Use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an

step 1 Let's look at this problem here. We're given the equation, Y equals X cubed, and are tasked with

Use the method in Example 3 to find (a) the slope of the...

Key Concepts

Tangent Line

A tangent line to a curve at a given point is the line that best approximates the curve near that point. It has the same slope as the curve at the point of tangency, meaning it reflects the instantaneous rate of change of the function at that point. Understanding tangent lines helps in approximating functions and analyzing their behavior locally.

Point-Slope Form

The point-slope form is a method used to express the equation of a line when the slope and a point on the line are known. This form, given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, is particularly useful in the context of finding the equation of a tangent line to a curve at a specific point.

Differentiation

Differentiation is the process in calculus that deals with finding the derivative of a function, which measures the function's instantaneous rate of change with respect to its independent variable. This concept is central in understanding how a function behaves locally near a point, making it a fundamental tool in solving problems related to slopes and rates of change.

Transcript

00:01 Let's look at this problem here.

00:03 We're given the equation, y equals x cubed, and are tasked with finding the slope of a line tangent to the graph of the curve of y equals x cubed at the point 28.

00:16 Here i've drawn the coordinate plane with a graph drawn in red representing y equals x cubed, the blue point on that red curve as the coordinates of 28, which of course x equals 2 ,8, and y equals 2.

00:31 Which is 8.

00:35 To find the slope of the line that's tangent to the curve and goes through the point 2 .8, we need a change in y over change in x, basic algebra, but we're only given one point, so we're looking for that delta y over delta x.

00:52 Therefore, we have to fabricate a point to help us and then use the limits that you're learning about to determine the slope of that particular our line.

01:02 If we add an arbitrary point, in the year i'll put it in green, right there, and what we've done is added a value to x, we'll call it h, and that represents the change in x here to, and then 2 plus h will be the x value of the green point.

01:31 The y value, because it's an x -cube function, the new y value, would be x plus h quantity squared since y equals x cubed i mean i said quantity squared i mean quantity cubed so since the equation of y equals x cubed and the new value of the green point is 2 plus h for x then the new y value would be 2 plus h quantity cubed now that we have two points we can easily determine the slope of the line that goes between those two points the only problem is that line isn't tangent to the blue point or tangent to the green point.

02:15 It's a secant line, which means it basically goes through both points and kind of goes inside the curve of the red line.

02:25 So that's not what we want, but that's how we have to start this particular problem.

02:32 So as i said a couple minutes ago, the coordinates of the green point now, now, 2 plus h for the x value, and then 2 plus h cubed for the y value.

02:57 Now in order to get your delta y over delta x, all we have to do is subtract, but because this is a cube value up here, okay, we have to expand that first.

03:13 So i'm going to make a little room over here on the left, this drawing, just so i can do some of the calculations here.

03:22 And what we're looking for is delta y, change in y, or change in x will give us the slope.

03:31 So then change in y, we just subtract the two ys here.

03:36 So if we expand 2 plus h cubed, we get 8 plus 12h plus 6h squared plus h cubed.

03:51 That is the green point the y value expanded 2 plus h times itself three times and then from there we have subtract the y value of the blue point which is just 8 you know make myself a little more room here so and then subtract 8 now we have the delta y and then we have to find the delta x so it's 2 plus h which is this value right here minus 2 from the blue point and the x and you accept see if i simplify these, i've got an 8 here and a minus 8 here so those kind of go away.

04:36 And i've got a 2 here and a minus 2 here so those kind of go away.

04:40 So what i'm left with is i'm left with 12h plus 6h squared plus h cubed all over h.

04:51 And obviously i can divide h into each one of those so that really leaves me with 12 plus 6h plus h...

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