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Use Desmos to graph the original function, and the tangent lines from questions 1 c and 1 e Take a screen shot of the graph:g) Summarize what you have done in this ...

Question

Use Desmos to graph the original function, and the tangent lines from questions 1 c and 1 e Take a screen shot of the graph:g) Summarize what you have done in this problem: What does the derivative represent? What do the equation of the tangent lines represent?2. Use the limit definition of a derivative to find the derivative of the function:h(c) = %8 + 1

Use Desmos to graph the original function, and the tangent lines from questions 1 c and 1 e Take a screen shot of the graph: g) Summarize what you have done in this problem: What does the derivative represent? What do the equation of the tangent lines represent? 2. Use the limit definition of a derivative to find the derivative of the function: h(c) = %8 + 1



Answers

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. $$g(x)=\frac{8}{x^{2}}$$

As we look at this problem, we have and define his X plus E to the X. Kind of looks interesting and what we care about is the ordered pair 01. Um And what we first need to do is figure out the derivative. Now with the calculator, I guess I'm doing part seat versus you can actually just type into this calculator G prime of zero. And that gives you the slope at that point. But let's assume we didn't know that. Well, what we would have to do is take the derivative of this function, which would be the derivative of access one and the derivative of E to the X. Is itself. And if you were to now plug in zero in for all of those X. Is uh I guess there's only one X to plug in zero. Each of the zero powers one plus one is two. So they did it correctly. Um So what's the equation of the tangent line? Well, I'm a big fan of point slope form where you do why it goes slope and then X minus the X coordinate plus the Y. Corbett. But if you look closely at the graph, it's kind of obvious that the y intercept is one, so it'd be a lot faster. Plus subtracting zero is not going to change anything to just write two X plus one, which is the correct answer they're looking for. And hopefully I've explained um everything you need to know to get this correct answer.

In the given question we have to sketch a graph or Effects it was to access were -2. Okay. We have to sketch affects. It was to access where -2. Under the point one Coma F one. We have to find a slope. Okay. No we know that. First of all initially why is he going to access? Whereas is a parabola. He's a parabola which opens upward so it is a standard parabola opening in the opposite direction. Here. In the given question there is access where -2 means simply you can say that if you want to sketch the graph then you have to modus this girl Along by access by -2. You have to shift along. Why access. Bye minus two. Okay so I am going to scratch it. This is X exist. This is why access and At the intersection of the origin here, U -1 Here you have -2. Okay, so simply we can say that we have to shift It on -2. This graph is shifted on -2. Oh I shouldn't have a graph that should look like me. This one. Okay, this is the girl. If I can say that FX It was to access Square -2. No, Here is a .1. And at this .1 this is the point which lies on the access rough. So this is the tangent here. I can say that this is the tangent line here. Okay, so I plotted in the graph and it is the tangent line. Now we have to find this law. So simply we are differentiating the function It provides us two weeks. Now we put the point, nobody has given us and after that we should find the slope that is to. Okay, thank you.

In this problem we are given fx as equal to eight over X. And this can can be written as eight X. To the poor minus one. Now to find the derivative of ethics which is f prime affects. We look at here we come look at this term by the power low minus one, comes and multiplies with minus one by eight By X. to the new power which is the old power -1 to get -2. And this can be rewritten as -8 over X squared. And this is the derivative of F X. Now to find the The gradient of the tangents at the two points, X is equal to -2 and X is equal to one. We are substituting these values into f prime of X. So first f prime of -2 gives us minus eight over minus two squared and we have minus 8/4 which gives us minus two as the first gradient of the tangent at minus two. The second gradient gradient of the tangent at f is equal to one. We do the same. His substitute one Into f prime affects. And here we get our solution as -8 and these two are the guardians at the given points of the tangent to the curve, Fx is going to eight over X.

Okay, so this question wants us to find the equation of a tangent line to this function at the point X equals eight. So to do this, we'll use the definition of derivatives to fund our slope. So plugging in the values we know one over X plus h minus on over acts and just to avoid using a complex fraction, are gonna bring the won over H over there. And now let's get a common denominator. So the first term needs an axe in the second term, needs an X plus age, and we still have the one of her age in front. And now you can see that this axe cancels with that ex so you just get negative h over X times, X plus h and then we have our factor of one over h and front in these Cancel out. So we just get the limit as X Sorry as H as H approaches zero of negative one over X Times X plus H, which is if you plug in zero, this h goes away. So it's just negative. One over x. Times X was zero, which is just negative. One over X squared equals F prime of X. So to find the slope at X equals eight, you just plug in eight into the derivative, which is negative. One over eight squared, which is negative, one divided by 64. And to see where that point ISS on the graph, you just plug in eight f of axe. So it's one over eight and now we have everything we need to construct our line. So we use the point slope equation. Why minus 1/8 is our point on the graph times negative one over 64 times X minus The point of tendency, which in this case is eight. And if we want to simplify that MX plus B form, we get negative won over 64 times acts plus 1/8 plus 1/8 which simplifies to 1/4. And there's our tangent line.


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