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Problem 58 Hard Difficulty

(a) If $ g(x) = x^{2/3} $, show that $ g'(0) $ does not exist.
(b) If $ a \neq 0 $, find $ g'(a) $.
(c) Show that $ y = x^{2/3} $ has a vertical tangent line at $ (0, 0) $.
(d) Illustrate part (c) by graphing $ y = x^{2/3} $.

Answer

a) $g^{\prime}(0)$ does not exist because limit does not exist
b) $f^{\prime}(a)=\frac{2}{3 a^{1 / 3}}$
c) see step for answer
d) ( Graph unavailable)

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Video Transcript

This is problem number fifty eight of the Stuart Calculus. A petition section too. Pointy party. If G of X equals X to the two third power show that g prime of zero does not exist. All right, let's use the definition here in green to solve a G prime ministerial. L a will be a hero. In this case, the limit urchins Nero of the function G of X, which is next to the third power minus. Evaluated it. Nero zero to that hearted, right. Thanks. Brightness A which is zero. Uh, here we just love the way, eh? To the two turns it over, Mermaid with zero so extra to third toe, my axe will be won over next to the one third. And as X approaches zero this right, This limit does not exist since we cannot divide by zero. So in party cheaper zero does not perfect. It is not zero Find g primary, eh? The reason the same definition. She probably equals or limited experts saying something is not zero of the function. Do you have X which is excellent to third minus a to the two thirds of the function value today divided by expert Estate two Simplify our answer little quicker. We're going to great D'You numerator and the denominator in terms. I'm the next to the one third and eight to one. So the first term act two thirds can be thought of exit the one third this weird and then aided her dirt's can be thought of as a Jew. The one third squared and then x minus c can be instead next to the one third cubed minus ated, one third Cute. And the reason we want to do this is because we want to convert the difference of squares in the new Maid and the difference of cubes in the nominator Ryan factor than the way that we see here. These air known formulas for the difference of cues and different squares, and that'LL allow us to simplify a lot faster. Limited's expertise paying Ah, uh, we're we're using these formulas where Capital A's except one third Capital B is the one third. So our difference of squares is going to be in the numerator is going to be a cylinder minus being, which is one third thanks are a plus B and denominators a little longer. We should have a minus being or X to one third minus. He took a one third. I want a pint by a squared or X to the two thirds was a Times B, which is the one they're attempting to the one parent out plus B squared or to the tube King. And here we see how this works out for us because we can remove our fact. Cancel these two terms because of this other equal, and our next step will be to no evaluate to limit as expert as a star. Every eight x becomes in a here we have a bigger one third April the one third that's too eight of the one third and then nominated. We have eighty two thirds plus eight of the one third time's A didn't want even to through this wall. And then we have another two two thirds. So that's three eight. Great. Now we can simplify a little more, and we see that are derivative. Definitely be to over three eight. The one third, and this is our final answer for party G prime of A to over three Kensi to the one third in the Dominator part scene show that y equals extra two thirds of the vertical pension at zero zero. So, essentially, we're going to use, um this results distributive. Um, because what the servant means is the slope of dysfunction and a vertical tension line has a very characteristics of so as a approaches zero, just like we showed. And, ah, part A but more correctly in part B, is it? It's going to be this limit as X approaches. Zero uh, the function that was given to us G of X minus zero over zero. Uh, and it's gonna be dysfunction to over three aides. The one third at a zero. And if a zero the denominator zero, and so this limit a purchase. Nero, our experts. Zero round goes towards infinity. So this is what we expect for a vertical tension Is that slope approaches and infinite value? Uh, the stuff gets more and more and more and more positive, and then it gets absolutely vertical. And that's what we expect to have. The fortune is that the slope are reaches an infinite undefined in number. Partying illustrate party by graphing. Why cause excessive too. So here we have a craft of that function accident two thirds. And we see that there is this corner. This cup custom here at X equals zero, and we see that the, uh, slopes of intention lines get more and more negative to the left of zero. And they're more and more positive as we get closer to zero from the right meaning There there is a vertical tension line here at the origin. So we've confirmed part. See my graphing part by cracking the function y equals two turns and I mean everything there we have shown is coming.