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Use the figure below fill in Lhe blanks the statements about tne! functionat point(2 95,5.03)Tangent line(a)(6)Question Attempts:-nd; ;mc daleMozilla that we can im...

Question

Use the figure below fill in Lhe blanks the statements about tne! functionat point(2 95,5.03)Tangent line(a)(6)Question Attempts:-nd; ;mc daleMozilla that we can improvc ycur experiencething

Use the figure below fill in Lhe blanks the statements about tne! function at point (2 95,5.03) Tangent line (a) (6) Question Attempts: -nd; ;mc dale Mozilla that we can improvc ycur experience thing



Answers

Fill in the blanks.
The slope of the tangent line to a graph at $(x, f(x))$ is given by ______ .

Okay, so there's questions basically asking which which formula can you get the limit of to find other slope of a line tension to any given point on a curve. So the answer here would be the difference quotient. And this is just a formula that that, if you take the ultimate of it, will eventually give you the slope, Um, of a curve at a given point if you if you were, simplify it enough.

From 63. So coordinates of a are for comma five. The derivative at four is gonna equal one half. So this is the slope. I think he's in so formula. We have why minus why coordinate is u goto one half x minus x court.

Let's do the same problem. But with from the different direction we still have the equation X cubed minus X. And we are given different set of points and the secret lines in order to approximate the slope of the tangent line at to Okay, All right, so it's the same process is just different numbers, and there is a point that I would like to make, So please be attention until the very end. Now we are going to start with the farthest point first. So from 2 to 3 Okay, the slope off that insolent 24 minus six, divided by three minus two, which is equal to 18. So in this particular case, it's the secret line from here to here. I'm sorry. It's not the best drawing, but that is going to be the slope off the secret life. Okay, now, the next one this value. You basically use this to that divided by this is not so. I already did the calculation. The slope is going to end up being 14.25 You can see it's getting a little bit less steep. Thanks. The slope off this line for me, it looks a little bit better the slope, but 2.5 third one I'm going to use to point wanted to. So it passes through this point. So it is going to be from here to there. That's the secret line that we're using and off 2.1 that one. It is going to be approximately equal to 11.6 now from the previous problem. We already went through the fact that em will end up equally. 11. And the difference between the previous problem is that from the previous problem, we were approaching two to from the left side of two, starting from 11.51 point nine. But in this case, we actually start from the right side three going to 2.52 point one and then eventually getting closer and closer to two. We can see that the slope starts steep, less deep, getting close to 11, and then eventually we reached 11, and that's basically what you're trying to do here. To approximate the slope that you have right here on the tangent line. The closer, the two points on the secret lines become the better the approximation to 11, and that's basically how you do these problems

For the income. We want to find a formula for the slope of the tangent line at a general point. So we're gonna use the formula mm tan to determine the points where the line is tangent to the horizontal. Yes. So what we're gonna do is to find this formula for the slope, we know that slope is equal to y tu minus Y one. Mhm Over X 2 -11. So in this case we're going to replace our wives with the fact that we have F of X two -1 A bex one over X. To my next column. The only issue with this though is that this gives us the second line. This is the kind of the average rate of change. So what we want to do if we call this delta X. We want delta X to get even smaller and smaller. So we get a tiny little change in the wind and a tiny little change in the X. So we're going to take the limit As Delta X goes to zero.


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