* Evaluate exponential functions.
* Find the equation of an exponential function.
* Use compound interest formulas.
* Evaluate exponential functions with base e.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
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
Economics
Curve Fitting
macroeconomics
Curve fitting helps in capturing the trend in the data by assigning a single function
across the entire range.
If the functional relationship between the two quantities being graphed is known to be
within additive or multiplicative constants, it is common practice to transform the data in
such a way that the resulting line is a straight line.(by plotting) A process of quantitatively
estimating the trend of the outcomes, also known as regression or curve fitting, therefore
becomes necessary.
For a series of data, curve fitting is used to find the best fit curve. The produced equation is
used to find points anywhere along the curve. It also uses interpolation (exact fit to the data)
and smoothing.
Some people also refer it as regression analysis instead of curve fitting. The curve fitting
process fits equations of approximating curves to the raw field data. Nevertheless, for a
given set of data, the fitting curves of a given type are generally NOT unique.
Smoothing of the curve eliminates components like seasonal, cyclical and random
variations. Thus, a curve with a minimal deviation from all data points is desired. This
best-fitting curve can be obtained by the method of least squares.
What is curve fitting Curve fitting?
Curve fitting is the process of constructing a curve, or mathematical functions, which possess closest
proximity to the series of data. By the curve fitting we can mathematically construct the functional
relationship between the observed fact and parameter values, etc. It is highly effective in mathematical
modelling some natural processes.
What is a fitting model?
A fit model (sometimes fitting model) is a person who is used by a fashion designer or
clothing manufacturer to check the fit, drape and visual appearance of a design on a
'real' human being, effectively acting as a live mannequin.
What is a model fit statistics?
The goodness of fit of a statistical model describes how well it fits a set of
observations. Measures of goodness of fit typically summarize the discrepancy
between observed values and the values expected under the model in question.
What is a commercial model?
Commercial modeling is a more generalized type of modeling. There are high
fashion models, and then there are commercial models. ... They can model for
television, commercials, websites, magazines, newspapers, billboards and any other
type of advertisement. Most people who tell you they are models are “commercial”
models.
What is the exponential growth curve?
Growth of a system in which the amount being added to the system is proportional to the
amount already present: the bigger the system is, the greater the increase. ( See geometric
progression.) Note : In everyday speech, exponential growth means runaway expansion, such
as in population growth.
Why is population exponential?
Exponential population growth: When resources are unlimited, populations
exhibit exponential growth, resulting in a J-shaped curve.
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.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* 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
* 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
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
2. Concepts & Objectives
⚫ Exponential Functions
⚫ Evaluate exponential functions.
⚫ Find the equation of an exponential function.
⚫ Use compound interest formulas.
⚫ Evaluate exponential functions with base e.
3. Properties of Exponents
⚫ Recall that for a variable x and integers a and b:
+
=
a b a b
x x x
−
=
a
a b
b
x
x
x
( ) =
b
a ab
x x
= b
a b a
x x
= =
a b
x x a b
5. Simplifying Exponents
⚫ Example: Simplify
⚫ 1.
⚫ 2.
⚫ 3.
−
3 2 2
2 5
25
5
x y z
xy z
( )
−
4
2 3
2r s t
−
2 3 1 2
5 6
y y
− − − −
= 3 1 2 2 2 5
5x y z
( ) ( )
−
=
4 2 4 3
4 4
2 r s t
( )( )
−
=
2 1
3 2
5 6 y
− −
= 2 4 3
5x y z
−
= 8 12 4
16r s t
=
1
6
30y
6. Exponential Functions
⚫ If a > 0 and a 1, then
defines the exponential function with base a.
⚫ Example: Graph
⚫ Domain: (–∞, ∞)
⚫ Range: (0, ∞)
⚫ y-intercept: (0, 1)
( )= x
f x a
( )=2x
f x
7. Exponential Functions (cont.)
Characteristics of the graph of :
1. The points are on the graph.
2. If a > 1, then f is an increasing function; if 0 < a < 1, then
f is a decreasing function.
3. The x-axis is a horizontal asymptote.
4. The domain is (–∞, ∞), and the range is (0, ∞).
( )= x
f x a
( ) ( )
−
1
1, , 0,1 , 1,a
a
8. Exponential Growth/Decay
⚫ A function that models exponential growth grows by a
rate proportional to the amount present. For any real
number x and any positive real numbers a and b such
that b ≠ 1, an exponential growth function has the form
where
⚫ a is the initial or starting value of the function
⚫ b is the growth factor or growth multiplier per unit x
( ) x
f x ab
=
9. Exponential vs. Linear
⚫ Consider companies A and B. Company A has 100 stores
and expands by opening 50 new stores a year, so its
growth can be represented by the function
Company B has 100 stores and expands by increasing
the number of stores by 50% each year, so its growth
can be represented by the function
( ) 100 50
A x x
= +
( ) ( )
100 1 0.5
x
B x = +
10. Exponential vs. Linear (cont.)
⚫ A few years of growth for these companies are
illustrated in the table:
Year Company A Company B
0 100 + 50(0) = 100 100(1+0.5)0 = 100
1 100 + 50(1) = 150 100(1+0.5)1 = 150
2 100 + 50(2) = 200 100(1+0.5)2 = 225
3 100 + 50(3) = 250 100(1+0.5)3 = 337.5
x A(x) = 100 + 50x B(x) = 100(1+0.5)x
11. Exponential vs. Linear (cont.)
⚫ Graphing the two functions over 5 years shows the
difference even more clearly.
B(x)
A(x)
12. Writing Exponential Functions
⚫ Given two data points, how do we write an exponential
model?
1. If one of the data points has the form (0, a) (the y-
intercept), then a is the initial value. Substitute a
into the equation y = a(b)x, and solve for b with the
second set of values.
2. Otherwise, substitute both points into two
equations with the form and solve the system.
3. Using a and b found in the steps 1 or 2, write the
exponential function in the form f(x) = a(b)x.
13. Writing Exponential Functions
Example: In 2006, 80 deer were introduced into a wildlife
refuge. By 2012, the population had grown to 180 deer.
The population was growing exponentially. Write an
exponential function N(t) representing the population (N)
of deer over time t.
14. Writing Exponential Functions
Example: In 2006, 80 deer were introduced into a wildlife
refuge. By 2012, the population had grown to 180 deer.
The population was growing exponentially. Write an
exponential function N(t) representing the population (N)
of deer over time t.
If we let t be the number of years after 2006, we can write
the information in the problem as two ordered pairs:
(0, 80) and (6, 180). We also have an initial value, so a = 80,
and we can use the process in step 1.
15. Writing Exponential Functions
Set up the initial equation (y = a(b)x) and substitute a and
the second set of values into it.
Thus, the function becomes
( )
6
6
1
6
180 80
180 9
80 4
9
1.1447
4
b
b
b
=
= =
=
( ) ( )
80 1.447
t
N t =
16. Writing Exponential Functions
Example: Find an exponential function that passes through
the points (‒2, 6) and (2, 1).
Since we don’t have an initial value, we will need to set up
and solve a system. It will usually be simplest to use the
first equation with the first set of values to solve for a, and
then substitute that into the second equation with the
second set of values to solve for b.
17. Writing Exponential Functions
Example: Find an exponential function that passes through
the points (‒2, 6) and (2, 1).
Thus, the function is
2
2
2
6
6
6
ab
a
b
a b
−
=
=
=
( )
2 2
4
4
1
4
1 6
1 6
1
6
1
0.6389
6
b b
b
b
b
=
=
=
=
( )
2
6 0.6389
2.4492
a =
=
( ) ( )
2.4492 0.6389
x
f x =
18. Writing Exponential Functions
Example: Find an exponential function that passes through
the points (‒2, 6) and (2, 1).
It may (or may not) surprise you to
learn that Desmos also offers us a
shortcut for this.
( ) ( )
2.4495 0.6389
x
f x =
19. Compound Interest
⚫ The formula for compound interest (interest paid on
both principal and interest) is an important application
of exponential functions.
⚫ Recall that the formula for simple interest, I = Prt, where
P is principal (amount deposited), r is annual rate of
interest, and t is time in years.
20. Compound Interest (cont.)
⚫ Now, suppose we deposit $1000 at 10% annual interest.
At the end of the first year, we have
so our account now has 1000 + .1(1000) = $1100.
⚫ At the end of the second year, we have
so our account now has 1100 + .1(1100) = $1210.
( )( )
= =
1000 0.1 100
I
( )( )
= =
1100 .1 110
I
21. Compound Interest (cont.)
⚫ Another way to write 1000 + .1(1000) is
⚫ After the second year, this gives us
( ) ( )
( )
+ + +
1000 1 .1 .1 1000 1 .1 ( )( )
= + +
1000 1 .1 1 .1
( )
= +
2
1000 1 .1
( )
+
1000 1 .1
22. Compound Interest (cont.)
⚫ If we continue, we end up with
This leads us to the general formula.
Year Account
1 $1100 1000(1 + .1)
2 $1210 1000(1 + .1)2
3 $1331 1000(1 + .1)3
4 $1464.10 1000(1 + .1)4
t 1000(1 + .1)t
23. Compound Interest Formulas
⚫ For interest compounded n times per year:
⚫ For interest compounded continuously:
where e is the irrational constant 2.718281…
= +
1
tn
r
A P
n
= rt
A Pe
24. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
25. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A P = 2500, r = .06,
n = 2, t = 10
26. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A
27. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A = $4515.28
28. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A
( )
= +
4 8
.048
15000 1
4
P
A = 15000, r = .048,
n = 4, t = 8
= $4515.28
29. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A
( )
= +
4 8
.048
15000 1
4
P
( )
15000 1.4648
P
= $4515.28
30. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A
( )
= +
4 8
.048
15000 1
4
P
( )
15000 1.4648
P
= $4515.28
31. Examples
1. If $2500 is deposited in an account paying 6% per year
compounded twice per year, how much is the account
worth after 10 years with no withdrawals?
2. What amount deposited today at 4.8% compounded
quarterly will give $15,000 in 8 years?
( )
= +
2 10
.06
2500 1
2
A
( )
= +
4 8
.048
15000 1
4
P
( )
15000 1.4648
P
= $10,240.35
P
= $4515.28
32. Examples
3. If $8000 is deposited in an account paying 5% interest
compounded continuously, how much is the account
worth at the end of 6 years?
4. Which is a better deal, depositing $7000 at 6.25%
compounded every month for 5 years or 5.75%
compounded continuously for 6 years?
33. Examples
3. If $8000 is deposited in an account paying 5% interest
compounded continuously, how much is the account
worth at the end of 6 years?
4. Which is a better deal, depositing $7000 at 6.25%
compounded every month for 5 years or 5.75%
compounded continuously for 6 years?
( )( )
=
.05 6
8000
A e
= $10,798.87
A
( )
= +
=
12 5
.0625
7000 1
12
$9560.11
A
( )( )
=
=
.0575 6
7000
$9883.93
A e