12/12/2023 0 Comments Calc abAnd so as we get closer andĬloser to X being equal to C, we see that our slope is actually approaching negative infinity. Then this expression would be the slope of this line. Let's say right over here, then this would be the slope of this line. Xs right over here? Well, for this point, X comma F of X, our slope, if we take F of X minus F of C over X minus C, that wouldīe the slope of this line. To take Xs to the right? So instead of our Xs being there, what if we were to take The derivative or this limit as we approach from the left, And in all of thoseĬases, it would be zero. And then let's get X even closer than that and find this slope. And then let me get a little bit closer, and let's get X a little bit closer and then let's find this slope. Of the limit right over here, you're essentially saying So that would be X, this would be the point X comma F of X, and then this is the point CĬomma F of C right over here. Is is a slope of a line between when X is some arbitrary value, let's say it's out here, Were trying to find this limit? Well, remember, all this Our function is defined atĬ, it's equal to this value, but you can see as Xīecomes larger than C, it just jumps down and Of a non-continuous function and then think about would weīe able to find this limit. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. Then you definitely will not be differentiable. What I just wrote down is, if you are not continuous, Other way around that if you're continuous, then Mean the other way around, and actually we'll lookĪt a case where it's not necessarily the case the It's differentiable, if we can find this limit, if we can find thisĭerivative at X equals C, then our function is alsoĬontinuous at X equals C. That I'm going to make is if F is differentiable, at X equals C, at X equals C, then F is continuous at X equals C. Will go a little bit more into the proof direction. So I'm now going to make aįew claims in this video, and I'm not going to To C, as our change in X gets closer and closer to zero? And we talk about that in other videos. See, well, what is that slope as X gets closer and closer It as our change in Y, if Y is equal to F of X,Īnd this is our change in X. Value of our function, or you could think of Looks a little bit strange, but all it is is it'sĬalculating the slope, this is our change in the And at first when you see this formula, and we've seen it before, it The derivative of our functionį at C is going to be equal to the limit as X approaches Z of F of X, minus F of C, over X minus C. Video, I'll write it as the derivative of our function at point C, this is Lagrange notation And there's multiple ways of writing this. And that is just a fancy way of saying does the function have aĭefined derivative at a point? So let's just remind ourselvesĪ definition of a derivative. Going to do in this video is explore the notion ofĭifferentiability at a point. Y = -x when x 0), the right hand limit is 1 (as x -> 0), therefore the limit at 0 does not exist!įor other functions that have more gentle curves then you get a more gradual shift toward the same limit near the top/bottom of a curve, mainly they approach 0 :) For the absolute value function it's defined as: The reason is because for a function the be differentiable at a certain point, then the left and right hand limits approaching that MUST be equal (to make the limit exist). There are some exceptions, especially for a function that has a very sharp curve, like y = |x|, these slopes one either side are completely opposite (-1 and 1), and so at the "bottom" there is no tangent. Yes, from either side of the hill/curve the tangents have different slopes, but as they each approach that "top" point, they should become equal to one another, again where the slope is equal to 0. 000001 inches then the tangent may change but that's not really the point. As you go up a hill the tangent is constantly changing, but there's still only "one" true tangent line at any exact point. Tops and bottoms of curves have a slope of 0, imagine driving a car and looking perfectly parallel to the ground.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |