REPRESENTING REAL-LIFE SITUATIONS USING EXPONENTIAL FUNCTIONS || GRADE 11 GENERAL MATHEMATICS Q1
Summary
TLDRThis educational video explores the concept of exponential functions, demonstrating their form and applications in real-life scenarios. It covers the definition of exponential functions, examples with bases greater than zero and not equal to one, and their use in modeling population growth, exponential decay, and compound interest. The script also explains how to construct a table of values for exponential functions and provides step-by-step calculations for various examples, including bacteria growth, radioactive decay, and investment growth. The video concludes with an introduction to the natural exponential function with base e, illustrating its application in temperature change.
Takeaways
- đ Exponential functions are mathematical functions of the form \( f(x) = b^x \) where \( b > 0 \) and \( b \neq 1 \).
- đ« The base of an exponential function cannot be one because it would result in a constant function regardless of the exponent.
- đ Examples of exponential functions include \( f(x) = 6^x \), \( f(x) = 16^x \), and \( f(x) = 3^x + 1 \).
- đą For exponential functions with negative bases, such as \( -4 \) raised to \( 1/2 \), the result is the square root of the base's reciprocal.
- đ A table of values for different functions like \( y = (1/3)^x \), \( y = 10^x \), and \( y = 0.8^x \) can be constructed by substituting various values of \( x \).
- đ± Applications of exponential functions include modeling population growth, where quantities can double or triple over certain time periods.
- đ Exponential decay, such as the half-life of radioactive substances, can also be modeled using exponential functions.
- đ° Compound interest is another real-life application where the amount of money grows exponentially over time.
- đą The natural exponential function uses the base \( e \) (approximately 2.71828), which is a fundamental constant in mathematics.
- đ A table of values for a function involving the natural base \( e \), like the temperature of a meat slab in an oven, can show how the temperature changes over time.
- đ ïž Using a scientific calculator is recommended to evaluate and check the results of exponential functions and their applications.
Q & A
What is an exponential function?
-An exponential function is a mathematical function of the form f(x) = b^x, where b is the base and x is the exponent. The base b must be greater than 0 and not equal to 1.
Why can't the base of an exponential function be 1?
-If the base b is equal to 1, then for any value of x, the function will always yield a result of 1, making it a constant function rather than an exponential one.
What are some examples of exponential functions mentioned in the script?
-Examples include f(x) = 6^x, f(x) = 16^x, and f(x) = (3^x) + 1.
Why is the function f(x) = x^3 not considered an exponential function?
-The function f(x) = x^3 is not an exponential function because the base is x, which is a variable, and not a constant greater than 0 and not equal to 1.
How do you construct a table of values for an exponential function?
-You substitute given values of x into the function and calculate the corresponding y values. This process is repeated for each x value to create the table.
What is the significance of the base being greater than zero in an exponential function?
-The base being greater than zero ensures that the function will yield positive values for any real number x, which is a requirement for an exponential function.
What is the formula for an exponential growth model in terms of population?
-The formula for an exponential growth model is y = y0 * (growth rate)^(t/time period), where y0 is the initial population, t is the time elapsed, and the growth rate is typically a factor greater than 1.
How is the concept of half-life used in exponential decay?
-In exponential decay, the half-life is the time it takes for half of the substance to decay. The remaining amount of substance after time t is given by y = y0 * (1/2)^(t/half-life period).
What is the formula for compound interest using an exponential function?
-The formula for compound interest is A = P * (1 + r)^t, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, and t is the time the money is invested for in years.
What is the natural base e used for in exponential functions?
-The natural base e, approximately equal to 2.71828, is a fundamental mathematical constant used as the base for the natural exponential function. It is commonly used in calculus and various scientific fields.
Outlines
đ Introduction to Exponential Functions
This paragraph introduces the concept of exponential functions, defining them as functions of the form f(x) = b^x, where b is greater than zero and not equal to one. It explains that the base b cannot be one, as it would make the function a constant. Examples of exponential functions are given, such as f(x) = 6^x, f(x) = 16^x, and f(x) = (3 + 1)^x. The paragraph also clarifies that f(x) = x^x is not an exponential function because the base is variable. It concludes with an instruction to complete a table of values for various x values using the given function y = (1/3)^x.
đą Evaluating Exponential Functions and Their Applications
The second paragraph delves into the evaluation of exponential functions with different bases, including 1/3, 10, and 0.8, for a range of x values. It demonstrates the process of substituting values into the functions to find corresponding y values, emphasizing the use of reciprocals for negative exponents. The paragraph also discusses the application of exponential functions in real-life scenarios such as population growth, exponential decay, and compound interest, providing a basic formula for population growth based on doubling time.
đż Exponential Growth in Bacterial Population
This paragraph focuses on the application of exponential functions to model the growth of bacterial populations. It uses an example where bacteria double every 100 hours, starting with an initial count of 20. The paragraph constructs an exponential model to represent the number of bacteria over time, y = 20 * 2^(t/100), and demonstrates how to calculate the number of bacteria at various time intervals, such as 100, 200, 300, and 400 hours.
âł Modeling Half-Life Decay with Exponential Functions
The fourth paragraph discusses the use of exponential functions to model the decay of radioactive substances over time, known as half-life decay. It provides a formula for calculating the remaining amount of a substance after a given time, y = y0 * (1/2)^(t/t_half), where y0 is the initial amount, t_half is the half-life, and t is the time elapsed. Examples are given to illustrate how to determine the remaining quantity of a substance after specific time periods, such as 30 days for a substance with a 10-day half-life and 600 years for one with a 400-year half-life.
đŠ Compound Interest and Natural Exponential Functions
The final paragraph explores the concept of compound interest, which is calculated using an exponential function. It explains how to calculate the future value of an investment using the formula A = P * (1 + r)^t, where P is the principal amount, r is the annual interest rate, and t is the time in years. An example is provided where an investment of 100,000 earns 6% interest compounded annually, with the future value calculated for five years. The paragraph also introduces the natural exponential function with base e, an irrational number approximately equal to 2.71828, and provides an example of how it can be used to model the heating of a meat slab in an oven.
Mindmap
Keywords
đĄExponential Function
đĄBase
đĄHalf-Life Decay
đĄCompound Interest
đĄPopulation Growth
đĄTable of Values
đĄNatural Exponential Function
đĄReciprocal
đĄInterest Rate
đĄPrincipal
đĄExponential Decay
Highlights
Introduction to exponential functions with the form f(x) = b^x, where b > 0 and b â 1.
Explanation of why b cannot equal 1 in an exponential function.
Examples of exponential functions with different bases.
Clarification that f(x) = x^3 is not an exponential function due to the variable base.
Demonstration of how to evaluate exponential functions with negative bases and exponents.
Instruction on constructing a table of values for exponential functions.
Example calculations for y = (1/3)^x with various x values.
Explanation of the exponential growth model in the context of population growth.
Real-life example of bacteria doubling in number every 100 hours.
How to calculate the number of bacteria after a certain time using an exponential model.
Application of exponential functions to model half-life decay.
Example of calculating remaining radioactive substance after a given time.
Introduction to compound interest as an application of exponential functions.
Explanation of how to calculate the future value of an investment with compound interest.
Use of the natural exponential function with base e in modeling real-life scenarios.
Example of modeling the heating of a meat slab in an oven using an exponential function.
Construction of a table of values to interpret the temperature increase of the meat slab over time.
Encouragement to use scientific calculators for evaluating exponential functions.
Closing remarks with a call to like, subscribe, and follow for more educational content.
Transcripts
[Music]
in this video we are going to
represent real life situation using
exponential
function an exponential function with
base b
is a function of the form f of x is
equal to b
raised to x or y is equal to b raised to
x
where b is greater than zero and b
should not be
equal to one so bucket indicative again
equals to one
because say once we sub uh once the base
is
one and any value of x like substituting
your
x still the answer is one so mugging
constant name function not nothing
initial exponential function
making constant
okay examples of exponential function
we have f of x is equal to 6 raised to
x f of x is equal to 16
raised to x and f of x is equal to 3
raised to x
plus 1 so so unan example in base
nothing that is six
and this is exponential function bucket
um base net in a
greater than zero and then hindi equals
one and so pangala wang function at n
the base is 16
sapangatung function at n the base is
three
itunes in the exponential
function so f of x is equal to x cube
bucket indica
exponential function because uh
jung base nothing is variable so
hinduisha exponential function
f of x is equal to 1 raised to x
so since you base not n is one
therefore hindusha exponential function
f of x is equal to x raised to x this is
also not
exponential function bucket in base
nation
negative numbers you base not n for
example melon tile
it based the negative four and your
exponent nothing is one half
we try to evaluate this so
negative four raised to one half kappa
and evaluate nothing negative four
raised to one half the answer is square
root of
negative four at independent negative
first
inside of radicand
hindi long uh hindi padding one and
game base mo greater than zero but not
equal to one
okay complete a table of values for x is
equal to negative three
negative two negative one zero one two
and three okay we have a given function
here
and then we substitute the given values
and
so i try to have a to construct a table
of values
using the given values of x okay
so we have negative three negative two
negative one zero one two
and three now okay we have first
function y
y is equal to one third raised to x this
is
exponential why because one third is
greater than zero attuned based not in a
ind equals one
so therefore y is equal to one third is
raised to x
and this is exponential function so on
gaga indiana
and substitute magnet is negative 3 in
our x so
exponent not in a negative numbers so
gagawing lang reciproc
so 3 raised to 3 that is 27 so same
process
negative 2 so since an exponent not in a
negative
so level reciprocal and then that will
become
three squared and the answer is nine
so three raised to negative uh one third
raised to negative one get the
reciprocal and three raised to one that
is
uh one so nothing reciprocal
exponent next one third raised to zero
that
is one so any number raised to zero the
answer is one
one third raised to one that is one
third
one third raised to two that is
one over nine because one times one that
is one
three times three that is nine and one
third
raised to three that is one over twenty
seven
so the second function that we have is y
is equal to ten raised to x
so y is equal to ten raised to x so
substituting nothing in my
negative three so that is one over one
thousand bucket
since negative your exponent nothing you
will coordinating your reciprocal that
will become one over ten
raised to negative at 10 raised to 3
since uh reciprocal so that will become
1
raised to 10 uh 1 over 10 raised to 3.
energy 1 over
1 000 same process negative to it that
will become 1 over 100
okay negative 1 that is 1 over 10 so 10
raised to the zeros
that answer is 1 10 raised to 1 10
10 raised to 2 or 10 squared is 100
10 cube is 1000 and the last function
that we have
is y is equal to 0.8 raised to x
so 0.8 is greater than zero
zero and not equal to one so therefore y
is equal to 0.8 raised to
x is x it's an exponential function so
substitute like nothing you might given
values than x
so apache enough to cheat nothing you
can use your scientific calculator
to check the answer so 0.8 raised to
negative to the answer is 1.5625
and then 0.8 raised to negative 1. so
since negative young it raises a program
so the answer is one point
twenty-five zero point eight raised to
zero the answer is one
and zero point eight raised to one that
is zero point eight zero point eight
raised to two that is zero point sixty
four
zero point eight raised to three that is
zero 0.512
so you can check this using your
calculator
later my uh gagamite calculator to
evaluate and to check our answer so the
pata pusing in video
calculator
for example number two we have f of x is
equal to three raised to
x evaluate f of two f of negative two
f of one half and f of zero point four
some gagavindang nathan papa little x
nathan
nama given values netanyahu negative to
one half 0.4
so if our x is two so f of two that will
become three squared
so three squared that is equal to nine
so f of negative two so again three
raised to negative two since negative
your exponent not
n get the reciprocal so that will become
one over three squared
and one over three squared that is one
over nine
next f of one half three raised to one
hops up again eval
and evaluate that into it will become
square root of three another we have
f of zero point four so that will become
three raised to zero point four
is equal to three raised to two over
five y two over five
kappa key convert not into zero point
first decimal so that will become four
over ten
so get the lowest term of four over ten
that is two over five
and then and evaluate nothing and three
raised to two over five that will become
fifth root of three squared and
threes this it will become fifth of
nine
okay the most common application in real
life of exponential function
and their transformation are population
growth
exponential decay and compound interest
so
if you uh real life application
ex exponential function so it says
exponential models and population growth
suppose a quantity y
doubles every t units of time if y sub
zero is the initial amount
then the quantity y after t units of
time is given by
y is equal to y sub zero times two by
two so young to d
is not a dependence given since double c
and so
out to yan that and then nothing y sub
zero that is the initial
amount raised to t over t so you capital
letter t that is the units of time
for example let t is equal to time in
days
at t is equal to zero there were
initially 20 bacteria
suppose that the bacteria double every
100 hours
give an exponential model for the
bacteria as a function of
t initially it is equal to zero so the
number of bacteria is 20
at the start so you number them bacteria
is 20 and then
magda double the double shot in every
hundred hours
so pagnetting the 100 are sodium
bacteria and the number of bacteria is
40 because 20 times to that is 40.
so pack that thing in 200 rs the
doubling of the monument 49 and somebody
getting 18 ah
so 2 squared that is 4 times 2 that is
80.
after 300 r are the doubling the
eighteen ion
that will become one sixty so upon uh
synonym to find that two raised to three
that is eight
times twenty that is one sixty and then
after 400
rs this is now uh 116 canadians
on the dome that is 320 now after 400
hours so t uh 400 hours 2 raised to 4
16 times 20 that is 320
or in this situation we can represent
using the exponential model
y is equal to 20 times 2 raised to t
over 100 so again sana kuwaiyan 20
that is the initial number of bacteria
italian
and then you do not want nothing because
the given is double
and the new one hundred nakhon and this
is the unit of time so
unit of time nothing detail is 100
hours so if we can check
nope neck nothing let's say check
nothing you 100 so 100 divide 100 so
that is 1.
2 raised to 1 is 2 times 20 that's still
40.
okay so this is not exponential model
for this situation
i'll give you another okay for another
example
at time t is equal to zero 500 bacteria
are in petri dish and this amount
triples every 15 days so we're going to
answer these
q questions given exponential models for
the situation
how many bacteria in the dish after 40
days
so using the exponential model kanina so
we're going to identify
the following so under your y subscribe
zero that is the initial amount or the
initial number
so i know by initial amount mapping jan
so that is 500 so therefore y
sub zero papadi 500
next you capital letter t that
is the unit of time so 11 units of time
not in jan
so that is 15 days so therefore that is
15. so papayta not into num15
so we can represent our exponential
model in this situation by
y is equal to 500 times 3 bucket 3
because of the word triples okay
raised to t over 15. so we can answer
now
the letter b question how many bacteria
in the dish after 40 days so
after 40 days so what we're going to do
is simply substitute
on the exponential models that we have
so 500 times 3 raised to 40 over 50
that is 9360
so that is equivalent for 900 9360
why uh we're going to
uh round up our answer into whole number
because
nothing is number of bacteria therefore
there will be 9 360 bacteria after 40
days
or using your scientific calculator you
can use
uh your scientific calculator to
using your calculator so
data 500 open parenthesis that is three
closed double spin the cube exponent
and then production var that is
so we can check our uh
answer kuntama okay the answer is nine
thousand
three hundred sixty okay thank you
sakasha then the download view
emulator
so another application of uh exponential
function is the half-life decay
so the half-life decay of a radioactive
substance it takes for half
of the substance to decay exponential
function and half-life if the half-life
of a substance is
t units and y sub zero
is the amount of the substance
corresponding to t
is equal to zero then the amount y of
substance remaining after t
units of time is given by y is equal to
y sub zero
times one half raised to t over
t or the unit of time so as you can see
palance
uh example nothing that's a population
growth
because of the half life okay for
example
suppose that the half-life of a certain
radioactive substance
is 10 days and there are 10 grams
initially
determine the amount of substance
remaining after
30 days okay so let's have first a
representation
in a certain time no so
at the start the amount of substance is
10 grams
so maga half life sha no naga hapsha
after uh 10 days every 10 days so
so in 10 days so 5 grams
so after 20 days so five grams yes
so that will become 2.5 grams and after
30 days so mahatma lit and that will
become 1.25 grams
so therefore the amount of substance
remaining after 30 days
is 1.25 grams so in this situation
we can represent using exponential model
by y is equal to 10
again 10 that is the initial amount or
initial number
times one-half one-half because of
half-life
raised to t over 10 and 10 is the
unit of time that is the 10 days now
from the given so another example the
half-life of a substance
is 400 years give an exponential model
for the situation
how much will remain after 600 years
if the initial amount was 200 grams
so on the initial amount not in detail
200 grams and then
what is the unit of time 400 years so
400 years in unit of time not in chin
so using this exponential model so y is
equal to 200
times one half raised to t over 400
so again 200 that is the initial amount
one half because of half-life
400 because that is the unit of time 400
years
so nothing young remaining
uh number of
substance so after 600 years
so t is equal to 600 sub digit log
naught is exponential model nut
and a y is equal to 200 times one half
raised to t over 400 and that will
become
y is equal to 200 times one one-half
raised to 600
over 400 that's that is equivalent to
70.71 grams okay
you can use your calculator okay
so again pretty nothing committing only
your calculator nothing
so that is opinion at 10
so we have 200
then open parenthesis of fraction lagena
1
and then down arrow that is 2
then close parenthesis
and then it open exponent that and so
like a dino production bar
and that is 600
okay down that is 400
so equal so that is 70.71
grams okay
another application of exponential
models
is no exponential function rather is
the compound interest a starting amount
of money called a principal can be
invested at a certain
interest rate that is earned at the end
of a given period of time
such as one year if the interest rate is
compounded
the interest earned at the end of the
period is added
the principal and this new amount will
earn
interest in the next period of time
the same process is repeated for each
succeeding period interest previously
earned
will also earn interest in the next
period
so by using this so using this um
exponential model we have p as the
principal
amount and r is the annual rate
and t is the uh time the amount after
time years is given so we can use this
exponential model for
compound interest for example
mrs dilacross invested 100 000
in a company that offers six percent
interest compounded annually
how much will this investment be worth
at the end of each year
for the next five years so using the
exponential model
a is equal to p times one plus r
raised to t so any principal amount not
in gen that is one hundred thousand
and the annual rate is six percent so
percent need we need to convert into
decimal
and our time the given time is five
years
so what you can do is substitute all the
given values in the
uh a is equal to p times one plus r
raised to t
so you can use that uh exponential model
so
we get uh this is equivalent to 132
822 pesos and 56 centavo
so you can use again your calculator to
check
okay so we can use the calculator to
check
that is 100 100
000. times
one point zero six
close parenthesis and then
up into tenulanto and then the game five
and then equals so 133
822.56 so we need to round off into two
decimal places now okay next
while an exponential function may have
voice bases
okay base says a frequently used base is
the irrational number
e whose value is approximately two point
seven one
eight two eight so in the internet tower
nothing natural exponential function
capacity base nut and i e
or that is approximately two point seven
one eight two eight
so calculator in your so
ito and then equals so
that is 2.71828
okay so that's the value of the natural
number or the
letter e not n i'll give you an example
a large slob of meat is taken from the
refrigerator and placed in a preheat
heated oven the temperature t of the
slab t
minutes after being placed in the oven
is given by t
is equal to 170 minus 165
e raised to negative 0.06 t
degrees celsius construct a table of
values
for the following value of t 0 10
20 30 40 50 60 and
interpret your results round off values
to the nearest integer
so by using this t is equal to 170
minus 165 e raised to negative zero
point zero zero six
t so we can substitute the value of d
zero ten twenty thirty forty fifty and
sixty so you can use your calculator
okay to check that answer so and sabi
dito
interpret your results so round off your
values to the near side so what will be
our interpretation based on the result
data so the slope of meat is increasing
in temperature at roughly the same
rate okay the somalian scientific
calculator
in scientific calculators
downloads the app store
is a cellphone so you can download this
called business
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