23. Intro to Derivatives & Area Under the Curve
Tangent Lines & Derivatives
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So in a recent video, we learned all about tangent lines, and how to calculate the slope of a tangent line using this equation. Now recall that this is just a line that touches the curve of your function at exactly one point, that's what a tangent line is. And in this video, we're going to now learn how to solve some other problems associated with tangent lines, specifically finding their equations. Now if this sounds scary, don't sweat it, because it turns out the process for doing this is actually related to concepts that we've already learned up to this point. So without further ado, let's just jump right into an example to to see what these kinds of problems look like. So here we're asked to find the equation of the line tangent to this function, f of x is equal to 3x squared minus 4 at x is equal to negative 2. So how can we go about solving this? Well, what I'm going to do is do this by the steps. And our first step is going to be to plug our x value of interest, which we call c, we discussed that in the previous video, into our function f of x to get our y value f of c. Now I can see here that our c value is going to be negative 2 since that's the x value of interest that we're looking at. So since our x value is negative 2, what we need to do is figure out what f of c is, which is gonna be f of negative 2. And f of negative 2, while plugging it into this function, will have 3 times negative 2 squared minus 4. Now, 3 times negative 2 squared turns out to be 12, and 12 minuteus 4 is 8. So what we end up with is 8 for our f of c value. So now that we found c and f of c, what we can do is move on to step 2, which is plugging things into this equation for the slope of a tangent line. Now we already figured out that the c here is negative 2, so this is going to stay consistent for this limit. But as for the rest of the equation, well, we need to plug our function in for f of x. F of x, we can see, is 3x squared minus 4. Now this whole thing is going to be minus f of c, which we just calculated to be 8. And all of this will be divided by x minus c, which is the same thing as x minus negative 2 or x plus 2. So this is what we get for our equation. Now from here, what I'm going to do is simplify what we have on top since I can see I can take this negative 4 and subtract off 8. That's going to give me negative 12, so we're going to have 3x squared minus 12 all over x plus 2. Now at this point, what I'm going to do is try applying my limit by pulling negative 2 in for x. But I notice that if I do that, I'm going to end up with negative 2 +2 on bottom, which is 0. You cannot divide by 0. You cannot have 0 in the denominator of a fraction. So we're going to need to find some sort of other strategy here to evaluate this limit. Now what I'm going to do is take this 3 and factor it out of both these values because I can see that 3 would go on the outside, leaving us with just x squared there, and 12 divided by 3 is 4. So this whole thing will become x squared minus 4, and then all of that will be divided by x plus 2. Now I still have this x plus 2 in the bottom, so I can't plug my negative 2 in yet, but what I notice is that we have x squared minus 4 on top. X squared minus 4 is a difference in squares, because we could also write this as x squared minus 2 squared, because 2 squared is 4, all divided by x plus 2. Since we have this difference in squares, this whole thing could be expanded out to be x plus 2 times x minus 2, and again this is all divided by x plus 2. Now from here, the x+2s are going to cancel, meaning all that I'm going to have is 3 times x minus 2. Now notice this x+2 in the denominator went away, so I can now plug my value of negative 2 in. So what we're going to have is 3 times negative two minus 2. Now negative two minus 2 is negative 4, and 3 times negative 4 is negative 12. So that there is the slope of the tangent line. So notice that we've gone ahead and evaluated this limit, and we did have to factor and cancel. But that's very common when dealing with these types of limits. You're going to have to do that step. So now that we found what our what our slope of the tangent line is, we now need to figure out what the equation of our tangent line is. But we can actually do this because notice that we have an x and y value up here, and we have a slope. When we have these values, we can just use point slope form. So that's the strategy for solving these types of problems. You first wanna calculate the slope, and then you wanna plug everything into point slope form to get your equation. So let's just go ahead and do this. So we have y minus our y value, which is going to be y minus this value of 8. And then it's going to be equal to our slope, which we just calculated to be negative 12. This was our slope. And then it's going to be times x minus our x value, which is negative 2, or we could just write this as x plus 2, because minus a negative would be plus. So now that we have everything plugged into point slope form, our last step is going to be to solve this equation for y. So let's go ahead and do that. Now the first thing I'm going to do is distribute this negative 12 into the parentheses. So what we're going to end up with is negative 12 times x, which is negative 12 x, and then negative 12 times 2, which is negative 24. Now from here, I'm going to add 8 on both sides of this equation, which will get the negative 8 and the positive 8 to cancel on the left. So that will give me y by itself which is what I'm looking for, and then we'll have negative 12 x, and then negative 24 +8 is negative 16. So y equals negative 12x minus 16 is the solution to this problem, and that's how you can find the equation of the tangent line. So this is how you can solve these types of problems. Hope you found this video helpful, and let's try getting some more practice with this.
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