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BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
§4.1 ax
Functions
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §3.5 → Applied Optimization
Any QUESTIONS About HomeWork• §3.5 → HW-17
3.5
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 3
Bruce Mayer, PE Chabot College Mathematics
§4.1 Learning Goals
Define exponential functions Explore properties of the natural
exponential function Examine investments involving
continuous compounding of interest
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 4
Bruce Mayer, PE Chabot College Mathematics
Exponential Function
A function, f(x), of the form
f x ax , a 0 and a1,
is called an EXPONENTIAL function with BASE a.
The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 5
Bruce Mayer, PE Chabot College Mathematics
Recall Rules of Exponents
Let a, b, x, and y be real numbers with a > 0 and b > 0. Then
ax ay axy ,ax
ayax y ,
ab x axbx ,
ax y axy ,
a0 1,
a x 1
ax
1
a
x
.
yxaa yx thenif
Product Rule
Quotient Rule
Product to a Power Rule
Power to a Power Rule
Zero Power Rule
Negative Power Rule
Equal Powers Rule
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 6
Bruce Mayer, PE Chabot College Mathematics
Evaluate Exponential Functions
Example a. Let f x 3x 2. Find f 4 . Solution a. f 4 34 2 32 9
Example b. Let g x 210x. Find g 2 .
Solution b. g 2 210 2 21
102 21
100 0.02
b. g 2 210 2 21
102 21
100 0.02
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 7
Bruce Mayer, PE Chabot College Mathematics
Evaluate Exponential Functions
Example
Solution
c. Let h x 1
9
x
. Find h 3
2
.
c. Let h 3
2
1
9
3
2 9 1
3
2 93
2 27
c. Let h 3
2
1
9
3
2 9 1
3
2 93
2 27
-1 -0.5 0 0.5 1 1.5 20
1
2
3
4
5
6
7
8
9
x
y =
f(x)
= (
1/9
)x
MTH15 • Bruce Mayer, PE
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 8
Bruce Mayer, PE Chabot College Mathematics
Solve Exponential Equation
Solve the following for x
Using the Transitive Property
Need to state 2187 in terms of a Base-3 to a power
Using the Equal Powers Rule
187 2when3 32
xfxf x
xfxf x 187 23 32
73187 2
7333 2732
xx
2442 xxx
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 9
Bruce Mayer, PE Chabot College Mathematics
Example Graph y = f(x) =3x
Graph the exponential fcn: ( ) 3 .xf x
Make T-Table,& Connect Dots
x y
01
–12
–23
13
1/39
1/927
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
4
3
6
2
5
1
-1
-2
78
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 10
Bruce Mayer, PE Chabot College Mathematics
Example Graph Exponential
Graph the exponential fcn:
Make T-Table,& Connect Dots
1( ) .
3
xf x
x y
01
–12
–2–3
11/33
1/99
27 • This fcn is a REFLECTION of y = 3x
3xy
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
4
3
6
2
5
1
-1
-2
78
1( )
3
xy f x
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 11
Bruce Mayer, PE Chabot College Mathematics
Example Graph Exponential
Graph the exponential fcn: y1
2
x
.
Construct SideWays T-Table
x −3 −2 −1 0 1 2 3
y = (1/2)x 8 4 2 1 1/2 1/4 1/8
Plot Points and Connect Dots with Smooth Curve
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 12
Bruce Mayer, PE Chabot College Mathematics
Example Graph Exponential
As x increases in the positive direction, y decreases towards 0
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 13
Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
A. The domain of f(x) = ax is (−∞, ∞).
B. The range of f(x) = ax is (0, ∞); thus, the entire graph lies above the x-axis.
C. For a > 1 (e.g., a = 7)i. f is an INcreasing function; thus, the graph
is RISING as we move from left to right
ii. As x→∞, y = ax increases indefinitely and VERY rapidly
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 14
Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn Properties
Let f(x) = ax, a > 1, a ≠ 1. Then iii. As x→−∞, the values of y = ax get
closer and closer to 0.
D. For 0 < a < 1 (e.g., a = 1/5 = 0.2)i. f is a DEcreasing function; thus, the graph
is falling as we scan from left to right.
ii. As x→−∞, y = ax increases indefinitely and VERY rapidly
iii. As x→ ∞, the values of y = ax get closer and closer to 0
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 15
Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
E. Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus:
i. – The Basis of the Equal Powers Rule
ii. f has an inverse
2121 xxaa xx
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 16
Bruce Mayer, PE Chabot College Mathematics
Exponential Fcn Properties
Let f(x) = ax, a > 0, a ≠ 1. Then
F. The graph f(x) = ax has no x-intercepts • In other words, the graph of f(x) = ax
never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = ax to equal 0
G. The x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 17
Bruce Mayer, PE Chabot College Mathematics
ExponentialFcn ≠ PowerFcn
The POWER Function is the Variable (x) Raised to a Constant Power; e.g.:
• Note that PolyNomials are simply SUMS of Power Functions:
The EXPONENTIAL Function is a Constant Raised to a Variable Power (x); e.g.:
9/4087.0137 xxxx
1238 xxxx
x
xxx
99
57310
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 18
Bruce Mayer, PE Chabot College Mathematics
ExponentialFcn ≠ PowerFcn
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-10
-8
-6
-4
-2
0
2
4
6
8
10
x
y =
f(x)
MTH15 • Power & Exponential Fcns
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
xy 3
3xy
The Exponential is NEVER Negative
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 19
Bruce Mayer, PE Chabot College Mathematics
Example Bacterial Growth
A technician to the Great French MicroBiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.
Assume that the bacteria count B(t) is modeled by the equation
B t 20002t ,• Where t is time in hours
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 20
Bruce Mayer, PE Chabot College Mathematics
Example Bacterial Growth
Given Bacterial Growth Equation B t 20002t ,
Find:a) the initial number of bacteria,
b) the number of bacteria after 10 hours; and
c) the time when the number of bacteria will be 32,000.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 21
Bruce Mayer, PE Chabot College Mathematics
Example Bacterial Growth
a) INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn:
B0 B 0 200020 20001 2000
b) At Ten Hours Sub t = 10 into Eqn:
b. B 10 2000210 2,048,000
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 22
Bruce Mayer, PE Chabot College Mathematics
Example Bacterial Growth
c) Find t when B(t) = 32,000
Thus 4 hours after the starting time, the number of bacteria will be 32k
32000 20002t
16 2t24 2t
4 t
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 23
Bruce Mayer, PE Chabot College Mathematics
The Value of the Natural Base e The number e, an irrational number, is
sometimes called the Euler constant. Mathematically speaking, e is the fixed
number that the expression
approaches e as n gets larger & larger
The value of e to 15 places:
e = 2.718 281 828 459 045
n
n
11
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 24
Bruce Mayer, PE Chabot College Mathematics
The “Natural” base e The most “common” base for people is
10; e.g., 7.3x105
However, analysis of physical; i.e., Natural, phenomena leads to base e
Check the Definition Graphically
0.495% less than the actual e-Value
718.21
1lim
n
n ne
0 10 20 30 40 50 60 70 80 90 1001
1.25
1.5
1.75
2
2.25
2.5
2.75
n
y =
f(n
) =
(1
+ 1
/n)n
MTH15 • e Value
0 1 2 3 4 5 6 7 8 9 101
1.25
1.5
1.75
2
2.25
2.5
2.75
n
MTH15 • e Value
BMay er • 16Jul13BMay er • 16Jul13
215297048138294 2.100
11
100
100
e
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 25
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 16Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = 0; xmax = 100; ymin = 1; ymax = 2.75;% The FUNCTIONx = linspace(xmin,xmax,1000); y = (1 +1./x).^x;% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(1, 2, 1)plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}n'), ylabel('\fontsize{14}y = f(n) = (1 + 1/n)^n'),... title(['\fontsize{16}MTH15 • e Value',]),... annotation('textbox',[.75 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer • 16Jul13','FontSize',7)hold onplot([xmin, xmax], [2.7182818, 2.7182818], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin:10:xmax]); set(gca,'YTick',[ymin:.25:ymax])hold off%%xmin1 = 0; xmax1 = 10; ymin1 = 1; ymax1 = 2.75;% The FUNCTIONn = linspace(xmin,xmax,1000); z = (1 +1./n).^n;% % The ZERO Lines%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(1, 2, 2)plot(n,z, 'LineWidth', 4),axis([xmin1 xmax1 ymin1 ymax1]),... grid on, xlabel('\fontsize{14}n'),... title(['\fontsize{16}MTH15 • e Value',]),... annotation('textbox',[.75 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'BMayer • 16Jul13','FontSize',7)hold onplot([xmin1, xmax1], [2.7182818, 2.7182818], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin1:1:xmax1]); set(gca,'YTick',[ymin1:.25:ymax1])
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 26
Bruce Mayer, PE Chabot College Mathematics
The NATURAL Exponential Fcn
The exponential function
with base e is so prevalent in the sciences that it is often referred to as THE exponential function or the NATURAL exponential function.
f x ex
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 27
Bruce Mayer, PE Chabot College Mathematics
Compare 2x, ex, 3x Several
ExponentialFunctionsGraphically• Note that
EVERY Exponetial intercepts the y-Axisat x = 1
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 28
Bruce Mayer, PE Chabot College Mathematics
Example Graph Exponential
Graph f(x) = 2 − e−3x
SOLUTIONMake T-Table,Connect-Dots
22
1.951
1
−18.09
−401.43
y = f(x)
0
−1
−2
x
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 29
Bruce Mayer, PE Chabot College Mathematics
Exponential Growth or Decay
Math Model for “Natural” Growth/Decay:
A t A0ekt
A(t) = amount at time t A0 = A(0), the initial, or time-zero, amount
k = relative rate of • Growth (k > 0); i.e., k is POSITIVE • Decay (k < 0); i.e., k is NEGATIVE
t = time
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 30
Bruce Mayer, PE Chabot College Mathematics
Exponential Growth
An exponential GROWTH model is a function of the form
00 keAtA kt
where A0 is the population at time 0, A(t) is the population at time t, and k is the exponential growth rate • The doubling time is the amount of time
needed for the population to double in size
A0
A(t)
t
2A0
Doubling time
kteAA 0
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 31
Bruce Mayer, PE Chabot College Mathematics
Exponential Decay
An exponential DECAY model is a function of the form
00 keAtA kt
where A0 is the population at time 0, A(t) is the population at time t, and k is the exponential decay rate • The half-life is the amount of time needed
for half of the quantity to decay
A0
A(t)
t½A0
Half-life
kteAA 0
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 32
Bruce Mayer, PE Chabot College Mathematics
Example Exponential Growth
In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1 percent.
Using the model on the previous slide, estimate the population of the world in the years
a) 2030
b) 1990
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 33
Bruce Mayer, PE Chabot College Mathematics
Example Exponential Growth
SOLUTION a) Use year 2000 as t = 0 Thusfor 2030 t = 30
A0 6
k 0.021
t 30
A t 6e 0.021 30
A t 11.265663
The model predicts there will be 11.26 billion people in the world in the year 2030
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 34
Bruce Mayer, PE Chabot College Mathematics
Example Exponential Growth
SOLUTION b) Use year 2000 as t = 0 Thusfor 1990 t = −10
The model postdicted that the world had 4.86 billion people in 1990 (actual was 5.28 billion).
A0 6
k 0.021
t 10
A t 6e 0.021 10
A t 4.8635055
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 35
Bruce Mayer, PE Chabot College Mathematics
Compound Interest Terms
INTEREST ≡ A fee charged for borrowing a lender’s money is called the interest, denoted by I
PRINCIPAL ≡ The original amount of money borrowed is called the principal, or initial amount, denoted by P• Then Total AMOUNT, A, that accululates in
an interest bearing account if the sum of the Interest & Principal → A = P + I
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 36
Bruce Mayer, PE Chabot College Mathematics
Compound Interest Terms
TIME: Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest amount, within a specified period. This period is called the time (or time-period) of the loan and is denoted by t.
SIMPLE INTEREST ≡ The amount of interest computed only on the principal is called simple interest.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 37
Bruce Mayer, PE Chabot College Mathematics
Compound Interest Terms
INTEREST RATE: The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r.
Unless stated otherwise, it is assumed the time-base for the rate is one year; that is, r is thus an annual interest rate.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 38
Bruce Mayer, PE Chabot College Mathematics
Simple Interest Formula
The simple interest amount, I, on a principal P at a rate r (expressed as a decimal) per year for t years is
I Prt.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 39
Bruce Mayer, PE Chabot College Mathematics
Example Calc Simple Interest
Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6%
a) How much interest-$’s will she receive?
b) How much money will she receive at the end of five years?
SOLUTION a) Use the simple interest formula with:
P = 8000, r = 0.06, and t = 5
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 40
Bruce Mayer, PE Chabot College Mathematics
Example Calc Simple Interest
SOLUTION a) Use Formula
I Prt
I $8000 0.06 5 I $2400
SOLUTION b) The total amount, A, due her in five years is the sum of the original principal and the interest earned
AP IA$8000 $2400
A$10, 400
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 41
Bruce Mayer, PE Chabot College Mathematics
Compound Interest Formula
AP 1 rn
nt
A = $-Amount after t years P = Principal (original $-amount) r = annual interest rate (expressed as a
decimal) n = number of times interest is compounded
each year t = number of years
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 42
Bruce Mayer, PE Chabot College Mathematics
Compare Compounding Periods
One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future-value amount, A, after one year if the interest is compounded:
a) Annually.
b) SemiAnnually.
c) Quarterly.
d) Monthly.
e) Daily.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 43
Bruce Mayer, PE Chabot College Mathematics
Compare Compounding Periods
SOLUTIONIn each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, is changing. Since t = 1, nt = n∙1 = n.
a) AnnualAmount: AP 1
r
n
n
A100 1 0.05 $105.00
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 44
Bruce Mayer, PE Chabot College Mathematics
Compare Compounding Periods
b) Semi Annual Amount:
AP 1r
n
n
A100 10.05
2
2
$105.06
AP 1r
4
4
A100 10.05
4
4
$105.09
c) Quarterly Amount:
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 45
Bruce Mayer, PE Chabot College Mathematics
Compare Compounding Periods
d) Monthly Amount: AP 1
r
12
12
A100 10.05
12
12
$105.12
AP 1r
365
365
A100 10.05
365
365
$105.13
e) Daily Amount:
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 46
Bruce Mayer, PE Chabot College Mathematics
Continuous Compound Interest
The formula for Interest Compounded Continuously; e.g., a trillion times a sec.
A = $-Amount after t years P = Principal (original $-amount) r = annual interest rate (expressed as a
decimal) t = number of years
rtPeA
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 47
Bruce Mayer, PE Chabot College Mathematics
Example Continuous Interest
Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.
SOLUTION: Convert 8-yrs & 3-months to 8.25 years. P = $8300 and r = 0.075 thenuse Formula
APert
A$8300e 0.075 8.25
A$15, 409.83
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 48
Bruce Mayer, PE Chabot College Mathematics
Compare Continuous Compounding
Italy's Banca Monte dei Paschi di Siena (MPS), the world's oldest bank founded in 1472 and is today one the top five banks in Italy
If in 1797 Thomas Jefferson Placed a Deposit of $450k in the MPS bank at an interest rate of 6%, then find the value $-Amount for the this Account in 2010; 213 years Later
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 49
Bruce Mayer, PE Chabot College Mathematics
Compare Continuous Compounding
SIMPLE Interest
AP Prt P 1 rt A$450,000 1 0.06 213 A$6.201 million.
YEARLY Compounding
AP 1 r t $450,000 1 0.06 213
A$1.105 1011
A$110.5 million.
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 50
Bruce Mayer, PE Chabot College Mathematics
Compare Continuous Compounding
Quarterly Compounding
Continuous Compounding
GigaBucks) (145.3 B 3.145$
103.145$4
0.061$450,0001
9
4213
A
A
rPA rt
GigaBucks) (159.8 B 8.159$
108.159$
e$450,0009
21306.0
A
A
PeA rt
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 51
Bruce Mayer, PE Chabot College Mathematics
Account $Value for $450k invested at 6% Interest for 213 Years
159.80
145.30
0.11
0.01
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
Co
nti
nu
ou
sQ
ua
rte
rly
Ye
arl
yS
imp
le
Inte
res
t C
om
po
un
din
g
Account Value ($B)M55_Sec9_1_Compare_Compounding_0810.xls
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 52
Bruce Mayer, PE Chabot College Mathematics
Effective Interest rate → APR
To help people compare simple, MultiPeriod-compounded, and continuous-compounded Interest rates, ALL advertised interest rates are stated in the effective Annual Percentage Rate, or APR or re
APR is the simple annual interest, re , that produces the same Change in $-Value in ONE year
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 53
Bruce Mayer, PE Chabot College Mathematics
Effective Interest rate → APR
APR Defined• For MultiPeriod Compounding
at k times per year• For Continuous
Compounding– Where r is the stated, or nominal, interest rate
When Assessing a Loan or a Savings Instrument the Consumer should consider ONLY the APR for comparisons
k
rre 1
1 re er
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 54
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §4.1• P71 → Beer-Lambert (absorption) Law
See AlsoENGR45
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 55
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
Very GoodCar Loan
RateBut what about the Purchase
$-Price, and Loan Fees?
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 56
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
–
srsrsr 22
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 57
Bruce Mayer, PE Chabot College Mathematics
ConCavity Sign Chart
a b c
−−−−−−++++++ −−−−−− ++++++
x
ConCavityForm
d2f/dx2 Sign
Critical (Break)Points Inflection NO
InflectionInflection
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 58
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 59
Bruce Mayer, PE Chabot College Mathematics
BMayer@ChabotCollege.edu • MTH15_Lec-18_sec_4-1_Exponential_Fcns.pptx 60
Bruce Mayer, PE Chabot College Mathematics
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