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Let FP be the parabola with equationL be the line with equation yV3and M be the line with equation~Var - 3_ Put AB = (-2v3,3) andV3C = (- ~1)- V3Show that the line ...

Question

Let FP be the parabola with equationL be the line with equation yV3and M be the line with equation~Var - 3_ Put AB = (-2v3,3) andV3C = (- ~1)- V3Show that the line L is tangent to Pat the point A and the line M is tangent to P at the point

Let FP be the parabola with equation L be the line with equation y V3 and M be the line with equation ~Var - 3_ Put A B = (-2v3,3) and V3 C = (- ~1)- V3 Show that the line L is tangent to Pat the point A and the line M is tangent to P at the point



Answers

The line $y=a x+b$ is tangent to the graph of $y=x^{3}$ at the point $P=(-3,-27) .$ Find $a$ and $b.$

Problem number 43. So with the tangents I'm is you go to to function is a go to expert of one half or problems. You could have one half expert negative one half. So what a of prime a would be one half age for our number one half for Pia's. You go do 1/16 coming on fourth than solving for B. We get B is gonna want a

Okay, so here's the curve. Two minutes. Expect two. Nice excuse. Excuse me. And then we want the slope of the line contingent to this curve at one king. So we just do the same thing we look at one plus age could the value of the curve. It wants h and then subtract just a curve evaluated at one and then all over age and then simplified cell and hope that we can see what happens when h gets really, really small as this as that secret mind kind of converges into the tangent line. So we have two. This is a This is a my name You'll cubed so we can use kind of Pascal's triangle. I mean, you could just distribute it out, but this should be ah, one plus three age plus three h squared plus h cute. Get every season. Binomial expansion has got triangle. Or you could just kind of Ford allowed on this. Terms minus two minus one. Give just one. Paula, her h. Okay, so this is two minus one minus ones. That zero. And then all of these terms become negatives. It's negative. Three age minus three squared, minus H cubed of age, which, if h is not zero, we can cancel in age from each turn. Negative three minus three h Aren't they cute? And we see that is H goes to zero. This is going to zero. This is going to zero and we're just left with negative three. So we conclude that the slope is negative three and the tension line. We have a slope of negative three. Then take away the X coordinate at my corner.

Okay, so here's the parabola. So the curve is X squared plus X. Let me erase this. Oh okay. And we get this curve. Why prime is the derivative to take the derivative of X squared plus X. The degree of the first coefficient we dropped down. So we make it the coefficient And we decrease the degree by one. So this was X where now it becomes X. And at the square drops down to being multiplied by. And then for this one you do the same thing. This is X to the first power. Okay. If you if you just imagine that this when we see an X. There it's always just X. To the first power. We're gonna drop that down and multiply it by X. To the zero. Which is just one. So now we have the derivative of this curve of this function. Well what is the derivative? This is the instantaneous rate of change. And the instantaneous rate of change is the slope of the tangent line. We can calculate the slope of the tangent line at any point on the curve or in this case it's not gonna be on the curve, but the slope of the line that's tangent. Okay, let's keep going. So it wants to know This point here. To negative three isn't a point on the graph, but we need to find the lines that are tangent to the curve that goes through this point. If you try to imagine that this is a lovelier drawing and it goes off infinitely this way infinitely. This way We can drop two lines through the curve that will go through this point. Okay. And that's because of where it is for part B. Well, we have a different point 27 but we'll get to that later. two, is not on the graph. When you plug to into this formula, you will get six. So 2 -3 is somewhere down here. We're going to use formulas for the slope here and we're also later, when we need to solve for the equation, we're going to use this formula for the equation of the line. But for now to use this we need two points. We need a Y. Two, X. M and X two, Y. Two and X one, Y one. So let's say that this one X two, Y two. This is X one, Y one. We plug we use this formula to find him. This is an arbitrary point on this curve. X is the first point and then the output is X squared plus X. We subtract this by our Y one all over this, X minus this X and this. We're gonna set equal to the derivative to find when the point these two, these two points what the slope would be between these two points. Which is the line. When we use algebra, we're going to simplify this equals this to get this. I don't really have a lot of rooms so I can't really solve the algebra part. But that that's all algebra. Okay. The only calculus was really this derivative part in understanding and interpreting the meetings. So when we solve this we get to zeros. So the slope is equal to the derivative at point X equals five and X equals negative one. Now that we know the X values on the curve that our lines tangent to the parabola that passed through this point two negative three. We can use it to get two points and then use those two points with algebra to get the equation of the line. So 30 minus negative 3/5 minus two equals 11. That is the slope. So why minus? Why? One times the slope times x minus x one. And this is the first equation that we will get. The second equation that we will get is when X is equal to negative one, we're going to get this point negative 10 Again, I showed you at the beginning that this is a zero of this function. So why minus Y one over X? Oh minus um X one. And we get -1 is the slope. This is all algebra. Why minus negative three equals the slope times x minus minus two. This is the equation of this. When we simplify this is the equation of the second line, We get two lines. So the we want to know about another point. We see that when we get this point down here to negative three, we have two lines tangent. What about this .27? It's up here. You cannot get a line that is tangent. Imagine that this continues off this way. This continues off. There's no way to get a line that is tangent. You can get a line that goes through probably twice. But you you won't get a line that is tangent. So algebraic lee. We will see this by trying to solve it the same way we solved the first problem. The first part. So we have this this parabola, This derivative of this Parabola and now a new .27. Let's try to take the why two minus y one over x minus x one equals the derivative. When we simplify this equation, I say we take X -2 and multiply it here, multiply this out foil it out. We get x squared plus x minus seven equals two, X squared minus three. X minus two. Subtract x squared. Subtract X. Subtract I mean add seven and we will get zero equals x squared minus four X plus five. There are no real solutions to this um quadratic equation and we know because of the discriminate. If you want to verify this is algebra B squared minus four A C. If that results in a number that is less than zero, then we know that this has no real solutions and that is it for this question.

Hi. Is this clear? So with section 3.3, Number 68 on stories by Accomplice book. And we're gonna find equations where it both lines crossed the point to common negative. Three. Your attention to the purple A y equals X squared plus x. So I'd say the question of the tension is one minus negative. Three X minus two. This is using the points, the corn. It points to a common negative three. So my eyes and time X minus two minus three, then differentiating the formulas. Why calls? That's where. Minus plus that's We know that the slope of the problem is going to pay two weeks. Waas one. So what we're going to do is wear want plugging two X plus one into our M tunes. That's nice too minus three, and we're going to set a equal to the equation off our problem. It's for the point of intersection and then this will give us the use of backs equals five. Like that one. The M is equal to a weapon. Negative. So you get our equations. You know why equals alive? Intense? When is 25 mimes? Because you got a one minus one, and this is for part eight or B s is to draw a graph smooth. When you draw your own label that scenes all do our problem. And green, it's going to like this more equations. Two other line in blue, huh? It's a bit waffling, but this point here things presents to comma negative three so we could verify that there is no line through the 0.27 about his tensions with problem.


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