College Algebra
Tutorial 42: Exponential Functions
Learning Objectives
Introduction
Tutorial
Definition of
Exponential Function
The function f defined by
where b > 0, b1, and the exponent x is any real number, is called an exponential function.
Also note that in this definition, the base b is restricted to being a positive number other than 1.
The two main exponent keys
found on calculators
are (you will only have one of these):
Check to see if you have one of the two main type of
exponent keys
(you won’t have both). If you don’t have one check the
other.
If you don’t see either, look in the reference manual that came with
the
calculator to see which key it is.
If you have the caret top key let's practice
taking 15 and raising
it to the 5th power. To do this you would type in 15^5 and press
your enter or = key. If you got 759375,
you
entered it in correctly. If not, try again. If you still
can't
get it look in your reference manual that came with the
calculator.
The key that look
like
OR are
most
common in business and scientific calculators, but can be found on
other
types of calculators.
On most business and scientific calculators, the exponent key looks
like
or or very similar to this .
If you have this key let's practice taking 15 and raising it to the 5th power. In this situation, you first type in your base then you activate your exponent key and then you type in your exponent and then press enter or equal. Go ahead and try it out by finding 15 raised to the 5th power. Type in 15, then press or and then type in 5 and press the enter or equal key. You should have gotten 759375 as your answer. If not, try again. If you still can't get it look in your reference manual that came with the calculator.
e is approximately 2.718281828...
e has a value attached it (similar to pi). e is approximately 2.718281828...
This base is used in economic analysis and problems involving natural growth and decay.
At this point, we are just going to learn how to find the value of e raised to an exponent using the calculator.
e KeyThe two main e keys found on calculators are (you will only have one of these):
Check to see if you have one of the two main type of e keys (you won’t have both). If you don’t have one check the
other. If you don’t see either, look in the reference manual that
came with the calculator to see which key it is.
On most graphing calculators in order to raise e to a power you must press the e key
first,
then press your exponent key ^, and then enter in your
exponent.
If you have the e key (with no exponent
showing)
and the caret top key let's practice taking e and
raising it to the 5th power. To do this you would press the e key, then press the exponent key ^, and then type in 5. It
should look like this on your screen: e^5. If you got
148.41316..., you entered it in correctly. If not, try again. If
you still can't get it look in your reference manual that came with the
calculator.
The key that looks
like is most common in business and scientific calculators, but can be found
on other types of calculators.
On most business and scientific calculators, the e function key looks like or very similar to this. So check for this key - note that
the difference between this and the one above is that this key has a
variable
exponent showing on the key - the above key only has an e (no
exponent. If you have this key let's practice taking e and raising it to the 5th power. In this situation, you
first
type in your exponent and then you activate your key. Go ahead and try it out by finding e raised to the 5th power. Type in 5 and then press .
You should have gotten 148.41316... as your answer. If not, try
again. If
you still can't get it look in your reference manual that came with the
calculator.
Graphing calculators:
Most graphing calculators allow you to put in the whole formula before
you press enter. In fact you are able to see it all. If you
are going to plug in the whole formula at one time, just make sure you
are careful. Pay special attention to putting in the parenthesis
in the right place.
Business and scientific calculators:
On most business and scientific calculators you will have to put the
formula in part by part. Work your way inside out of the
parenthesis. DO
NOT round until you are at the end. As you go step by step,
don't
erase what you have on your calculator screen, but use it in the next
step,
so you will have have the full decimal number. The examples are
set
up to show you how to piece it together - it goes step by step.
All calculators:
DO NOT round until you get to the final answer.
You will
note on a lot of the examples that I put dots after my numbers that
would
keep going on an on if I had more space on my calculator. Keep in
mind that your calculator may have fewer or more spaces than my
calculator
does - so your calculator may have a slightly different answer
than
mine due to rounding. It should be very close though.
Make sure that you go through
these examples
with your calculator to check to make sure you are
entering in everything ok.
If you are
having problems check your reference book that came with your
calculator
or ask your math teacher about it.
Be careful that you use the order of operations when working a problem like this. We need to deal with the exponent first BEFORE we multiply by 4. It is really tempting to cross the 4 on the outside with the 4 in the denominator of our base. But the 4 in the denominator is enclosed by a ( ) with an exponent attached to it, so we have to deal with that first before getting the 4 on the outside involved.
Let’s find some order pairs. Again, we will use the same input values for x we used in examples 3 and 4 and find the corresponding output values for this exponential function.
where,
S = compound or
accumulated amount
P = Principal
(starting
value)
r = nominal rate (annual
% rate)
n = the number of compound
periods per
year
t = number of years
Also sometimes r represents a periodic rate where in the above formula it is a nominal rate. In that type of formula you would not see r being divided by n, but you would still need to do that if your r represents a periodic rate. It just isn't shown as part of the formula.
Just keep in mind that no matter how the formula looks the concept is the same.
$15000 for 14 years at an annual rate of 5% compounded monthly.
Also note that different calculators round to different place values. So when you are putting this into your calculator keep in mind that yours may be rounding to a different place than mine, so it may be slightly different for the last digit to the right of the decimal. Also, don’t round anything until you get to the final answer. For example if you round to 2 decimal places in the first step, then your final answer may be off. You want to keep as close to the numbers as possible, so go with whatever your calculator gives you and then round when you write your final answer.
*Plug in values shown above into compound form.
*Find number inside the ( ) first
*Raise the ( ) to
the 168th power
*Multiply
The compound amount is the total amount that is in the account. How do you think we are going to get the interest?? Well we have the principle which is the beginning amount and we have the compound amount which is the end result. Looks like, if we take the difference of the two, that will give us how much interest was earned from beginning to end. What do you think?
Compound amount - principle: 30162.39 - 15000 = 15162.39
So our compound interest is $15162.39.
Wow, our money doubled and then some - of course it compounded 168 times.
$20500 for 15 years at an annual rate of 7.5% compounded semiannually.
*Plug in values shown above into compound form.
*Find number inside the ( ) first
*Raise the ( ) to
the 30th power
*Multiply
The compound amount is the total amount that is in the account. How do you think we are going to get the interest?? Well we have the principle which is the beginning amount and we have the compound amount which is the end result. Looks like, if we take the difference of the two, that will give us how much interest was earned from beginning to end. What do you think?
Compound amount - principle: 61858.16 - 20500 = 41358.16
So our compound interest is $41358.16.
where,
S = compound or
accumulated amount
P = Principal
(starting
value)
r = nominal rate (annual
% rate)
t = number of years
Note that this formula may look different than the one from your math book. Sometimes an A is used for compound or accumulated interest instead of an S.
Just keep in mind that no matter how the formula looks the concept is the same.
*Raise e to the 18th power
*Multiply
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1b: Approximate the given number using a calculator. Round to four decimal places.
Practice Problems 2a - 2b: Graph the given function.
Practice Problems 3a - 3b: Find the a) compound amount AND b) the compound interest for the given investment and rate.
Practice Problem 4a: Find the accumulated value for the given investment and rate.
Need Extra Help on these Topics?
The following are webpages that can assist you in the topics that were covered on this page.
http://www.purplemath.com/modules/graphexp.htm
This webpage goes over graphing exponential functions.
http://www.sosmath.com/algebra/logs/log4/log4.html#exponential
This website will help you with the definition of and graphing
exponential
functions. Stop when it gets to
logarithmic
functions, we will go over that in another tutorial.
http://www.ping.be/~ping1339/exp.htm#Exponential-function
This website will help you with the definition of and graphing
exponential
functions. Stop when it gets to
logarithmic
functions, we will go over that in another tutorial.
Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.
Last revised on March 21, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.