Calculus: Graphing Exponential Functions

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Special Functions
Exponential Functions
Graphing Exponential Functions Page [1 of 2]
Okay. Well, now it's time to think about functions that are actually exponential functions, which means that the variable is in the exponent, which is dramatically different than what we've been looking at so far. So first, I thought we'd begin with a little review of exponents, just to remind you of how all this stuff fits together here.
Let's suppose that N is some number. Take a look, for example at N, raised to the A power, times N raised to the B power. Then what does that equal? Well, since these bases are the same, the rule is that I can just add the exponent. And so what I see here is N to the power A plus B. So that actually can be a very useful rule, which I'm going to use an awful lot, as you'll see.
Now, what if we actually replaced the A by, by a variable. So for example, suppose we create a function -- let's say F of X equals N raised to the X power. So, for example, I could look at a function, maybe, I'll call it G of X which equals two to the X. So, notice that the roles have switched, usually we look at X to the two power, or X squared. But here what we're looking at is the number two raised to a variable power. And this is the function, where when you put something into the machine, what it does is, it takes that thing you put in, takes two and raises it to that power.
So for example, what is G? Let's actually do some examples together. What's G of three? Well, that would be two cubed, which would be eight. What's G of one? Well, that would be two to the one, which would be two. What about G to the negative two? Well, that's a little tricky. But, that would be two raised to the negative two power and remember the negative sign in the exponents, pushes me downstairs, and so what I actually see is one over two squared, which is four. So I see one fourth. What would G of zero be? G of zero would be - well, two to the zero, which is one. So you can see that the variable, the thing that is changing is the exponent, is the exponent.
What does the graph of such a function look like? Let's actually take a look at how to graph these exponents, exponential functions. So if we look at G of X equals two to the X, we could actually plot all those points that we just saw. So, if we plot those points, let's see what we see here. So let's see, when X is zero, we've already seen that G of zero, two to the zero, is one. So, when x is zero, we're at one. Put a little dot there. At one, two to the one is two, so at one, we're at two. At two, we're already at eight. Look how quickly this function grows. At two, we're already at eight, grows quite rapidly.
What about the negative values? How about negative one? Well, two to the negative one power would be one over two or a half, so in fact, a negative one, we're located at a half. And we've already seen that at negative two, I'm at a quarter. That's even closer, and if you graph these, and connect the, connect the lines, what you see is a curve that goes very, very, quickly up. And then gently lands and seems to want to head toward the negative X axis, but never touches it. And it approaches that negative X axis, but like I said it's asymptotic to it. It never touches it. And that's the graph of G of X, two to the X, and this, the standard graph of an exponential function.
Now what happens if I, if I make this number in here smaller? For example, what if I put in here -- let's say, instead of two, one and a half, or three halves to the X? Well, if you think about it, if I put a smaller number in, when I raise it to powers, well, they'll get bigger, but they'll get bigger at a slower rate than this does. For example, at one that would be three halves, whereas this was two, so this would actually be over here. What happens is, if you put a smaller value for, in for, the, the base, you get a graph that looks like this. It grows gradually here, and oops, and more gradually here. Let me see if I can fatten that up there. Right, that might be, that might be the graph of three halves to the X. The point is, that the smaller this number is the more gradual this curve is, the larger the number is -- for example, if I put a three in there, three to the X, that's going to grow very quickly. For example, at one, it's already a three, at two it's already at nine. So in fact, if I put in, if I put in, a bigger number, then it's even more steep comes down, it comes below. Again, never touching the axis but much steeper, this might be three to the X. So the general form of these curves is of this kind.
What by the way, if I put in, I put a number say oh, I don't know, how about like negative three? Can I look at that? Can I look at this function here? B of X equals negative three to the X. Well, let's think about that? What happens if I evaluate this at, at a half? Well, that would be minus three to the one half power, which remember one half power means square root. So, I'd be taking the square root of minus three, but the square root of minus three doesn't exist. There is no number whose square is negative three. So in fact, this is undefined when X equals a half. It's also undefined when X equals a fourth, or X equals a sixth, or X equals three eighths. Anytime I have that, an even denominator here in the X; in fact, this is not going to exist. And thus, we never talk about functions like this. This is a bad function that's why I called it B. This is a bad function.
We always look at bases that are positive. So the general form that we look at is the following, in fact let me write this down here. F of X equals some number, some fixed number to the X, where that number is positive, where that number is positive. And, and how does the graph look? Well, it depends upon the size of that number. Let me try to recap this for you. So F of X equals N, some fixed number to the X, where N has to be positive, otherwise, you don't look at this function. And the graph basically looks like this. Well, for numbers they all pass through the point 01. And for large numbers, they look like this. This is for large N, large N. You put in a smaller number; slightly smaller number then it, then it grows more gradually. This is a smaller N, smaller N.
What if I put in a really small N? Like for example, let's try to put in like a half. Well, if you put in a half, and actually graph--plot some points, you actually see that the graph actually goes the complete opposite way. Let's think about that, for example one half to the first power would be a half. So at one, I'd put a dot at a half. And one half squared is actually a fourth, so then I'd put a little thing at a fourth. Whereas one half to the minus one, one half raised to the minus one power, that flips, and so I see a two. So it's just this picture flipped. So in fact, I get this kind of picture. This would be for N, that are smaller than one. This is smaller N bigger than one and this is large N. So the picture always looks something like this, and that gives you a sense of the exponential function.
Now, maybe, first of all, why would we even need the exponential function? Well, it turns out the exponential function is critical, when you want to study growth and nature. And in fact, maybe you've heard of the very special number E, the very special number E. This number is used all the time in studying growth. For example, if you have money in a bank account that actually is generating interest - well, the interest is accumulating a certain way; E is involved. If you, if you look at how something grows, population, so forth, reproduction, reproduction, it's all E. So you might be saying, "Well, where does the number E come from and what exactly does it equal?" It turns out that E comes from a calculus idea - in particular, finding derivatives.
So what we're going to do next actually is discover exactly how to figure out what that value E is, and what it means, and the power of it. So, we're going to take a look at that, with the exponential idea, up next when we take a look at how to find derivatives of these kinds of functions. Okay. Bye, bye.