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Given the function defined by the equation $s(x)=\frac{e^{x}-e^{-x}}{2},$ (a) Show it is an odd function, Determine on which intervals the function is (b) increasing and decreasing, (c) concave upward and downward, (d) determine any relative extrema (e) Sketch the graph. (f) What, if any, are the extrema of the function? (Note: this function arises frequently in engineering applications and is call the hyperbolic sine, defined by $\sinh (x)=\frac{e^{x}+e^{-x}}{2}$ )

a)(b) inc for all $x$(c) $\mathrm{CU}$ for $x>0,$ CD for $x<0$(d) nonee)(f) none (0,0) is an inflection point

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Missouri State University

Oregon State University

Harvey Mudd College

Baylor University

Lectures

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Okay. So we're given this function F of X is four X times E. To the negative three X. And the first part is we want to determine which on which intervals the function is increasing or decreasing to do that. We have to find the first derivative. So I take the derivative of this function. No, it's the product of four X. And each of the negative three X. So we can use the product rule. We have the first record of the first function which is 4X. Which is four times a second function. Which is easy to negative three X plus the first function for X times the derivative of the second function. Yeah, because it's an exponential derivative. An exponential function is the exponential itself. And then because it's a negative three X in the exponent we have to buy the chain rule Multiplied by -3. Okay, so if I simplify here, I'm going to get four E. To the negative three X minus 12 X. E. To the negative three X. And we want to find the critical points. In other words where this drug is equal to zero. Now to do this, notice that Even the negative three acts as a common factor to both terms. So if I factor that out, I'm going to have each and they give three X. And then in brackets I'm going to have what's left over which is four And then the -12 acts. And we're trying to solve this equation. When is it equal to zero? Now we were the two factors. The parks are two factors equal to zero. So either one of these factors can be zero. But the exponential factor is never zero. It's just property of exponents. That E to any power is never zero for any X. But this this factor could be zero and it's zero. When Well we can solve here -12. X is -4. And if you divide by negative 12 will end up with X equals a third. So that's our critical point. So now what we want to do is do a number line for the first derivative. So here's my number line and we're going to mark out our critical point 1/3 and the derivative, the first rate is zero at this point. Uh That's what we just figured out. But on either side it could be positive or negative. Now, what you wanna do is you want to test points so you want to go into this derivative, Call the derivative, is this expression here and substitute a value of X that's larger than a third or and smaller than a third. What you're going to end up noticing is that if you pick the next value that's larger than the third, any X. Value, you're going to see that this expression this whole expression for the derivative is going to be negative. So we're gonna put a negative sign here and then you'll see similarly, it'll be positive. You pick a X value that's less than third. So what this tells us is that the function is increasing to the left of X equals third. So it's increasing on the interval from negative infinity up to a third and it's decreasing On the interval 1 3rd, deposit infinity. Another way of writing this is it's increasing for X is less than a third and it's decreasing for all X. Greater than the third. Okay, that's part a part B says we want to find where it's concave upwards or concave down and to do that we need the second derivative. So we're going to use this expression for the second derivative. That's all right. Sorry, this expression for the first derivative and then take the driver of this to have a second relative. So down here again, notice that the first river is a product of two functions four minus 12 X. And E to the negative three X. So we use the product rule, The first relative is just negative 12 Because the derivative of 4 -12 x is -12. The second function is E. To the negative three X. Plus the first function. And then again, we want to find the druid of each the negative three X. So that's this times negative three. And now to simplify. Be a bit careful here. So let's copy down the first part, The -3 is going to go into the brackets just. And so we're gonna end up with plus negative 12 Plus 36 x. And then now we again we can combine both factors because they both have each the negative three X. So you can sort of factor that out from both terms to simplify And you end up with -24, That's 36 x. Each of the next three X. Okay. And that's our second derivative. So we want to find where the inflection points inflection points are when the second derivative changes sign and then once we find the inflection points, we can figure out on which intervals it's concave upward, concave down. So do that mean to figure out where the second river is equal to zero. And again, similarly this factor here is never zero. So really it's only going to be zero if this factor is zero. And so if this factor is zero, If you divide by 36 you get uh 2/3. Okay? And then similarly we can create a number line for the second derivative. So here remember line, here's two thirds. And again, when X is 2/3 we just saw that the second river is equal to zero. But on the other side it could be Greater or less than zero, positive or negative. And if we substitute the value of X, that's greater than 2/3 you're going to see that the second road is if you plug it in here it's going to be positive And similar if you plug in, the value of X is less than 2/3, maybe zero. You're gonna see that's going to be end up being negative. So what this means is that for X is greater than 2/3. It's concave up and for X is less than two. There's concrete down. So that's what we can say is our answer. I can't give up for on two thirds to infinity and concave down From negative infinity to 2/3. Okay then we want to find parts CS for any relative extra MMA So to figure that out, we have to use the first derivative. We can actually uh look at our work a little bit up here. So notice uh Ekstrom occurs when the function goes from increasing to decreasing or from decreasing to increasing. In this case, you can see that X equals 1/3. The first relative goes from positive to negative. In other words, the function f goes from increasing to decreasing and so x equals 1 3rd is going to be a relative maximum. So this is part C here. Well, excuse one third is a relative maximum. But we want to find the y coordinate of the maxim also, So I'll say a max or let's say a relative max at X equals one third. And the y coordinate of those max comes from substituting the value of X into The original function. So that means four x times e to the -3 x. And if you do that you'll end up with four times the third times E. To the negative three times one third. That gives us 4/3 times E. To the negative one. And so we can write that as 4/3. And so the relative max so let's keep the colors. The same relative max is one third 4/3. That's the coordinates of our relative max. Okay And finally we'll part d. We want to sketch the graph so um we don't have all the information that we might need but let's just try to get a sketch here. Tracks and why? Then we know at 1/3 and four divided by three E. There is a relative maximums that's probably what we should have here. So maybe I'll put here's one. Yeah let's make it pretty big here. So here's one. So he'll be one third two thirds If you do four divided by three E. That's about 0.5. So in the y coordinate and here's let's see here's one, Here's 1 1/3 about .5 that's a relative maximum. Um So the curve is gonna go something like this. Now we also have an inflection point which you'll notice occurs at X equals two thirds. And reflection point is when the second road of changes sign. So if you quickly evaluate the function at that point C. 42 3rd Each of the -32/3. Maybe I'll just then you end up with eight divided by three E squared. Which is about 0.36. Okay, so that means that at 2/3 which is about here is an inflection point. Okay, so that means the graph is going to go down curving downwards and then it's gonna start curving upwards Now to finish the sketch, we're gonna need to observe a few more things. Well, one thing is that the original function, which is this function here uh has a y intercept and an X intercept of zero. If I plug in zero, I'm going to get zero out. That means the point 0, 0 is going to be on the graph. The origin. Also a few more things as X gets very large, positive as X goes out deposit infinity here. You're going to notice that the original graph um maybe I'll copy down here why equals four X. Eastman egg A three X. You can see that it is a four X. That's the polynomial apart. But it's got an exponential part negative exponential. And so as X gets large, each of this negative, large number is going to take this down to zero. So it's going to have a small toad At y equals zero as X gets large. On the other hand, would access large, negative. So in this direction the exponential terms even get very large and this extreme is going to be negative. So this combination of X term, that's negative and a large exponential term which is positive is going to end the graph is concave down, it's gonna motor the raft down to negative infinity. And that's pretty much what the graph looks like. Uh At least a rough sketch. You can use a graphing calculator or other programs get a more accurate sketch and then finally, party says what if any are the extreme of the function? Well, we can see here that there's no minimum because the graph just keeps going down on the left side here, but there is a maximum, which is this point here, which is our local maximum. Ah It is the local maximum. It's also an absolute maximum for this function.

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