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

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

Key Concepts

Point-Slope Form

The point-slope form is a method for writing the equation of a line when a point on the line and its slope are known. The formula is expressed as y - y? = m(x - x?), where m represents the slope and (x?, y?) is the point on the line.

Chain Rule

The chain rule is a differentiation technique used when dealing with composite functions. It allows the derivative of a function that is composed of an inner function and an outer function to be found by differentiating the outer function evaluated at the inner function and then multiplying by the derivative of the inner function.

Derivative

The derivative of a function conveys the instantaneous rate of change at a given point and represents the slope of the tangent line to the curve at that point. It is typically found using limit definitions or established differentiation rules.

Tangent Line

A tangent line is a straight line that touches a curve at a specific point and has the same slope as the curve at that point. Finding the tangent line involves determining the slope via the derivative and then using a point on the curve to write the line's equation.

Transcript

00:01 Alright, so here we have a curve, y equals 7 minus x squared, and we want to find the slope of the curve, or more specifically the slope of the line tangent to the curve at the point to, okay, and we can verify that 2 -3 is a point on this curve.

00:26 So what do we do? well, suppose we wanted the slope of the second line, or the average rate of change, we would need an interval.

00:37 And that's basically what we're going to do, but we're going to allow that integral to shrink to this point two.

00:44 Okay, so what we're going to do is we're going to let h be a positive number, and we're going to think of h being a really small positive number, because we're going to want this interval to be really small.

00:56 And we're going to find the average rate of change on the interval to minus h to 2 plus h.

01:06 And then allow that to shrink in around 2.

01:13 So the average rate of change is, well, first we evaluated the right endpoints of 7 minus 2 plus h squared minus 7 minus 2 minus h squared, all over 2 plus h plus h squared, all over 2 plus h minus 2 minus 2 minus 2 minus minus h.

01:47 Okay, and then we just have some algebra to do to see what happens.

01:51 And again, then eventually we want to set h to essentially be zero to find kind of the instantaneous rate of change.

01:59 But we really only know how to do that by looking at sort of the sikent line and then squeezing it in around two.

02:06 Okay, so let's see what we get here.

02:09 So this previous expression, let's expand out seven minus...

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