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Tpoints ! SApCalcBr1 2.4.035. Use the definition of a derivative to find f"x) ad f"(x): f(x) = 4 + 6x x2f"(x)f"(x)Then graph f,f' , and f o...

Question

Tpoints ! SApCalcBr1 2.4.035. Use the definition of a derivative to find f"x) ad f"(x): f(x) = 4 + 6x x2f"(x)f"(x)Then graph f,f' , and f on a common screen and check to see if your

Tpoints ! SApCalcBr1 2.4.035. Use the definition of a derivative to find f"x) ad f"(x): f(x) = 4 + 6x x2 f"(x) f"(x) Then graph f,f' , and f on a common screen and check to see if your



Answers

Use the definition of a derivative to find $ f'(x) $ and $ f''(x) $. Then graph $ f $, $ f' $, and $ f'' $ on a common screen and check to see if your answers are reasonable.

$ f(x) = 3x^2 + 2x + 1 $

For the given problem, we're going to use the definition of the derivative to find F prime of X and F double prima max. So we're gonna consider ffx which equals one plus four X minus x squared. Yeah. Machine fine. We have premium tax using this definition here, which will call G fx Is going to equal -2 X what's for. And then we can plug this now into this definition of the derivative, making this X plus H. We'll see the forest cancel. And what will be left with is each of X, which is the second derivative. And that's going to be a negative too. So you see how these um graphs are related, This is slipping zero right here And then we see that this is just the line. So it has a continuous slope of -2.

In the given question the intuition of Graph is a fax. It was to access where -6 sets. That's for okay. It is the commission. And we have to find the tangent. Oil commodities. We're tangent is horizontal. So first of all, we have to find the derivative. We have to find the derivative. They deserve districts. So simply I have to differentiate all the items. That is access where minus the real deal with facts. Does derivative of constant that is all. So the Liberty of Access Square is two X -6 derivative of X is one. Okay. And that he would have won 20. So finally You get your derivative that is two X -6. Now we know that this delivered to represents the slope of the tangent also. And in the given given opportunities said that 10% is very gentle. Okay. And gentle, horizontal. So I can say that if the line is horizontal, then it's sloppy the question judo. So there's no most people's 20 from here, immediate taxi close to trade. And if we put this street in the even function we get three square minus six. Multiplied three. Bless for. Okay This provides me 9 -18 4 nine minus 18 plus four nine plus four. It were to 13 13 minus I think it was 2 -5. So I can say that the coordinates of this point are three comma minus five. Finally these are the coordinates. And this was your deliberating. Now we have to prove this by graphing utility. So here I am taking the function that is excess square. Okay. And minus of 66 Let's food. Okay here I'm making the point that is legal minus five high level that went with knowing not in the ignition of attendant line and that is why I minus what I want. Yes. My in place five. It must do little. We can see that and then line is original. Okay. Thank you

In the given question we have to When the river do of the function. And the function is if x equals two access world minus four X plus three. After that you'll be able to find. Calling herself point at with the tangent line is for a gentle. So first of all, I am finding the deliver to we know that derivative of access square. Okay. We have to differentiate. There's all the dump status where -4. That's plus three. I can represent it as a derivative of X squared minus four into delivered to your fix plus Did You? two of 3. The re radio taxi square is two weeks Deloted to access one minus formal deployed with one and derivative any constantly zero. So finally figured the value of derivatives That illegals to works -4. Okay, notice that that we have to find the point At which 10 guarantees originally 10 10 days for a 20. We know that since 10 minutes for United. So it's slope must week was 20 and this delegated represents slope of the tangent. This represents global attendant. Okay, So the slope must be to us to zero from here. We should get Actually it was two and if you put this too in the function two square minus four multiply two plus three and then Discomforts Me for -8. Last three. That is it was 2 -1. Okay, so finally I can say that the coordinator of the blind side Access to and wise -1. Okay, No, we have to do with the graphing utility to prove all this. So I am looking here I'll get up. That is a cholesterol. Okay, thanks. It was true. I have to do where with me? I am doing that square access square and after that -4 x last three minus of Cool. Thanks. And lots of three. Okay. I've been that I'm floating the ground can given to us which were that we did it at the top of my mind as well. Okay. You comma minus one. Look at that point, I'm leveling it. Here is the point and now I'm running attendant at that point that is By minutes violent means my plus one it was to slope and slow beats zero again. Say that The slope is zero. So it was too little. Really like x minus or actually that is to Okay who in the instagram we're going to see that And that point we heard the line which is horizontal. Okay, Okay.

This is a problem. Number fifty four of the Stuart Calculus eighth Edition Section two point eight. Use the definition of a derivative to find at prime of X and F double primary Kes, then Graff. Time enough. Double prime on a common screen and check to see if your answers are reasonable and the function F it's given us at of X equal to execute minus X minus three X name. So let's start with and paramedics by using the derivative ah, definition of a derivative. The prime of X is given as the limit as each of purchase. Zero of the function hearing Next Hugo Industry X Evaluated Expo Sage. So the text plus age quantity cubed minus three times his age. And then we're going to subtract the function execute ministry and this on the right of right teach. Okay. Our next step is to expand the numerator. Separate all the terms X plus h Cubed is binomial cubed will give us X cubed plus three expert H plus three x h squared not plus age cubed. Then we distribute the negative three in the next term, making three expense three each and then we should make it appear negative, X cubed Class three X and they're all divided by each. Hey, we have a next cute how you hear minus and execute high here. Ah, a positive three x here and a negative directs there that were canceled and then each remaining term has it one age values. So the agent the denominator counsels with one h and each of the remaining terms. And that leaves us with for Lim as each approaches zero out of the function three x squared less three x h pompous h squared minus three. And if we take a look here each term, that has an a jewel that being negligible since ages approaching zero. So three x h and H squared will go to zero and the resulting terms are are derivative three X squared minus three. Now, for the second derivative, we do the same procedure except we are using the function have paramedics instead of ethics so that the second rate of double prime is the limited. H approaches zero of primal axe evaluated at X plus H three times X plus H squared minus three and there were subjecting the function minus three X squared minus three all the better for age. Next step is to simplify the numerator. Here's a binomial squared so we'LL have X squared plus two Ex age plus eight squared or not by three. Gives us three X squared class six X h plus three age squared minus three And then over here, we're going to subtract Are we're going to This should be the negatives from the ministry X squared plus three This all the h ah, now we canceled three extorted with negative three x squared And then? Then they got three and the positive three. And then agent the denominator comes around with one inch of each of the remaining terms leaving us with the limit is H approaches zero of six. Eight x plus three each and his age approaches zero three to purchase zero, leaving us with just six. X as our derivative, our secondary of F double prime of acts. So we have determined deaf prime and asked about crime I'm to recall after vexes x cubed minus tree x f Prima vex is at three x squared my street. Enough double primer vexes six x So the next part is to grab all three of these air to show that the answers seem reasonable. So the original function of X is shown in purple executed minus three x the derivative, or that it's three expert ministrations in red, and it threw her after that is six x, which is shown in blue. So the original function in purple is a cubic function. It ah is increasing up until its maximum point here, meaning that it's derivative which is shown in red must be all positive, which is true, but then as the derivative zero, meaning that service crosses the X axis. Then afterwards it's decreasing until it's minimums. All negative in that region followed a pie, and afterwards it's increasing, which means that it's rude is all positive, so that seems pretty consistent. On the drill you have prime in red. It is always decreasing up until its minimum. Here on DH. It's derivative. Here in blue is showing just that since function have prime is decreasing, double crime must be negative. And then it's ah, derivative of a derivative of brain eyes equal to zero here at X equals zero, which is true. And then afterwards it's all positive derivative, thought positive, meaning that the function of peace increasing half their X equals zero, which is true. So the graph confirms that all of these functions are consistent with each other as F f F prime and double prime.


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