2011 Calculus AB free response #6b

Part B. For x is not equal to 0, express f prime of x as a piecewise-defined function. Find the value of x for which f prime of x is equal to negative 3. So the first thing you might be wondering is, why did we even have to take out x is equal to 0, or why is the derivative not going to be defined there? And that's just because you're going to see that the derivative is going to be something different when we approach x is equal to 0 from the left, versus when we approach x is equal to 0 from the right. And that's why they just took it out of there for us. But let's just figure out what that derivative is for all of the other values of x. So f prime of x is equal to, so for x is less than 0, we're going to take the derivative of this first case. So the derivative of 1 is just 0. The derivative of negative 2 sine of x. Well, the derivative of sine of x is just cosine of x. So it's going to be negative 2 cosine of x for x is less than 0. And then for x is greater than 0-- I'll do this in another color, I'll do it in orange-- we have this case right over here. And we'll just do the chain rule, derivative of negative 4x with respect to x is negative 4. And derivative of e to the negative 4x with respect to negative 4x is just e to the negative 4x. Sometimes you could say this is a derivative of the inside times the derivative of the outside, with respect to the inside. So either way, it's negative 4 e to the negative 4x for x is greater than 0. So we did the first part, we expressed f prime of x as a piecewise-defined function. We didn't define the derivative. I actually forgot a parenthesis here. We didn't define the derivative when x is equal to 0, because it's actually not going to be defined there. Now let's do the second part. Find the value of x for which f prime of x is equal to negative 3. And so if this wasn't piecewise-defined, you'd very simply just say, look, f prime of x is equal to negative 3. You would take whatever f prime of x is equal to, and you'd do some algebra to solve for it. But here you're like, which case do I use? I don't know if the x that gets us to negative 3 is going to be less than 0, or I don't know if it's going to be greater than 0. So I don't know which case to use. And one thing that we realize is to look at these functions a little bit and realize that cosine of x is a bounded function. Cosine of x can only go between positive 1 and negative 1. So negative 2 cosine of x can only go between positive 2 and negative 2. So it can never get to negative 3. So if anything's ever going to get to negative 3, it's going to have to be this part of the derivative, or this part of the derivative definition. So it's going to have to be this thing right over here. And hopefully there's some values of x greater than 0 where this thing right over here is equal to negative 3. So let's try it out. Negative 4 e to the negative 4x needs to be equal to negative 3. We can divide both sides by negative 4. We get e to the negative 4x is equal to negative 3/4 divided by negative 4 is 3/4. We could take the natural log of both sides, and we will get negative 4x is equal to the natural log of 3/4. And just to be clear, what I did here, you literally could put the natural log here, natural log there, and you could put the natural log there as well to see that step. This is saying, what power do I have to raise e to, to get e to the negative 4x? Well, obviously, I just need to raise e to the negative 4x power there. So this power is negative 4x. And then we just took the natural log of the right-hand side as well. And then to solve for x, we can divide both sides by negative 4. So you get x is equal to-- or we could multiply both sides by negative 1/4, either way-- negative 1/4 natural log of 3/4. And what we need to do is verify that this x. So we used this case right over here, but we have to make sure that we can use this case, that this x is greater than 0. And we might be tempted right when we look at this to say, wait, this looks like a negative number. But we have to remind ourselves that the natural log of 3/4, since 3/4 is less than e, the natural log of 3/4 is going to be a negative number. It's going to be e to some negative exponent. So since this is negative and this part right over here is negative, you have a negative times a negative, so this right over here is going to be positive. So this is a positive value right over here. So you would use this case right over here. So that's our answer-- x is equal to negative 1/4 natural log of 3/4. Or the derivative, we could write f prime of negative 1/4 times the natural log of 3/4 is equal to negative 3. And we're done.