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Let n be a natural number; that is,n = 1,2.3,Prove (In(r))"dt I(In(r))"-(In(z))"-'dx.Suppose h is continuous and differentiable function on |0,#...

Question

Let n be a natural number; that is,n = 1,2.3,Prove (In(r))"dt I(In(r))"-(In(z))"-'dx.Suppose h is continuous and differentiable function on |0,#] with(h(r) + 6"(2)) sin(r)dr = 3and h(a) = L Find the value of h(().Write 2e dr aS AH expression in Lerus odr;

Let n be a natural number; that is,n = 1,2.3, Prove (In(r))"dt I(In(r))"- (In(z))"-'dx. Suppose h is continuous and differentiable function on |0,#] with (h(r) + 6"(2)) sin(r)dr = 3 and h(a) = L Find the value of h((). Write 2e dr aS AH expression in Lerus o dr;



Answers

Let $h>0 .$ Suppose $f$ is continuous on $\left[x_{0}-h, x_{0}+h\right]$ and differentiable on $\left(x_{0}-h, x_{0}+h\right) .$ Show that if$$\lim _{x \rightarrow x_{n}} f^{\prime}(x)=L$$ then $f$ is differentiable at $x_{0}$ and $f^{\prime}\left(x_{0}\right)=L .$ HINT: Excrcise 42

Were given the equation V. Is equal to 80 H. Squared. And we're looking for D. V. D. T. When H. Is three. Okay so let's write down all the information that we're given. So we're given when H. Is three. So a chance to go to three D. H. D. T. is equal to one over 12. Okay so let's first find D. V. D. T. And we do that by taking the derivative of both sides of this equation. Um with respect to T. So the left side that would that would just be D. V. D. T. And then the right side would be 80 times two h. Times G. H. 80. Okay so this is 160 H. Times D. H. D. T. Okay so now we're ready to find what D. V. D. T. Is at this particular value. Okay so when steve, so we're looking for D. V. D. T. Evaluated at H. Is equal to three. Okay so we get this is 160 times three Times, and DHDT is 1/12 and then just multiply through and then simplify. And we should get this is equal for 40.

Our goal for this problem is to use the definition of continuity. Um And the properties of limits to show that the function is continuous at the given number A. So what we're going to do is um using the property of limits, we can take the limit of the top and the bottom We end up getting to -3, which is negative one Over two. So we get negative 1/2 as the limit. And then if we actually evaluate the function H. Of one, we end up getting negative one half. So based on that we end up seeing that the limit as X approaches um see laugh of ax equals or half of a really expected A is equal to half right? And we'll see this with the next problem to which is problem. Um Problem 14 we end up seeing that for this case our function is G. of x equals to route three -X. So if we have to square root reminder facts okay than when we plug in a specific value in this case um will end up seeing that The property still holds because the limit as X approaches a of Quebec's will equal five

This phone number sixty three of the Stuart Calculus eighth edition section two point five prove that if his continues at a if, and only if the limit is six. H approaches zero of the function April's Each is equal to that of a and if we were calling a function F is on ly. Continuous is continuous if and only if it's limit. As XO. Purchasing of the function is equal to have a mai. So we're going to use this. See if we can associate Thies, too. We're gonna let h equal explains saying. Ah, and if that's true, we can re writes this part and the limit as the limit of the function. April's a jewel. A procedure is the same. A saint ex or let me replace it. And that is a plus age which is experiencing. Cols Asperger's and then a We can ah keep the same over here. No, since ages explain to say we're going to show it as explains a approaching zero. And if we do some more some modifications here a man is a zero, and then here we can add it to both sides of this Aargh. So this would be the same as limited as expert as a or the function f of X because after today so essentially what we have is that at the beginning this is another way of saying the definition of continuity. It is written, written using ancient set of X. Using this conversion, H is equal to x one and saying so we were able to show that this is equivalent to the definition of continuity and thus it is continuous, Ari, for this reason.

Yeah, a didn't integral. If you want to pull its a double s off you into B. So the integration off the vegetable death is it's your Desi's here and the limiting it won't go, as you know, in vigorous And how if that's off X in could be X is equal to a pop. It seem lovely. Indicates off even double That's up X into being a little If Bess up takes no blood in the past Upper Limited. So the upper limit is cool minus and the lower limited in this h two days off one. So that way have given that Billy Joel it today. So who so that Billy walk it today's off. Who is high were minus and they get two days off one who were so the my man has to each TV. So the panel inside it here given question. Thank you


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