It might already have occurred to you that there is a problem with the idea of letting the exponent in a power expression be a variable with arbitrary real values. After all we have only defined powers with rational exponents.
The key fact that allows us to invent a definition for all real exponents is the fact that small changes in the exponent make only small changes in the result. This allows us to “fill in the gaps” by defining irrational powers as “limits” of rational ones.
This idea is elaborated at the beginning of Section 4.2 of the text.
Open the text now and read pages 291 and 292.
p291
After reading the authors’ construction of the graph of on page 291, repeat the same process (starting with integer values of x, adding some fractional points, and filling in) to draw graphs for and . Your graphs of , , and should each look roughly like one or other of the general patterns shown at the top of page 285.
This process would work similarly for any base a>0. What happens if the base is negative? Or zero? (for answer look on page 292)
p292
If a > 0, then so is any power of a. So the graph of stays always above the x-axis. But even though is always positive, if a less than 1 then high powers of a will be very small, and the graph will get very close to the x-axis on the far right.
On the other hand, if a > 1 then high powers of a will be large, but high negative powers will be the reciprocals of large numbers and so will give very small results. So in this case the graph of will get close to the x-axis on the far left. Either way, our exponential graph has the x-axis as an asymptote. Unlike the case for rational functions, these exponential functions have asymptotes on just one side – on the left if the base is greater than 1 and on the right if the base is less than 1.
Note also that if a>1 then is an increasing function (ie greater exponents give bigger results) but that if 0<a<1 then it is decreasing (greater exponents give smaller results). What happens if a=1? (the answer is at the end of the second last paragraph before the theorem on p292).
p292-4
As a consequence of the previous observation we see that for the simple exponential function with base a is one-to-one. This makes it possible to solve simple exponential equations as in the Illustration and Example#1 at the top of page 293.
Once we understand the general pattern it is easy to quickly graph simple exponential functions as in Examples#2&3 by just using the point at x=1 to get the scale right. And then shifting and scaling can be used to graph related functions as in Examples#4&5 on pg 294-5.
Note that the graph for Example#6 is not one-to-one. Although exponential functions of the form are always one-to-one if 0< , other functions involving exponentials need not be.
Our text, like most, uses the name “exponential function” just for cases with , but some others use it more generally and call a “simple exponential function”.
p295
The general behaviour of more complicated functions involving exponentials can often be figured out on the basis of whether the powers involved have positive or negative exponents. For example in the bell curve of Example#6, the exponent is always negative and becomes even more negative as . So the value of is always less than 1 and the graph has a horizontal asymptote at y = 0 on both sides.
Using this quick qualitative approach you might have drawn the correct overall shape without plotting as many explicit points as the text did – just the intercept and asymptotes might have been enough.
After considering Example#6, try using the quick qualitative approach to graph , then check or improve your graph by adding a few more explicit points (eg at x = -2, -2, 0, 1, 2) as the text did for the bell curve, and finally check your result with a graphing calculator or the Graph Explorer applet.
p296-299
When reading through the Applications on pages 296-299, note that in each case the rate of change of some quantity might be expected to be proportional to the quantity itself. For example, the rate of growth of a population should depend on the number of adults available to breed; the rate at which money grows in the bank depends on how much you have on deposit; or, in the last example, the amount of drug left in the pill might go down due to dissolving through the wall of the pill at a rate which might depend on the concentration of what’s left inside the pill. It will be important for you to be able to recognize such situations on the basis of a verbal description (even if they come from applications that have not been studied in this course) and to realize that exponential functions might be necessary in order to study them.
NOTE: In some of the applications (eg population growth) the quantity involved actually only takes on whole number values but the timing of the jumps (eg births) may be unpredictable or random. In such cases, the continuous exponential growth model might be thought of as an average over all possible cases. So if a model predicts say 12.5 deer on an island at some time it might be reasonable to expect anywhere from 10 with two pregnant to 13 with one newborn. In other cases, such as compound interest, the jumps occur at specific times and the formula only applies exactly at those times. To get the exact formula would require rounding down the time to the immediately preceding interest payment – which would be an application of the greatest integer function.
The main points to remember are as follows:
Exponential functions are functions of the form for .
(For irrational values of the argument they are defined by taking limits of cases with rational ones.)
For a>0 the value of is always positive, no matter what x is.
For every positive the exponential function with base a is one-to-one
(increasing if a>1 with a horizontal asymptote at y=0 on the left,
and decreasing with a horizontal asymptote at y=0 on the right if a<1).
What happens if a<0? If a=0? If a=1? (See answer #1 below)
The one-to-one property allows us to solve some simple exponential equations.
The general pattern for this is that if , then expression#1=expression#2.
Check that you understand this by solving for x in .(See answer #2 )
Note that each side must be of the form and the bases must be equal for this to work. Sometimes a bit of cleverness with power rules can get you to this form if that’s not quite what you are given. For example can you solve ? (See answer #3)
Once we understand the general pattern it is easy to quickly graph simple exponential functions by just using the points at x=0 and x=1 to get the scale right.
What are the y-values at these two points? (See answer #4)
Shifting and scaling can be used to graph related functions.
How are (i) and (ii) related to ?(See answer #5)
The general behaviour of more complicated functions involving exponentials can often be figured out on the basis of whether the powers involved have positive or negative exponents. For example in the bell curve of Example#6, the exponent is always negative and becomes even more negative as . So the value of is always less than 1 and the graph has a horizontal asymptote at y = 0 on both sides.
Using this quick qualitative approach you might have drawn the correct overall shape without plotting as many explicit points as the text did – just the intercept and asymptotes and maybe one other point (e.g. at x=1) might have been enough.
Try using the quick qualitative approach to graph , then check or improve your graph by adding a few more explicit points (eg at x = -2, -2, 0, 1, 2) as the text did for the bell curve, and finally check your result with a graphing calculator or the GraphExplorer applet.
Applications of exponential functions include radioactive decay, population growth, compound interest, and just about anything else where the same factor is applied repeatedly. This happens typically when the rate of change of some quantity is proportional to the current size of the quantity itself.
For example, the rate of growth of a population should depend on the number of adults available to breed; the rate at which money grows in the bank depends on how much you have on deposit; or, in the text’s last example on page 291, the amount of drug left in a pill might go down due to dissolving through the wall of the pill at a rate depending on the concentration of what’s left inside the pill. It will be important for you to be able to recognize such situations on the basis of a verbal description (even if they come from applications that have not been studied in this course) and to realize that exponential functions might be necessary in order to study them.
NOTE: In some of the applications (eg population growth) the quantity involved actually only takes on whole number values but the timing of the jumps (eg births) may be unpredictable or random. In such cases, the continuous exponential growth model might be thought of as an average over all possible cases. So if a model predicts say 12.5 deer on an island at some time it might be reasonable to expect anywhere from 10 with two pregnant to 13 with one newborn. In other cases, such as compound interest, the jumps occur at specific times and the formula only applies exactly at those times. To get the exact formula would require rounding down the time to the immediately preceding interest payment – which would be an application of the greatest integer function.
(See answer#6).
Check your understanding, and practice for speed, by working through some of the Exercises on pp299-304.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort.)
Rather than using the solution guide try to match the questions with text examples
Eg for #1-10 try comparing with Example#1
for #11-24 try comparing with Examples#2-4
for #25-28 try comparing with Example#5
for #29-48 try to find similar applications or question types among the Applications on pp295-299.
#49-66 give you an opportunity to practice using your calculator or computer on similar examples involving numbers you might not want to crank out “by hand”.
As a bare minimum you should do ##1,5,11,15,21,25,31,35,41and 45, and when done, compare your solutions with those in the student solutions guide. If your answers are different or you couldn’t do them without looking, check the guide to see what ideas you were missing and try again with some of ##3,13,23,33,and 43. And so on…
Your criterion for success should be to be able to complete such a set of 5 correctly without referring to the solutions before you are done.
1. If a is negative, then is not a real number. So is undefined as a real number when . The same is true whenever is rational with an even denominator and odd numerator – which includes, for example, every decimal ending with an odd digit.
And if x has an odd denominator, just increasing the numerator by 1 makes the result jump across to the other side of the x-axis. So there is no natural way to “fill in” the graph, and we shall not attempt to study exponential functions with negative base.
If then is undefined, as is also for all . And at x=0 we get another undefined expression . is well defined for x>0, but since its domain is restricted it is not usually included as an “exponential function”.
If , then for all x. This is perfectly well defined but is not one-to-one (in fact it is constant) and our text does not call it an “exponential function”.
3. and , so it’s the same as the previous one.
4. At x=0, y=1 and at x=1, y=a
5. The most natural correct answers would be
(i)Shift left by 3, and (ii) Squash horizontally to half its width.
But by the results of #3 above, for (i) it would also be correct to say
(i)Stretch vertically by a factor of 8
6. If interest is paid at the end of each compounding interval, then the correct formula would be