Exponential Function fx a x for any positive number a other than one Examples What are the domain and range of y 23 x 4 What are the roots of 0 5 25 x ID: 278330
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Slide1
Exponential FunctionsSlide2
Exponential Function
f(x) = a
x
for any positive number
a
other than one.Slide3
Examples
What are the domain and range of
y = 2(3
x) – 4?
What are the
roots of
0 =5 – 2.5
x
?Slide4
Properties of Powers (Review)
When multiplying
like bases
,
add exponents.
a
x
● ay = ax+yWhen dividing like bases, subtract exponents.
When raising a power to a power,
multiply exponents.
(a
x
)
y
=
a
xySlide5
Properties of Powers (Review)
When you have a
monomial
or a fraction raised to a power (with no add. or sub.), raise everything to that power.
orSlide6
Half-Life & Exponential Growth/Decay
The
half-life
of a substance is the time it takes for half of a substance to exist.
Mirrors the behavior of
Exponential Growth & Decay
functions.
Exponential Growth: y = kax, if a > 1k is the initial amount presenta is the rate at which the amount is growing
Exponential Decay: y = kax, 0 < a < 1k
is the initial amount presenta is the rate at which the amount is growingSlide7
Example
Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining?
After 20 days:
After 40 days:
IN GENERAL:
Models the mass of the substance after
t
days.
Therefore, let
graph, and find intersection.
t ≈ 46.44 daysSlide8
Exponential Growth/Decay
Example:
A population initially contains 56.5 grams of a substance. If it is increasing at a rate of 15% per week, approximately how many weeks will it take for the population to reach 281.4 grams?Slide9
Exponential Growth
Example:
How long will it take a population to triple if it is increasing at a rate of 2.75%?Slide10
The Number
e
Many real-life phenomena are best modeled using the number
e
e
≈ 2.718281828
e
can be approximated by:
Interest compounding continuously:
I = Pe
rt
, where P = initial investment,
r = interest rate (decimal)
t = time in years Slide11
Example Compounding Interest
A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded
a.) quarterly
b.) monthly
c.) continuouslySlide12
Suggested HW
Sec. 1.3 (#5, 7, 11, 19, 21-31 odd)
1.3 Web Assign Due Monday night