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

Tangent Line

A tangent line to a curve at a specific point is the line that just touches the curve at that point and has the same slope as the curve there. The concept of tangency is crucial because the slope of the tangent line is obtained by evaluating the derivative at that point, providing a linear approximation of the function.

Power Rule

The power rule is a key method in differentiation that simplifies finding derivatives of polynomial terms. It states that for any term of the form x^n, the derivative is n*x^(n-1). This rule is widely used because it is simple and directly applicable to any polynomial function.

Derivative

The derivative of a function is a fundamental concept in calculus that represents the function's instantaneous rate of change with respect to its variable. It tells you how the function's output is changing at any given point and is used to analyze rates of change, slopes of curves, and behaviors of functions.

Point-Slope Form

The point-slope form is a way of writing the equation of a line when you know one point on the line and its slope. This form is particularly useful in calculus for writing the equation of the tangent line, where the point is the point of tangency and the slope is given by the derivative of the function at that point.

Transcript

00:01 Okay, so for this problem we're going to use the method in example 3 to find the slope of the curve at the given point and an equation for the tangent line.

00:10 We're using the method of slope or example 3.

00:14 So in example 3 they assume that there's a point q that makes a secant line with p.

00:20 Now remember a secant line is where you have two points on the curve and you connect them.

00:27 That's the secant line and we're going to define this point as 2 plus h comma this long expression here to the right.

00:37 And all i've done here is put substituted 2 plus h in for x in the expression.

00:42 So notice here 2 plus h cubed.

00:45 Notice here 2 plus h squared.

00:48 This is the y coordinate of the point q.

00:53 And so then we can use point p and point q.

00:59 We can find the slope of the secant line.

01:03 We do some substitution and the reason we get h is because your x coordinate is 2 plus h and then you subtract 2.

01:13 So you're left with h.

01:17 You'll notice i substituted this big y thing in first and then subtracted 0 to make it a little more painless.

01:26 And then i simplified this expression to have h squared plus 3h.

01:31 Now this is going to be give me the slope of the secant line.

01:35 So now if we're just thinking about what happens if we change h.

01:43 So if h is negative 1 then the slope of my secant line would be negative 2.

01:52 And so it's coming down...

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