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For the function, find the point(s on the graph at which the tangent line has slope 12.y =025x2The point(s) is/are (Simplify your answer Type an ordered Paic Use co...

Question

For the function, find the point(s on the graph at which the tangent line has slope 12.y =025x2The point(s) is/are (Simplify your answer Type an ordered Paic Use comma to separate answers as needed )

For the function, find the point(s on the graph at which the tangent line has slope 12. y = 025x2 The point(s) is/are (Simplify your answer Type an ordered Paic Use comma to separate answers as needed )



Answers

Determine the coordinates of the points on the graph of $y=x^{3}+2$ at which the slope of the tangent is 12.

We're asked to find the equation of the tangent line that goes with our equation why and 0.12 So remember that, why prime will be the slope. And here we're looking at why prime at one. So our first step is just going to be to find my prime and I'm going to rewrite why just to make it a little bit easier to see what we're doing. So the first thing we're gonna do is pull down that one half, keep the inside here the same and now we just subtract one from our exponents. We have negative one half, but we're not done because we need to take the derivative of the inside there and multiply that too. So the derivative of X squared, we just bring down to and get two X. And for three X we just get three. So at this point we can go ahead and plug in one everywhere, there's an X. So we have two times one plus three and that's going to be over or two from our one half. And because we had a negative one half, that's wherever it goes in the bottom here. So I'm actually just going to write it like that and we have one squared plus three times one. So simplify this a little bit and we have five over two times the square root of four, which is to, so that equals 5/4. And remember this is our slope. So we have our slope, we can use the equation y minus y. One equals um times x minus X one. So we'll just plug in our coordinates from one to so why minus two equals 5/4 times X minus one. Want to play out, you get 5/4 X minus 5/4 and now we just need to add the two over and why equals 5/4 X plus 3/4. And this is the equation of your tangent line at 12

Okay, So in this problem work even the function. Why you people to to times square root of X so that to revert to will be sorry. Two times the derivative of the square root of ax is one over two of to times square root of X. So two and two will be canceled. So it will be one over square without X ends. Our slope at given point is one too. So we take X to be one so slow that one will be just one, as we have over, um, you question to the tension line, Which enough attention, like, which is why not equals, uh was I was acts plus some, Constantine. Now we're plugging our our 0.1, too. So we have to do is equal to one class t so t will be one has our equation of the tension line is excellent. Plus one

We need to find equation of lines changing to the graph off why he calls X into five minus X square. In order to find the slope, we first need to find the door abated. The derivative can be fined by using the product cool so D y by the x becomes D by D X off X into under rule five minus X square plus D by D X off 105 minus X square into x. Therefore, we get five minus X square raised to the power one by two plus D by D. X off five minus X square raised to the power off one by two into acts in order Find a derivative of this expression We If I change ALS, the inner function becomes u equals five minus x square the upper function. We can be rich and as you raised to the power off one by two de by the axe off five minus X square is to the power one by two cannot be written as do you buy DX into d. V by d. U by using chain goal. Next we find the derivatives of each of these expressions. We get negative two x into one by two U S to the power of minus one by to unzip defecation and substituting the value off you, we get minus X in Dubai minus X square, raised to the power of minus one by two. We put the value of this expression in the equation that we started off with to find the door of a of D Y. By D. X can now be written as five minus X square, raised to the power one by two plus minus acts in 25 minus X square raised to the power of minus one by two into X On simplification, we get by minus X square, raised to the power one by two minus X squared, divided by five minus X square. Praise to the power off one by two on For just simplification we get the final answer off the derivative, which is five minus two X square, divided by under route five minus X square. Since the derivative is actually the slope, we can evaluate the slope at any given point. So the slope at the 0.1 by two can be evaluated as we can denote it by m one and putting the while you as one in place off X. We get five minus two into one, divided by under route five minus one square hole under route, which gives us three by two on simplification. Next we find the slope at the point minus two minus two. So the second slope can be denoted by m two putting in the value minus two in place off X. We get the following the answer for em too. Thus, M two can be evaluated as negative free. The line change into the 0.12 can be written as four slope M one can be written as why minus Why not in tow em one into x minus x Not putting in the values we get. Why minus two equals three by two into X minus one On simplification we get why equals three by two acts plus one by two. The line changing to the point minus two four slope and two equals minus three can be written as why minus Why not into em to thank you X minus X Not putting in the values we get. Why minus minus two into negative three in tow. Asked minus negative two on simplification we get why equals negative three x Negative it By using the graphing utility, we can plot the given function y equals X into five minus x square. The first change int line Why equals three bite two into X plus one by two and the final Slow it ancient line Why equals minus three x minus eight.

Hello. If you have to find the providers and slip of the tenure land at the 10.1 conference, the question is given X minus two y to the power of X men is too white power of three. That is a constant to buy square monastery. The point is one comma one Okay, yeah. So we will differentiate it with respect to tax. Okay, so we can died three and two Ex minister Why Square multiplied by depends season of Health Minister, We will give one minus two develop only X costume for white develop on the X. Okay, so it will be three in two x minutes to y to the power of two minus six in two X minus two y to the power of two. Divide by DX. He wants to for by dy dx. So from here we can diet Divi upon leaders will be caused to train two x manage to white to the power of two upon six x managed to buy 20 power of two plus four way. Okay, this is the functional Valadez. Mhm. Yeah. So why does Well? Because 23 and two x managed to buy to the power of Group upon Six and two X minus two by two The power group plus four by Yeah, mhm. This is the function of ideas. X What? Mhm. Now we will find the value of by the slope of the tenure at the game point. So slow. Yeah. Okay. That is a cost to give. I've idea. Looks at the point one comma once. It will. Because 23 and 21 managed to fully square upon 61 minutes to telescope plus fall into one. Yeah, So slow. That is divi upon dia at point one comma one will be cost too. Candy upon. One managed to is mine as well. One has been wholly score is 16 plus four. That is three by 10. This is the slope. Yeah. The slope of the tangent will be three by 10. I hope you are natural. This is the answer. Thank you.


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