. Assignment Direction: Answer the problem below. As the lightning strikes, the time between the flash that we see and the thunder that we hear depends on the distance that we are from where the lightning struck. A table for this function is shown below. Distance (in kilometers), d 1 2 3 4 5 Time (in seconds), t 3 6 9 12 15
- 19. Questions: 1. Write a formula for this function. 2. Graph the data in the table by letting the x-axis represents the distance and the y-axis represents the time. 3. Why does it seem reasonable that the graph of this function should go through the origin?
- 20. Prepared by: DAREL F. CLAVERIA