Algebra 1, 5 3
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This document discusses functions and how they can be represented in three ways: as a function rule, as a table of ordered pairs, and as a graph. It provides examples of modeling functions using each representation. It also discusses using functions to model real-world situations like the costs of producing CDs for different numbers of copies. Functions can have linear graphs or non-linear graphs like parabolas or absolute value graphs. Learners are asked to complete tables, graph functions, compare costs at different quantities, and graph additional functions.
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Onto Functions
Surjective, Injective, Bijective, etc. If you have heard these terms but do not exactly know what these mean, this is the question for you. If you have not even heard these terms, then start now, hit wikipedia
Veranderen is bewegen
Presentatie tijdens het Hulshoff Seminar.
Over Hulshoff:
Hulshoff, een familiebedrijf met ruim 150 medewerkers, brengt haar klanten verder door facilitaire werkzaamheden volledig uit handen te nemen. Zo kunnen klanten van Hulshoff zich richten op de essentie van hun onderneming. Hulshoff zorgt continu voor innovatieve en passende oplossingen en vormt zo een betrouwbare en professionele serviceorganisatie.
Al sinds 1935 staat de balans tussen belangen op korte- en lange termijn voorop. Daarom werkt Hulshoff volgens de principes van Maatschappelijk Verantwoord Ondernemen (MVO) en hanteren we de eisen van het KVGM-systeem (Kwaliteit Veiligheid Gezondheid en Milieu).
De dienstverlening is onderverdeeld in drie segmenten; projectverhuizing, archiefbeheer en digitalisering.
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Onto Functions
Surjective, Injective, Bijective, etc. If you have heard these terms but do not exactly know what these mean, this is the question for you. If you have not even heard these terms, then start now, hit wikipedia
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Veranderen is bewegen
Presentatie tijdens het Hulshoff Seminar.
Over Hulshoff:
Hulshoff, een familiebedrijf met ruim 150 medewerkers, brengt haar klanten verder door facilitaire werkzaamheden volledig uit handen te nemen. Zo kunnen klanten van Hulshoff zich richten op de essentie van hun onderneming. Hulshoff zorgt continu voor innovatieve en passende oplossingen en vormt zo een betrouwbare en professionele serviceorganisatie.
Al sinds 1935 staat de balans tussen belangen op korte- en lange termijn voorop. Daarom werkt Hulshoff volgens de principes van Maatschappelijk Verantwoord Ondernemen (MVO) en hanteren we de eisen van het KVGM-systeem (Kwaliteit Veiligheid Gezondheid en Milieu).
De dienstverlening is onderverdeeld in drie segmenten; projectverhuizing, archiefbeheer en digitalisering.
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Similar to Algebra 1, 5 3
M166Calculus” ProjectDue Wednesday, December 9, 2015PROJ.docx
M166
“Calculus” Project
Due: Wednesday, December 9, 2015
PROJECT WORTH 50 POINTS –
1) NO LATE SUBMISSIONS WILL BE ACCEPTED
2) COMPLETED PROJECTS NEED TO BE LEGIBLE
I. Computing Derivatives (slope of curve at a point) of polynomial functions.
For each of the following functions in a.-e. below perform the following three steps:
1. compute the difference quotient
2. simplify expression from part 1. such that h has been canceled from the denominator
3. substitute and simplify
a.
b.
c.
d.
e. consider , using the results from parts a. through d.,
f. find a general formula for (steps 1 through 3 performed).
II. Show that
Consider the unit circle with in standard position in QI.
a. show that the area of the right triangle (see diagram) is
b. show that the area of the sector (see diagram) is
c. show that the area of the acute triangle (see diagram)
d. set up the inequality
e. multiply the inequality in part d. by . (direction of inequalities is unchanged)
f. take the reciprocal of each term from part e. The direction of the inequality must be reversed because .
g. plug in 0 for for only. The result should be
III. Show that
a. multiply by
b. use trigonometric identity to rewrite the numerator of the expression in part a. in terms of
c. factor the expression in part b. with one factor equal to . (find remaining factor).
d. use the fact that and substitute in the second factor (result is 0)
IV. Show that derivative of
a. find the difference quotient for
(use sum angle formula )
b. factor out of the two terms in the numerator with in part a
c. split up the expression in part b with each term over the denominator h
d. use identities to simplify part c. to
Thus you have shown that if .
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f
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)
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x
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=
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x
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2
3
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)
3
2
2
3
3
3
3
:
int
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tan
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Right
Area
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Area
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sin
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Acute
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MATH133: Unit 3 Individual Project 2B Student Answer Form
Name (Required): ____Michael Magro_________________________
Please show all work details with answers, insert the graph, and provide answers to all the critical thinking questions on this form for the Unit 3 IP assignment.
A version of Amdah ...
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
Joe Bob
Mon lab: 4:30-6:50
Lab 3
Exercise 1
(a) Create function M-file for banded LU factorization
function [L,U] = luband(A,p)
% LUBAND Banded LU factorization
% Adaptation to LUFACT
% Input:
% A diagonally dominant square matrix
% Output:
% L,U unit lower triangular and upper triangular such that LU=A
n = length(A);
L = eye(n); % ones on diagonal
% Gaussian Elimination
for j = 1:n-1
a = min(j+p.
MATH133 UNIT 2 Quadratic Equations Individual Project Assignm.docx
MATH133 UNIT 2: Quadratic Equations
Individual Project Assignment: Version 2A
Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols.
Problem 1: Modeling Profit for a Business
IMPORTANT: See Question 3 below for special IP instructions. This is mandatory.
Remember that the standard form for the quadratic function equation is
y
=
f
(
x
) =
ax
2
+
bx
+
c
and the vertex form is
y
=
f
(
x
) =
a
(
x
–
h
)
2
+
k
, where (
h
,
k
) are the coordinates of the vertex of this quadratic function’s graph.
You will use
P
(
x
) = −0.2
x
2
+
bx
–
c
where (−0.2
x
2
+
bx
) represents the business’ variable profit and
c
is the business’s fixed costs.
So,
P
(
x
) is the store’s total annual profit (in $1,000) based on the number of items sold,
x
.
1. Choose a value between 100 and 200 for
b
. That value does not have to be a whole number.
2. Think about and list what the fixed costs might represent for your fictitious business (be creative). Start by choosing a fixed cost,
c
, between $5,000 and $10,000, according to the first letter of your last name from the values listed in the following chart:
If your last name begins with the letter
Choose a fixed cost between
A–E
$5,000–$5,700
F–I
$5,800–$6,400
J–L
$6,500–$7,100
M–O
$7,200–$7,800
P–R
$7,800–$8,500
S–T
$8,600–$9,200
U–Z
$9,300–$10,000
4.Replace
b
and
c
with your chosen values in Parts 1 and 2 in
P
(
x
) = −0.2
x
2
+
bx
−
c
. This is your quadratic profit model function. State that quadratic profit model functions equation.
5. Next, choose 5 values of
x
(number of items sold) between 500 and 1,000. Think about the general characteristics of quadratic function graphs (parabolas) to help you with choosing these 5 values of
x
.
6. Plug these 5 values into your model for
P
(
x
), and evaluate the annual business profit given those sales volumes. (Be sure to show all of your work for these calculations.)
7. Use the 5 ordered pairs of numbers from 5 and 6 and Excel or another graphing utility to graph your quadratic profit model, and insert the graph into your Word answer document. The graph of the quadratic function is called a
parabola
.
8. What is the vertex of the quadratic function graph? (Show your work details, or explain how you found the vertex.)
9. What is the equation of the line of symmetry? Explain how you found this equation.
10. Write the vertex form for your quadratic profit function.
11. Is there a maximum profit for your business? If so, how many items must be sold to produce the maximum profit, and what is that maximum profit? If your quadratic profit function has a maximum, show your work or explain how the maximum profit figure was obtained.
12. How would knowing the number of items sold that produces the maximum profit help you to run your business more effectively.
13. Analyze the results of these profit cal.
Computational Assignment Help
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3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
MATH133 UNIT 2 Quadratic EquationsIndividual Project Assignment.docx
MATH133 UNIT 2: Quadratic Equations
Individual Project Assignment: Version 2A
Name (Required): __________________________________________________
Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. Handwritten scanned work is not acceptable for AIU Online.
Problem 1: Modeling Profit for a Business
IMPORTANT: See Question 3 below. This is mandatory.
Remember that the standard form for the quadratic function equation is y = f (x) = ax2 + bx + c and the vertex form is y = f (x) = a(x – h)2 + k, where (h, k) are the coordinates of the vertex of this quadratic function’s graph.
You will use P(x) = -0.2x2 + bx – c where (-0.2x2 + bx) represents the business’s variable profit and c is the business’s fixed costs.
So, P(x) is the store’s total annual profit (in $1,000) based on the number of items sold, x.
1. (List your chosen value for between 100 and 200.)
2. (List what the fixed costs might represent for your fictitious business, and be creative; also list your chosen value for c from the table below).
If your last name begins with the letter
Choose a fixed cost between
A–E
$5,000–$5,700
F–I
$5,800–$6,400
J–L
$6,500–$7,100
M–O
$7,200–$7,800
P–R
$7,800–$8,500
S–T
$8,600–$9,200
U–Z
$9,300–$10,000
3. Important: By Wednesday night at midnight, submit a Word document with only your name and your chosen values for b and c above in Parts 1 and 2. Submit this in the Unit 2 IP submissions area. This submitted Word document will be used to determine the Last Day of Attendance for government reporting purposes.
4. (State that quadratic profit model function’s equation by replacing and with your chosen values.)
5. (Choose five values of (number of items sold) between 500 and 1000. Insert those -values in the table.)
6. Plug these five values into your model for and evaluate the annual business profit given those sales volumes. (Be sure to show all your work for these calculations; complete the table below.)
7. Use the five ordered pairs of numbers from 5 and 6, and Excel or another graphing utility, to graph your quadratic profit model and insert the graph into your Word answer document. The graph of a quadratic function is called aparabola. (Insert graph below.)
8. (Show work details or explain how you found the vertex. Write the vertex in ordered-pair form: .)
9. (Write the explanation and the equation of the line of symmetry.)
10. (Write your quadratic profit function in vertex form, where is the vertex of this quadratic function’s graph. Show the details of how you found this equation.)
11. (State the maximum profit (if any), and show how you determined how many items must be sold to give the maximum profit.)
12. (State how knowing the number of items sold that produces the maximum profit help you to run business more effectively.)
13. (Give an analysis of the results of these profit calculations, and give some ...
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docx
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
Math 116 pres. 5
CHAPTER 6 Real Numbers: Completing a Mathematical System
6.6 Making the Connections: Is There REALly a Completion to the Number System?
M210 Songyue.pages__MACOSX._M210 Songyue.pagesM210-S16-.docx
M210 Songyue.pages
__MACOSX/._M210 Songyue.pages
M210-S16-class-5.pdf
M 210 More drawing in LATEX February 23, 2016
Arcs
An arc of radius r from point (p,q) with initial angle ϕ (phi) and final angle ψ (psi) is drawn in
TikZ by the code
LATEX\draw (p,q) arc (phi:psi:r);
Note that TikZ does not require the center (a,b), which is often easier to determine that the arc’s
initial point. Polar coordinates (relative to the arc’s center) can be used to express the initial point
in terms of the arc’s center.
(a,b)
(p,q)
ϕ
ψ
r
p = a + r cosϕ
q = b + r sinϕ
In tikzpicture environment enter
\draw ({a+r*cos(phi)},{b+r*sin(phi)}) arc (phi:psi:r);
for the specific values of a, b, r, ϕ (phi) and ψ (psi).
LATEX
According to the TikZ’s manual, by default arcs are drawn in positive (counter-clockwise) direction,
assuming initial angle ϕ to be smaller than ψ. However, I have been able to draw arcs in negative
(clockwise) direction by choosing an initial angle larger than final angle.
Precise Label Positioning
The relative positions above, below, left, right, above left, above right, below left, and
below right, provide positioning relative to the node (indicated by the dot) as illustrated in the
following figures.
below
above
left right
above left
below left
above right
below right
While it is possible to adjust the separation between label and node, it is not that easy to adjust
the angle (the above provide label placement only for angles in multiples of 45◦). The following will
work to place the label (that is its center) at the the indicated position: precise distance s at relative
angle θ from the position of the node.
node
lab
el
s
node coordinates (nx,ny)
label coordinates (lx, ly)
θ
To place label at the above mentioned position use the code (where θ is theta degrees):
M 210 — February 23, 2016 page 44
LATEX\draw ({nx + s*cos(theta)}, {ny + s*sin(theta)}) node {label};
To rotate the label as in the above illustration, use code:
LATEX\draw ({nx + s*cos(theta)}, {ny + s*sin(theta)}) node {\rotatebox{theta}{label}};
Special Triangles
In trigonometry the triangles with degrees 45-45-90 and 30-60-90 provide exact values for the trigono-
metric functions. It is quite easy to obtain the exact values of angles of 15 and 75 degrees from those
of 30 degrees by half- and double-angle formulas and other trigonometric identities, however, these
values can also be obtained from the following figure containing an equilateral triangle in a square
with sides equal to 2.
1
The Regular Pentagon. Use trigonometric functions to draw a regular pentagon with its lower
side along the line segment from (−1,0) to (1,0). Start by drawing this line segment.
LATEX\begin{center}
\begin{tikzpicture}[scale=2]
\draw[line width=1.2pt] (-1,0) -- (1,0);
\end{tikzpicture}
\end{center}
Extend the line segment on both sides by adding the following code.
LATEX\draw[line width=1.2pt] (1,0) -- ({1+2*cos(72)},{2*sin(72)});
\draw[line width=1.2pt] ...
Me 443 4 plotting curves Erdi Karaçal Mechanical Engineer University of Gaz...
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MATLABgraphPlotting.pptx
Trigonometric Function, Complex Numbers & M Files in MATLAB/ Graph & Plotting in MATLAB
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
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