Title: State exponential growth or exponential decay
1State exponential growth or exponential
decay (no calculator needed)
a.) y e2x b.) y e2x c.) y
2x d.) y 0.6x
k lt 0, exponential decay
k gt 0, exponential growth
0ltblt1 so decay, but reflect over y-axis, so
growth
bgt1 so growth, but reflect over y-axis, so decay
2Characteristics of a Basic Exponential Function
Domain Range Continuity Symmetry Boundedness
Extrema
( - ?, ? )
( 0, ? )
continuous
none
Asymptotes End Behavior
b 0
y 0
none
3(No Transcript)
4Question
Use properties of logarithms to rewrite the
expression as a single logarithm.
log x log y 1/5 log z log x log 5 2 ln x
3 ln y ln y ln 3 4 log y log z ln x ln
y 4 log (xy) 3 log (yz) 1/3 log x 3 ln (x3y)
2 ln (yz2)
5Change of Base Formula for Logarithms
6the exponential function
the natural base
2.718281828459 (irrational, like ?)
Leonhard Euler (1707 1783)
f(x) a e kx for an appropriately chosen real
number, k, so ek b
exponential growth function
exponential decay function
7State exponential growth or exponential
decay (no calculator needed)
a.) y e2x b.) y e2x c.) y
2x d.) y 0.6x
k lt 0, exponential decay
k gt 0, exponential growth
0gtbgt1 so decay, but reflect over y-axis, so
growth
bgt1 so growth, but reflect over y-axis, so decay
8Rewrite with e approximate k to the nearest
tenth.
a.) y 2x b.) y 0.3x
e? 2
e? 0.3
y e0.7x
y e1.2x
9Characteristics of a Basic Logistic Function
Domain Range Continuity Symmetry Boundedness
Extrema
( - ?, ? )
( 0, 1 )
continuous
about ½, but not odd or even
Asymptotes End Behavior
B 0, b 0
y 0, 1
none
10Based on exponential growth models, will Mexicos
population surpass that of the U.S. and if so,
when?
Based on logistic growth models, will Mexicos
population surpass that of the U.S. and if so,
when?
What are the maximum sustainable populations for
the two countries?
Which model exponential or logistic is more
valid in this case? Justify your choice.
11Logarithmic Functions
inverse of the exponential function
logbn p
bp n
logbn p iff bp n
find the power
5
2? 32
0
3? 1
½
4? 2
1
5? 5
½
2? ?2
12Basic Properties of Logarithms (where n gt 0, b gt
0 but ? 1, and p is any real number)
Example
logb1 0 because b0 1
log51 0
logbb 1 because b1 b
log22 1
logbbp p because bp bp
log443 3
blogbn n because logbn logbn
6log611 11
13Evaluating Common Log Expressions
Without a Calculator
2
1.5132176
log 100
0.22914
1/7
undefined
8
10 log 8
14Solving Simple Equations with Common Logs and
Exponents
Solve
log x 1.6
10 x 3.7
x log 3.7
x 10 1.6
x 0.57
x 0.03
15Evaluating Natural Log Expressions
Without a Calculator
1/3
3.443
7
0.9416
log e7
5
e ln 5
undefined
16Solving Simple Equations with Natural Logs and
Exponents
Solve
ex 6.18
ln x 3.45
x ln 6.18
x e 3.45
x 31.50
x 1.82
17Logarithmic Functions
0.91 ln x
0.91 ln x
- reflect over the x-axis
- vertical shrink by 0.91
18Graph the function and state its domain and range
f(x) log4x
0.721 ln x
Vertical shrink by 0.721
f(x) log5x
0.621 ln x
Vertical shrink by 0.621
f(x) log7(x 2)
0.514 ln (x 2)
Vertical shrink by 0.514, shift right 2
f(x) log3(2 x)
0.091 ln ((x 2)
Vertical shrink by 0.091 Reflect across
y-axis Shift right 2
19Logarithmic Functions
one-to-one functions
u v
2x 25
x 5
log22x log27
x log27
isolate the exponential expression
take the logarithm of both sides and solve
20Newtons Law of Cooling
An object that has been heated will cool to the
temperature of the medium in which it is placed
(such as the surrounding air or water). The
temperature, T, of the object at time, t, can be
modeled by
where Tm temp. of surrounding medium
T0 initial temp. of the object
Example A hard-boiled egg at temp. 96? C is
placed in 16? C water to cool. Four (4) minutes
later the temp. of the egg is 45? C. Use
Newtons Law of Cooling to determine when the egg
will be 20? C.
21Compound Interest
Interest Compounded Annually
A P (1 r)t
A Amount P Principal r Rate t Time
Interest Compounded k Times Per Year
A P (1 r/k)kt
k Compoundings Per Year
Interest Compounded Continuously
A Pert
22Annual Percentage Yield
23Annuities
Future Value of an Annuity
R Value of Payments i r/k interest rate
per compounding n kt number of payments
11 (p. 324)
14,755.51
24Annuities
For loans, the bank uses a similar formula
25Annuities
If you loan money to buy a truck for 27,500,
what are the monthly pay-ments if the annual
percentage rate (APR) on the loan is 3.9 for 5
years?