The right-sided and left-sided derivatives of a function at a ...

Question

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{'} (a) = \lim_{h \to 0^{+}} \frac{f (a + h) - f (a)}{h}$ and $f_{-}^{'} (a) = \lim_{h \to 0^{-}} \frac{f (a + h) - f (a)}{h}$ , respectively, provided these limits exist. The derivative $f^{'} (a)$ exists if and only if $f_{+}^{'} (a) = f_{-}^{'} (a)$ .
Compute $f_{+}^{'} (a)$ and $f_{-}^{'} (a)$ at the given point $a$ .
$f (x) = ∣ x - 2 ∣$ ; $a equals 2$

Compute $f_{+}^{'} (a)$ and $f_{-}^{'} (a)$ at the given point $a$ .

$f (x) = ∣ x - 2 ∣$ ; $a equals 2$

Accepted Answer

Welcome back, everyone. In this problem, the following formulas for F of A from the left and F of A from the right represent the left and the right-sided derivatives of a function at a point A respectively. F A from the left equals the limit as H approaches 0 from the left of F of A plus H minus F of A or divided by H, while F A from the right equals the limit as H approaches 0 from the right of F A plus H minus F of A all divided by H. Consider F of X to be equal to the absolute value of X minus 8. Find F A from the left and F A from the right at A equals 8. A says F of 8 from the left is 1, while F of 8 from the right is -1. B says they are -1 and 1 respectively. C 0 and -1 and D 0 and 1. Now, what is going to help us to figure out our left and right sided derivatives of F A at A equals 8? Well, we're already given the definitions for our derivatives, right? And now, we also know that F of X equals the absolute value of X minus 8. So if we can use that to figure out F of A plus H and FFA, we can substitute them into our definitions to solve for our derivatives from both sides. Now Starting with F of A plus H, OK. That simply means that we're going to put this as. F of 8 plus H, because A equals 8. I should probably make a note first here that since A equals 8, OK. Then F of A plus H is going to be FF 8 plus H. And now anywhere we see X, we're going to substitute 8+ H. So now this is going to be equal to the absolute value of 8 plus H minus 8, and that is equal to the absolute value of H. That is F of A plus H. Likewise, F of A is going to be equal to F of 8. And that's the absolute value of 8 minus 8, which equals 0. Know that we have values for FFA plus H and FFA, we can go ahead and solve our derivatives from both sides. Let's start. Let me come down here and let's let's put this in red. Let's start with our derivative, OK, from the left. Of it So that means it's gonna be equal to the limit as H approaches 0 of F of A plus H, which we know is the absolute value of H, minus F of A, which is 0, all divided by H. No, H minus 0 equals H, so this is the limit as H approaches 0 of the absolute value of H divided by H. And now, what do we know about this absolute value? Well, here, since it's approaching 0 from the left, OK, let me just add those to our limit here. My apologies, I forgot them. But since it's approaching 0 from the left, OK, then 4. OK, let me just put this here. For H being negative, approaching it from the left, then we know that our absolute value of H will be negative H. This is a very important point because no, that means we can rewrite this as the limit as H approaches 0 from the left, OK. Of negative h divided by H and that it will be equal to the limit as H approaches 0 of -1, which is going to simply be equal to -1. Those are derivative from the left equals -1. What about are derivatives from the right? Well, again, we have our limit as H approaches 0 from the right of the absolute value of H minus 0. OK. We know that's gonna leave all divided by H, my apologies. We know that's gonna leave us with the limit as H approaches 0 of the absolute value of H divided by H. But no, here, let me, where can I write this? Let me just write this down here. No, we also know that for H greater than 0, it's approaching from the right, then the absolute value of H will be a positive H. Thus we can write this as the limit as H approaches 0 of H divided by H. From the right and thus that's gonna be the limit as H approaches 0 from the right of 1 H divided by H equals 1, which is going to be equal to 1. Thus, our our left sided derivative, when A equals 8 is going to be -1, while our right side derivative is going to be positive 1. If we look back at our answer twice, that tells us B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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