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Problem 30 Hard Difficulty

(a) If $ G(x) = 4x^2 - x^3 $, find $ G'(a) $ and use it to find equations of the tangent lines to the curve
$ y = 4x^2 - x^3 $ at the points $ (2, 8) $ and $ (3, 9) $.

(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Answer

a) At $(2,8) : y=4 x$ At $(3,9) : y=-3 x+18$
b) see solution

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Video Transcript

Okay I had some of this already worked out but I had to redo this video because I also need to show you the graph. Um so let me explain what's going on here. We have a function G of X equals for expert minus execute. And we want to find G prime of a. The derivative of this function evaluated at A. So if G of X is for x squared minus X cubed G prime of X, the derivative of four X squared is eight X minus the derivative of execute which is three X squared. So chief prime of X equals eight x minus three X squared cheap, remove A is going to be eight times a minus three times a square. Next what we want to do is we want to come up with the equation of the tangent line. So the line that is tangent to this graph, the graph of this function at this point. Now. Uh so to find the equation of a tangent line passing through this point or more precisely actually touching the graph of G of X at this point. We need to use the point slope form of the equation of the line. Why minus Y one equals M times x minus x one. So uh this will be the equation of our tangent line. So this tangent line is going to touch the graph of this function, the curve at this point. So we used the coordinates of this point and we label them X one, Y one. So that's going to be the X one and Y one in our equation of our line, all that remains is to find the slope of the tangent line. The slope of the tangent line is equal to the derivative of the function. So since the tangent line is going to touch the function at this 0.2 comma eight, the tangent line is going to touch, the function is going to be tangent to the graph of the function when X is too. So the slope of the tangent line is going to be the derivative of the function when X is two, I'll say that again. That's super important. The tangent line has a slope M mr slope of the tangent line to slope of the tangent line equals the derivative of the function at that particular x value. So, since this line is going to be tangent two, G fx at the point to comment eight. That means this line is going to be tangent to the function G of X when x is too. So the slope of the tangent line em is going to be the derivative when x is too G prime of a is a a minus three A square. So the derivative uh for two G prime of two is simply eight times two minus three times two squared That comes out equal to four. So now we know what X one is and why one and M. So now we just substitute those into our point slope form of the equation of the line, Why -J. one y -8 Equals M. The slope, the slope of the tangent line is going to be four Times Parentheses X -X. x -2. So you have Y -8 equals four times parentheses X minus 24 times x minus two is four X minus eight. But if we add this eight to both sides we will actually get y equals four X. So now I'm going to show you on the dez most graphing calculator. I'm going to show you the graph of G of X. I'm going to show you uh the graph of this tangent line, Y equals four X. And you're going to see that this line, Y equals four X. Will be tangent to the curve G fx at the point to commentate. So uh the red curve is G. Of X. The blue line is our tangent line. You can see that this blue tangent line, this blue line is tangent to the red curve. It's just touching it right there at the point to commentate. So The blue line whose equation is four x is tangent to the graph of four X squared minus X cubed at the point. To commentate. All right. So next we have to redo this all over again for the 0.3 Common nine. So we now want to find a new equation of a new tangent line. Uh That is going to be tangent to G F X at the 90.3 common nine. So find the equation of the line. That is tangent to G F x At the .3 Common nine. Well, immediately we know that we're going to label the three as X one And the nine as Why one. And we know that the slope of the tangent line is equal to the derivative for this particular X value. So the tangent line passing through this point is going to have a derivative of G prime of three. Okay, this is the most important part. So I'll explain this one more time. We want to find The equation of a line that is tangent to GFX at the .3 Common nine. This tangent line will have a slope that's equal to the derivative of the function at this X value in all the words to slow pay attention line is going to be the derivative. Okay, G prime evaluated for this X value to slow pay attention line is going to be G prime of three. Now, G prime of a is eight times a minus three times a square. So G prime of three. Just plug three and 40 a. You're going to get eight times three minus three times a squared minus three times three squared. All right, this is our little a value. Okay, you plug in three in for a into G prime of a expression. Eight times a minus three X squared eight times three -3 times three squared. All right, what does this come out to be? eight times 3 is 24 minus three times three squared three squared is nine times three is 27. So we have 24 minus in 27 which is negative three. And so the equation of the tangent line, 2° of G of X At the .3 common nine is going to have the form. Uh Why- Why 1? The equation of any line can be given by this point slope form. Why minus? Why one equals M times X minus X. One. Let's get ourselves some space. So why minus? This is gonna be once we plug in for X one and Y one and M We're going to have the equation of the change in line to the graph of G of X at this point. So now what we have to do is just substitute in the values why one is nine. Why minus Why one? Why minus nine equals M. The slope detention line is gonna be negative three Ties, Princes X -X one X -3. So why -9 equals let's do negative three times these terms. And the princes negative three times X is negative three X negative three times Think of this as negative three positive night. If we add 9 to both sides of this equation uh minus nine plus nine will cancel. We'll have why Equals -3 X. And then plus nine plus nine plus 18. So here is the equation of the line. That would be tangent to G. Fx At the .3 Common nine. So next we're going to go back and graph our G. Of X function. And this uh equation of the tangent line. And you will see that this line will be tangent to our curve G F X. At this 0.3 common nine. Okay, so here on the desk, most graphing calculator. You see our curve G. F. X. That's our red curve G. Of X. You see the tangent line? The green line. You can see that our line is tangent To our function G. FX at the .3 Common nine. Okay. This line the line with this equation is tangent to the curve to function GFX at the .3 Common nine. Okay, here's our lying being tangent To the function GFX at the .3 common