Download Exponential Functions: A Comprehensive Guide with Examples and Applications and more Exams Calculus in PDF only on Docsity!
Exponential Functions
In this chapter, a will always be a positive number. For any positive number a > 0, there is a function f : R! (0, 1 ) called an exponential function that is defined as f (x) = a x^. For example, f (x) = 3 x^ is an exponential function, and g(x) = ( 174 ) x^ is an exponential function. There is a big di↵erence between an exponential function and a polynomial. The function p(x) = x 3 is a polynomial. Here the “variable”, x, is being raised to some constant power. The function f (x) = 3 x^ is an exponential function; the variable is the exponent.
Rules for exponential functions
Here are some algebra rules for exponential functions that will be explained in class. If n 2 N, then a n^ is the product of n a’s. For example, 3^4 = 3 · 3 · 3 · 3 = 81
a 0 = 1
If n, m 2 N, then
a mn = m
p a n^ = ( m
p a) n
a x^ =
a x
The rules above were designed so that the following most important rule of exponential functions holds:
a x^ a y^ = a x+y
Another variant of the important rule above is
a x a y^
= a x y
And there is also the following slightly related rule
(a x^ ) y^ = a xy
Examples.
(^12) = 2
p 4 = 2
- 7 ^2 · 7 6 · 7 ^4 = 7 2+6 ^4 = 7 0 = 1
- 10 ^3 = 1013 = 10001
- 15
6 155 = 15^
101 = 3 2 = 9
2
(8)
2 3
( 3
p
- 2
taller and taller; and when moving to the left, the graph becomes shorter and shorter, shrinking towards, but never touching, the x-axis.
Not only does the graph grow bigger as it moves to the right, but it gets big in a hurry. For example, if we look at the exponential function whose base is 2, then
f (64) = 2 64 = 18, 446 , 744 , 073 , 709 , 525 , 000
And 2 isn’t even a very big number to be using for a base (any positive number can be a base, and plenty of numbers are much, much bigger than 2). The bigger the base of an exponential function, the faster its graph grows as it moves to the right. Moving to the left, the graph of f (x) = a x^ grows small very quickly if a > 1. Again if we look at the exponential function whose base is 2, then
f ( 10) = 2 ^10 =
The bigger the base, the faster the graph of an exponential function shrinks as it moves to the left. 181
taller and taller; and when moving to the left, the graph becomes shorter and shorter, shrinking towards, but never touching, the x-axis.
Not only does the graph grow bigger as it moves to the right, but it gets big in a hurry. For example, if we look at the exponential function whose base is 2, then f (64) = 2^64 = 18, 446 , 744 , 073 , 709 , 525 , 000 And 2 isn’t even a very big number to be using for a base (any positive number can be a base, and plenty of numbers are much, much bigger than 2). The bigger the base of an exponential function, the faster its graph grows as it moves to the right. Moving to the left, the graph of f (x) = ax^ grows small very quickly if a > 1. Again if we look at the exponential function whose base is 2, then
f ( 10) = 2 ^10 =
The bigger the base, the faster the graph of an exponential function shrinks as it moves to the left.
taller and taller; and when moving to the left, the graph becomes shorter and shorter, shrinking towards, but never touching, the x-axis.
Not only does the graph grow bigger as it moves to the right, but it gets big in a hurry. For example, if we look at the exponential function whose base is 2, then f (64) = 2^64 = 18, 446 , 744 , 073 , 709 , 525 , 000 And 2 isn’t even a very big number to be using for a base (any positive number can be a base and plenty of numbers are much, much bigger than 2). The bigger the base of an exponential function, the faster it grows. Moving to the left, the graph of f (x) = ax^ grows small very quickly if a > 1. Again if we look at the exponential function whose base is 2, then
f ( 10) = 2 ^10 =
The bigger the base, the faster the graph of an exponential function shrinks as we move to the left. 4
Base less than 1 (but still positive). If a is positive and less than 1, then we we can show from our rules of inequalities that a n+1^ < a n^ for any n 2 Z. That means that
· · · > a ^3 > a ^2 > a ^1 > a^0 > a^1 > a^2 > a^3 > · · ·
So the graph of f (x) = a x^ when the base is smaller than 1 slopes down as it moves to the right, but it is always positive. As it moves to the left, the graph grows tall very quickly.
One-to-one and onto
Recall that an exponential function f : R! (0, 1 ) has as its domain the set R and has as its target the set (0, 1 ). We see from the graph of f (x) = a x^ , if either a > 1 or 0 < a < 1, that f (x) is one-to-one and onto. Remember that to check if f (x) is one-to-one, we can use the horizontal line test (which f (x) passes). To check what the range of f (x) is, we think of compressing the graph of f (x) onto the y-axis. If we did that, we would see that the range of f (x) is the set of positive numbers, (0, 1 ). Since the range and target of f (x) are the same set, f (x) is onto. 182
Base less than 1 (but still positive). If a is positive and less than 1, then we we can show from our rules of inequalities that an+1^ < an^ for any n ⌅ Z. That means that · · · > a ^3 > a ^2 > a ^1 > a^0 > a^1 > a^2 > a^3 > · · · So the graph of f (x) = ax^ when the base is smaller than 1 slopes down as it moves to the right, but it is always positive. As it moves to the left, the graph grows tall very quickly.
One-to-one and onto
Recall that an exponential function f : R ⇥ (0, ⇤) has as its domain the set R and has as its target the set (0, ⇤). We see from the graph of f (x) = ax, if either a > 1 or 0 < a < 1, that f (x) is one-to-one and onto. Remember that to check if f (x) is one-to-one, we can use the horizontal line test (which f (x) passes). To check what the range of f (x) is, we think of compressing the graph of f (x) onto the y-axis. If we did that, we would see that the range of f (x) is the set of positive numbers, (0, ⇤). Since the range and target of f (x) are the same set, f (x) is onto. 146
Base less than 1 (but still positive). If a is positive and less than 1, then we we can show from our rules of inequalities that an+1^ < an^ for any n ⌅ Z. That means that · · · > a ^3 > a ^2 > a ^1 > a^0 > a^1 > a^2 > a^3 > · · · So the graph of f (x) = ax^ when the base is smaller than 1 slopes down as it moves to the right, but it is always positive. As it moves to the left, the graph grows tall very quickly.
One-to-one and onto
Recall that an exponential function f : R ⇥ (0, ⇤) has as its domain the set R and has as its target the set (0, ⇤). We see from the graph of f (x) = ax, if either a > 1 or 0 < a < 1, that f (x) is one-to-one and onto. Remember that to check if f (x) is one-to-one, we can use the horizontal line test (which f (x) passes). To check what the range of f (x) is, we think of compressing the graph of f (x) onto the y-axis. If we did that, we would see that the range of f (x) is the set of positive numbers, (0, ⇤). Since the range and target of f (x) are the same set, f (x) is onto.
importance of the number e becomes more apparent after studying calculus, but we can say something about it here.
Let’s say you just bought a new car. You’re driving it o↵ the lot, and the odometer says that it’s been driven exactly 1 mile. You are pulling out of the lot slowly at 1 mile per hour, and for fun you decide to keep the odometer and the speedometer so that they always read the same number. After something like an hour, you’ve driven one mile, and the odometer says 2, so you accelerate to 2 miles per hour. After driving for something like a half hour, the odometer says 3, so you speed up to 3 miles an hour. And you continue in this fashion. After some amount of time, you’ve driven 100 miles, so you are moving at a speed of 100 miles per hour. The odometer will say 101 after a little while, and then you’ll have to speed up. After you’ve driven 1000 miles (and here’s where the story starts to slide away from reality) you’ll have to speed up to 1000 miles per hour. Now it will be just around 3 seconds before you have to speed up to 1001 miles per hour. You’re traveling faster and faster, and as you travel faster, it makes you travel faster, which makes you travel faster still, and things get out of hand very quickly, even though you started out driving at a very reasonable speed of 1 mile per hour.
If x is the number of hours you had been driving for, and f (x) was the distance the car had travelled at time x, then f (x) is the exponential function with base e. In symbols, f (x) = e x^. Calculus studies the relationship between a function and the slope of the graph of the function. In the previous example, the function was distance travelled, and the slope of the distance travelled is the speed the car is moving at. The exponential function f (x) = e x^ has at every number x the same “slope” as the value of f (x). That makes it a very important function for calculus. For example, at x = 0, the slope of f (x) = e x^ is f (0) = e 0 = 1. That means when you first drove o↵ the lot (x = 0) the odometer read 1 mile, and your speed was 1 mile per hour. After 10 hours of driving, the car will have travelled e 10 miles, and you will be moving at a speed of e 10 miles per hour. (By the way, e 10 is about 22, 003.)
Exercises
For #1-11, write each number in simplest form without using a calculator, as was done in the “Examples” in this chapter. (On exams you will be asked to simplify problems like these without a calculator.)
1.) 8 ^1
2.) ( 1743 ) 3 ( 1743 ) ^3
3.) 125 ^
(^13)
4.) 100 ^5 · 100 457 · 100 ^50 · 100 ^400
5.) 4 ^
(^32)
1001
(^23)
8.) 97 ^16 97
(^36)
297 3300
(^47) )
(^144)
For #23-31, match the numbered functions with their lettered graphs.
23.) 2 x^ 24.) ( 13 ) x^ 25.) 2 x^ + 1
26.) ( 13 ) x^ 1 27.) ( 13 ) (x 1)^ 28.) 2 x
29.) 5(2 x^ ) 30.) ( 13 ) x^ 31.) 2 (x+1)
A.) B.) C.)
D.) E.) F.)
G.) H.) I.)
187
I
I^
—^
—
V
I
—
V
—
I^
—^
—
V
I
I^
—^
—
V
I
—
V
I
I^
—^
—
V
I
V
I
For #32-40, match the numbered functions with their lettered graphs.
32.) e x^ 33.) ( 12 ) x^ 34.) e x^ 1
35.) ( 12 ) x^ + 1 36.) ( 12 ) (x+1)^ 37.) e x
38.) 2(e x^ ) 39.) ( 12 ) x^ 40.) e (x 1)
A.) B.) C.)
D.) E.) F.)
G.) H.) I.)
00 GO
C
C
00 GO
C C^
00 GO
C^ C
00 GO
C
C C
