[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.20,0:00:01.03,Default,,0000,0000,0000,,- [Instructor] We're told to consider
Dialogue: 0,0:00:01.03,0:00:02.83,Default,,0000,0000,0000,,the curve given by the equation.
Dialogue: 0,0:00:02.83,0:00:04.07,Default,,0000,0000,0000,,They give this equation.
Dialogue: 0,0:00:04.07,0:00:07.21,Default,,0000,0000,0000,,It can be shown that the\Nderivative of y with respect to x
Dialogue: 0,0:00:07.21,0:00:09.61,Default,,0000,0000,0000,,is equal to this expression,\Nand you could figure that out
Dialogue: 0,0:00:09.61,0:00:11.18,Default,,0000,0000,0000,,with just some implicit differentiation
Dialogue: 0,0:00:11.18,0:00:14.16,Default,,0000,0000,0000,,and then solving for the\Nderivative of y with respect to x.
Dialogue: 0,0:00:14.16,0:00:15.90,Default,,0000,0000,0000,,We've done that in other videos.
Dialogue: 0,0:00:15.90,0:00:18.89,Default,,0000,0000,0000,,Write the equation of the horizontal line
Dialogue: 0,0:00:18.89,0:00:22.96,Default,,0000,0000,0000,,that is tangent to the curve\Nand is above the x-axis.
Dialogue: 0,0:00:22.96,0:00:25.96,Default,,0000,0000,0000,,Pause this video, and see\Nif you can have a go at it.
Dialogue: 0,0:00:25.96,0:00:28.30,Default,,0000,0000,0000,,So let's just make sure\Nwe're visualizing this right.
Dialogue: 0,0:00:28.30,0:00:30.25,Default,,0000,0000,0000,,So let me just draw a\Nquick and dirty diagram.
Dialogue: 0,0:00:30.25,0:00:33.80,Default,,0000,0000,0000,,If that's my y-axis, this is my x-axis.
Dialogue: 0,0:00:33.80,0:00:35.86,Default,,0000,0000,0000,,I don't know exactly what\Nthat curve looks like,
Dialogue: 0,0:00:35.86,0:00:38.35,Default,,0000,0000,0000,,but imagine you have some type of a curve
Dialogue: 0,0:00:38.35,0:00:41.63,Default,,0000,0000,0000,,that looks something like this.
Dialogue: 0,0:00:41.63,0:00:44.13,Default,,0000,0000,0000,,Well, there would be two tangent lines
Dialogue: 0,0:00:44.13,0:00:47.56,Default,,0000,0000,0000,,that are horizontal based\Non how I've drawn it.
Dialogue: 0,0:00:47.56,0:00:49.28,Default,,0000,0000,0000,,One might be right over there,
Dialogue: 0,0:00:49.28,0:00:50.66,Default,,0000,0000,0000,,so it might be like there.
Dialogue: 0,0:00:50.66,0:00:54.12,Default,,0000,0000,0000,,And then another one might\Nbe maybe right over here.
Dialogue: 0,0:00:54.12,0:00:56.01,Default,,0000,0000,0000,,And they want the equation\Nof the horizontal line
Dialogue: 0,0:00:56.01,0:00:59.34,Default,,0000,0000,0000,,that is tangent to the curve\Nand is above the x-axis.
Dialogue: 0,0:00:59.34,0:01:00.18,Default,,0000,0000,0000,,So what do we know?
Dialogue: 0,0:01:00.18,0:01:05.18,Default,,0000,0000,0000,,What is true if this\Ntangent line is horizontal?
Dialogue: 0,0:01:05.57,0:01:08.55,Default,,0000,0000,0000,,Well, that tells us that, at this point,
Dialogue: 0,0:01:08.55,0:01:11.50,Default,,0000,0000,0000,,dy/dx is equal to zero.
Dialogue: 0,0:01:11.50,0:01:13.89,Default,,0000,0000,0000,,In fact, that would be true\Nat both of these points.
Dialogue: 0,0:01:13.89,0:01:15.94,Default,,0000,0000,0000,,And we know what dy/dx is.
Dialogue: 0,0:01:15.94,0:01:19.19,Default,,0000,0000,0000,,We know that the derivative\Nof y with respect to x
Dialogue: 0,0:01:19.19,0:01:22.34,Default,,0000,0000,0000,,is equal to negative\Ntwo times x plus three
Dialogue: 0,0:01:22.34,0:01:26.28,Default,,0000,0000,0000,,over four y to the third\Npower for any x and y.
Dialogue: 0,0:01:26.28,0:01:28.86,Default,,0000,0000,0000,,And so when will this equal zero?
Dialogue: 0,0:01:28.86,0:01:30.97,Default,,0000,0000,0000,,Well, it's going to equal\Nzero when our numerator
Dialogue: 0,0:01:30.97,0:01:34.06,Default,,0000,0000,0000,,is equal to zero and\Nour denominator isn't.
Dialogue: 0,0:01:34.06,0:01:36.45,Default,,0000,0000,0000,,So when is our numerator going to be zero?
Dialogue: 0,0:01:36.45,0:01:38.52,Default,,0000,0000,0000,,When x is equal to negative three.
Dialogue: 0,0:01:38.52,0:01:42.73,Default,,0000,0000,0000,,So when x is equal to negative three,
Dialogue: 0,0:01:42.73,0:01:45.85,Default,,0000,0000,0000,,the derivative is equal to zero.
Dialogue: 0,0:01:45.85,0:01:48.87,Default,,0000,0000,0000,,So what is going to be\Nthe corresponding y value
Dialogue: 0,0:01:48.87,0:01:50.78,Default,,0000,0000,0000,,when x is equal to negative three?
Dialogue: 0,0:01:50.78,0:01:52.65,Default,,0000,0000,0000,,And, if we know that, well, this equation
Dialogue: 0,0:01:52.65,0:01:54.60,Default,,0000,0000,0000,,is just going to be y\Nis equal to something.
Dialogue: 0,0:01:54.60,0:01:56.48,Default,,0000,0000,0000,,It's going to be that y value.
Dialogue: 0,0:01:56.48,0:01:57.55,Default,,0000,0000,0000,,Well, to figure that out,
Dialogue: 0,0:01:57.55,0:01:59.48,Default,,0000,0000,0000,,we just take this x equals negative three,
Dialogue: 0,0:01:59.48,0:02:01.60,Default,,0000,0000,0000,,substitute it back into\Nour original equation,
Dialogue: 0,0:02:01.60,0:02:03.23,Default,,0000,0000,0000,,and then solve for y.
Dialogue: 0,0:02:03.23,0:02:04.29,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:02:04.29,0:02:08.08,Default,,0000,0000,0000,,So it's going to be negative three squared
Dialogue: 0,0:02:08.08,0:02:10.23,Default,,0000,0000,0000,,plus y to the fourth
Dialogue: 0,0:02:10.23,0:02:14.18,Default,,0000,0000,0000,,plus six times negative\Nthree is equal to seven.
Dialogue: 0,0:02:14.18,0:02:15.57,Default,,0000,0000,0000,,This is nine.
Dialogue: 0,0:02:15.57,0:02:17.75,Default,,0000,0000,0000,,This is negative 18.
Dialogue: 0,0:02:17.75,0:02:19.99,Default,,0000,0000,0000,,And so we're going to get y to the fourth
Dialogue: 0,0:02:19.99,0:02:23.47,Default,,0000,0000,0000,,minus nine is equal to seven,
Dialogue: 0,0:02:23.47,0:02:25.38,Default,,0000,0000,0000,,or, adding nine to both sides,
Dialogue: 0,0:02:25.38,0:02:28.91,Default,,0000,0000,0000,,we get y to the fourth\Npower is equal to 16.
Dialogue: 0,0:02:28.91,0:02:30.93,Default,,0000,0000,0000,,And this would tell us that y is going
Dialogue: 0,0:02:30.93,0:02:33.40,Default,,0000,0000,0000,,to be equal to plus or minus two.
Dialogue: 0,0:02:33.40,0:02:36.10,Default,,0000,0000,0000,,Well, there would be then\Ntwo horizontal lines.
Dialogue: 0,0:02:36.10,0:02:37.88,Default,,0000,0000,0000,,One would be y is equal to two.
Dialogue: 0,0:02:37.88,0:02:40.81,Default,,0000,0000,0000,,The other is y is equal to negative two.
Dialogue: 0,0:02:40.81,0:02:43.33,Default,,0000,0000,0000,,But they want us, the equation\Nof the horizontal line
Dialogue: 0,0:02:43.33,0:02:47.55,Default,,0000,0000,0000,,that is tangent to the curve\Nand is above the x-axis,
Dialogue: 0,0:02:47.55,0:02:51.13,Default,,0000,0000,0000,,so only this one is going\Nto be above the x-axis.
Dialogue: 0,0:02:51.13,0:02:52.00,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:02:52.00,0:02:54.03,Default,,0000,0000,0000,,It's going to be y is equal to two.