State “exponential growth” or “exponential decay” (no calculator needed)
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Pre-Calculus 1/3/2007 State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e 2x b.) y = e –2x c.) y = 2 –x d.) y = 0.6 –x k > 0, exponential growth k < 0, exponential decay b>1 so growth, but reflect over y-axis, so decay 0<b<1 so decay, but reflect over y-axis, so grow
description
State “exponential growth” or “exponential decay” (no calculator needed). a.) y = e 2x b.) y = e –2x c.) y = 2 –x d.) y = 0.6 –x. k < 0, exponential decay. k > 0, exponential growth. 0
Transcript of State “exponential growth” or “exponential decay” (no calculator needed)
Pre-Calculus
1/3/2007
State “exponential growth” or “exponential decay”(no calculator needed)
State “exponential growth” or “exponential decay”(no calculator needed)
a.) y = e2x b.) y = e–2x
c.) y = 2–x d.) y = 0.6–x
k > 0, exponential growth k < 0, exponential decay
b>1 so growth, but reflect over y-axis, so decay
0<b<1 so decay, but reflect over y-axis, so growth
Pre-Calculus
1/3/2007
Characteristics of a Basic Exponential Function: Characteristics of a Basic Exponential Function:
Domain:
Range:
Continuity:
Symmetry:
Boundedness:
Extrema:
Asymptotes:
End Behavior:
( - , )
( 0, )
continuous
none
b = 0
none
y = 0
lim f(x) x =
lim f(x) x - = 0
Pre-Calculus
1/3/2007
Pre-Calculus
1/3/2007
QuestionQuestion
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)
Pre-Calculus
1/3/2007
Change of Base Formula for Logarithms
Change of Base Formula for Logarithms
bln
xlnor
blog
xlog
blog
xlogxlog
a
ab
Pre-Calculus
1/3/2007
“the” 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
Pre-Calculus
1/3/2007
State “exponential growth” or “exponential decay”(no calculator needed)
State “exponential growth” or “exponential decay”(no calculator needed)
a.) y = e2x b.) y = e–2x
c.) y = 2–x d.) y = 0.6–x
k > 0, exponential growth k < 0, exponential decay
b>1 so growth, but reflect over y-axis, so decay
0>b>1 so decay, but reflect over y-axis, so growth
Pre-Calculus
1/3/2007
Rewrite with e; approximate k to the nearest tenth. Rewrite with e; approximate k to the nearest tenth.
a.) y = 2x b.) y = 0.3x
y = e0.7x y = e–1.2x
e? = 2 e? = 0.3
Pre-Calculus
1/3/2007
Characteristics of a Basic Logistic Function: Characteristics of a Basic Logistic Function:
Domain:
Range:
Continuity:
Symmetry:
Boundedness:
Extrema:
Asymptotes:
End Behavior:
( - , )
( 0, 1 )
continuous
about ½, but not odd or even
B = 0, b = 0
none
y = 0, 1
lim f(x) x = 1
lim f(x) x - = 0
Pre-Calculus
1/3/2007
Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when?
Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when?
Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when?
Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when?
What are the maximum sustainable populations for the two countries?
What are the maximum sustainable populations for the two countries?
Which model – exponential or logistic – is more valid in this case? Justify your choice.
Which model – exponential or logistic – is more valid in this case? Justify your choice.
Pre-Calculus
1/3/2007
Logarithmic Functions Logarithmic Functions inverse of the exponential function
logbn = p
bp = n
logbn = p iff bp = n
find the power
2? = 32= 5
3? = 1= 0
4? = 2= ½
5? = 5= 1
2? = 2= ½
Pre-Calculus
1/3/2007
Basic Properties of Logarithms(where n > 0, b > 0 but ≠ 1, and p is any real number)
Basic Properties of Logarithms(where n > 0, b > 0 but ≠ 1, and p is any real number)
logb1 = 0 because b0 = 1
logbb = 1 because b1 = b
logbbp = p because bp = bp
blogbn = n because logbn = logbn
Example
log51 = 0
log22 = 1
log443 = 3
6log611 = 11
Pre-Calculus
1/3/2007
Evaluating Common Log ExpressionsEvaluating Common Log Expressions
log 100 =
10 log 8 =
Without a Calculator:
log 32.6 =
log 0.59 =
log (–4) =
With a Calculator:
log 710 =
2
1/7
8
1.5132176
–0.22914…
undefined
Pre-Calculus
1/3/2007
Solving Simple Equations with Common Logs and ExponentsSolving Simple Equations with Common Logs and Exponents
Solve:
10 x = 3.7
x = log 3.7
log x = – 1.6
x ≈ 0.57
x = 10 –1.6
x ≈ 0.03
Pre-Calculus
1/3/2007
Evaluating Natural Log ExpressionsEvaluating Natural Log Expressions
log e7 =
e ln 5 =
Without a Calculator:
ln 31.3
ln 0.39
ln (–3)
With a Calculator:
ln 3e = 1/3
7
5
≈ 3.443
≈ – 0.9416
= undefined
Pre-Calculus
1/3/2007
Solving Simple Equations with Natural Logs and Exponents
Solving Simple Equations with Natural Logs and Exponents
Solve:
ln x = 3.45
x = e 3.45
ex = 6.18
x ≈ 31.50
x = ln 6.18
x ≈ 1.82
Pre-Calculus
1/3/2007
Logarithmic Functions Logarithmic Functions
≈ 0.91 ln x
•vertical shrink by 0.91
xln3ln
1
3ln
xln xln
)3/1ln(
1
)3/1ln(
xln ≈ – 0.91 ln x
•reflect over the x-axis•vertical shrink by 0.91
xlogxlog bb/1
Pre-Calculus
1/3/2007
f(x) = log4x
f(x) = log5x
f(x) = log7(x – 2)
Graph the function and state its domain and range:
f(x) = log3(2 – x)
0.721 ln x
0.621 ln x
0.514 ln (x – 2)
0.091 ln (–(x – 2)
Vertical shrink by 0.721
Vertical shrink by 0.621
Vertical shrink by 0.514, shift right 2
Vertical shrink by 0.091Reflect across y-axisShift right 2
Pre-Calculus
1/3/2007
Logarithmic Functions Logarithmic Functions
one-to-one functions
u = v
isolate the exponential expression
81.22ln
7ln
take the logarithm of both sides and solve
2x = 25
x = 5
log22x = log27
x = log27
Pre-Calculus
1/3/2007
Newton’s Law of CoolingNewton’s Law of CoolingAn 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 Newton’s Law of Cooling to determine when the egg will be 20 C.
ktm0m eTTT)t(T
Pre-Calculus
1/3/2007
Compound Interest Compound Interest
Interest Compounded Annually Interest Compounded Annually
A = P (1 + r)t
A = Amount P = Principal r = Rate t = Time
Interest Compounded k Times Per Year Interest Compounded k Times Per Year
A = P (1 + r/k)kt
k = Compoundings Per Year
Interest Compounded ContinuouslyInterest Compounded Continuously
A = Pert
Pre-Calculus
1/3/2007
Annual Percentage YieldAnnual Percentage Yield
Annual Percentage YieldAnnual Percentage Yield
APY = (1 + r/k)k – 1
Compounded ContinuouslyCompounded Continuously
APY = er – 1
Pre-Calculus
1/3/2007
Annuities Annuities
R = Value of Paymentsi = r/k = interest rate per compoundingn = kt = number of payments
Future Value of an AnnuityFuture Value of an Annuity
n(1 i) 1FV R
i
# 11(p. 324) $14,755.51$14,755.51
Pre-Calculus
1/3/2007
Annuities Annuities
R = Value of Paymentsi = r/k = interest rate per compoundingn = kt = number of payments
Present Value of an AnnuityPresent Value of an Annuity
n1 (1 i)PV R
i
For loans, the bank uses a similar formula
Pre-Calculus
1/3/2007
Annuities Annuities
n1 (1 i)
PV Ri
12(5)0.039
1 112
27500 R0.03912
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?
1 0.8231
27500 R0.00325
R $505.29
