WEBVTT

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- [Instructor] We're told to consider

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the curve given by the equation.

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They give this equation.

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It can be shown that the
derivative of y with respect to x

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is equal to this expression,
and you could figure that out

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with just some implicit differentiation

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and then solving for the
derivative of y with respect to x.

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We've done that in other videos.

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Write the equation of the horizontal line

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that is tangent to the curve
and is above the x-axis.

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Pause this video, and see
if you can have a go at it.

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So let's just make sure
we're visualizing this right.

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So let me just draw a
quick and dirty diagram.

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If that's my y-axis, this is my x-axis.

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I don't know exactly what
that curve looks like,

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but imagine you have some type of a curve

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that looks something like this.

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Well, there would be two tangent lines

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that are horizontal based
on how I've drawn it.

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One might be right over there,

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so it might be like there.

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And then another one might
be maybe right over here.

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And they want the equation
of the horizontal line

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that is tangent to the curve
and is above the x-axis.

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So what do we know?

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What is true if this
tangent line is horizontal?

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Well, that tells us that, at this point,

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dy/dx is equal to zero.

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In fact, that would be true
at both of these points.

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And we know what dy/dx is.

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We know that the derivative
of y with respect to x

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is equal to negative
two times x plus three

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over four y to the third
power for any x and y.

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And so when will this equal zero?

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Well, it's going to equal
zero when our numerator

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is equal to zero and
our denominator isn't.

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So when is our numerator going to be zero?

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When x is equal to negative three.

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So when x is equal to negative three,

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the derivative is equal to zero.

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So what is going to be
the corresponding y value

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when x is equal to negative three?

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And, if we know that, well, this equation

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is just going to be y
is equal to something.

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It's going to be that y value.

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Well, to figure that out,

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we just take this x equals negative three,

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substitute it back into
our original equation,

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and then solve for y.

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So let's do that.

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So it's going to be negative three squared

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plus y to the fourth

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plus six times negative
three is equal to seven.

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This is nine.

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This is negative 18.

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And so we're going to get y to the fourth

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minus nine is equal to seven,

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or, adding nine to both sides,

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we get y to the fourth
power is equal to 16.

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And this would tell us that y is going

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to be equal to plus or minus two.

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Well, there would be then
two horizontal lines.

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One would be y is equal to two.

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The other is y is equal to negative two.

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But they want us, the equation
of the horizontal line

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that is tangent to the curve
and is above the x-axis,

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so only this one is going
to be above the x-axis.

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And we're done.

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It's going to be y is equal to two.