Equations of tangent lines by definition ( 2 ) a. Use...

Equations of tangent lines by definition ( 2 )
a. Use definition (2) (p. 129 ) to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
$$f(x)=x^{3} ; P(1,1)$$

Key Concepts

Derivative as a Limit

This concept involves finding the derivative of a function by taking the limit of the difference quotient as the interval approaches zero. It provides the slope of the tangent line at a given point, capturing the instantaneous rate of change of the function.

Difference Quotient

The difference quotient is the expression (f(x + h) - f(x))/h used to measure the average rate of change of the function over an interval. Its limit as h approaches zero is the foundation of the derivative and the slope of the tangent line.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at only one point and has the same slope as the function at that point. It serves as a linear approximation of the function near the point of tangency.

Point-Slope Form

This is a method for writing the equation of a line when a point on the line and its slope are known. It is particularly useful in deriving the equation of a tangent line by directly substituting the point of tangency and the calculated slope.

Transcript

00:01 Hello.

00:02 So today we're going to take a given function and a specified point and we're going to use the alternative definition for finding the slope to find the slope of the function at the specified point as well as the equation of a tangent line at that point so you see that i provided the alternative definition to the right and right off of that we are going to take our a and set that equal to our given x not and the point which is one in this case because the point is just x not and why not? so then we have x cubed.

00:40 So using this for our function, we have 1 plus h squared, or cubed, and then we have minus 1 cubed, all over h, and the limit as h goes to 0.

00:59 Well, 1 plus h cubed, you might not be familiar with this expansion, so let's break it down to things we know.

01:04 So we've already done an example where we have 1 plus h squared.

01:10 So let's take this and break it down into 1 plus h squared times 1 plus h.

01:16 And that would be to the first power because when you have a something that is cubed, you can actually break it down to three parts such that it is 1 plus h times 1 plus age times 1 plus h.

01:35 So now it's looking a little more.

01:37 Straightforward, something we can deal with, something that's manageable.

01:41 So if we just focus on this part right here, we'll solve that first, or expand it first.

01:48 So one times one is one, and then one times h is h, h times h times h, so we have a two h, then h times h is squared.

01:59 So that's our first part, and then we can multiply this, and i like to put that on the smaller component on the left side.

02:06 It's easier for me.

02:07 So, so 1 times 1 is 1, 1 times 2h, 2h, 1 times h squared, then we have h times 1, which is h, h times 2h, which is 2h squared, and then h times h squared, which is h cubed.

02:25 So that is actually our expanded version of that, and we have minus 1.

02:30 So the 1s cancel, and then if we simplify that numerator, i like to start with the highest exponent...

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