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PreCalc CH 12.3 The Tangent Line Problem.notebook
1
April 09, 2013
The Tangent Line ProblemThe blue segments below are tangent to the red curve, which means the segments usually intersect the curve at only one point.
The Tangent Line Problem
We can find the rate of change of the curve by finding the slope of the blue segments.
The Tangent Line ProblemThe difference in finding slope now as compared to the past is that we will find slope using only one point, whereas in the past we needed two points.
The Tangent Line ProblemIn the next five slides, notice that the distance between the x values of the coordinates is decreasing and approaching zero. We are going to find the slope of a line tangent to the curve at x=1 by using only one point in the last slide whereas in the first four slides we are approximating the slope of the line tangent to the curve by using two points. As the difference in the x values approaches zero, the slope of the line tangent to the curve approaches 2.
The Tangent Line Problem
x1
x2
x2 x1 = 2.00 1.00 = 1.00
The Tangent Line Problem
x1
x2
x2 x1 = 1.50 1.00 = 0.50
PreCalc CH 12.3 The Tangent Line Problem.notebook
2
April 09, 2013
The Tangent Line Problem
x1
x2
x2 x1 = 1.10 1.00 = 0.10
The Tangent Line Problem
x1
x2
x2 x1 = 1.02 1.00 = 0.02
The Tangent Line Problem
x1
x2
x2 x1 = 1.00 1.00 = 0
The Tangent Line ProblemIn the past we have used the equation below for slope.
We have also used difference quotients in the past, but now we will use them to find slope.
The value of h is the distance between x2 and x1.
The Tangent Line Problem
The two limits below follow from the difference quotient.
This is the important one
The Tangent Line Problem
Definition of the Slope of a Graph
The slope of the graph of at the point is equal to the slope of its tangent line at and is given by:
provided this limit exists.
PreCalc CH 12.3 The Tangent Line Problem.notebook
3
April 09, 2013
The Tangent Line ProblemFind the slope of the line tangent to the
graph at a certain point.If find the slope of the line tangent to the graph at the point .
The Tangent Line Problem
The Tangent Line Problem The Tangent Line Problem
The Tangent Line Problem The Tangent Line ProblemSince we know a point on the tangent line and the slope of the tangent line, we can write the equation of the tangent line using the point slope form.
PreCalc CH 12.3 The Tangent Line Problem.notebook
4
April 09, 2013
The Tangent Line Problem
We can rewrite this in slope intercept form:
The Tangent Line Problem
Now we will find a general formula for the slope of any line tangent to a graph.
The Tangent Line Problem The Tangent Line Problem
The Tangent Line Problem The Tangent Line ProblemFor the function below, we now have a formula to find the slope of the tangent line at any point on the graph.
What is the slope of the tangent line at x=5?
What is the slope of the tangent line at x=4?
What is the slope of the tangent line at x=0?
PreCalc CH 12.3 The Tangent Line Problem.notebook
5
April 09, 2013
The Tangent Line Problem
What we have found when we find a general formula for the slope of the tangent line is called the derivative.
The Tangent Line Problem
Definition of the Derivative:The derivative of at is given by
provided the limit exists.
The Tangent Line ProblemThe derivative can be denoted in different ways, but they all represent the same thing.
If
then the derivative can be written as:
The Tangent Line Problem
We have seen that when the line tangent to the curve has a slope of zero, there exists either a minimum or maximum, but we can not always be sure which one we are looking at.
The Tangent Line ProblemFind the locations where the tangent line is horizontal, given the following conditions.
The Tangent Line ProblemThe tangent line is horizontal when the slope is zero.
So we are going to set the derivative equal to zero.
PreCalc CH 12.3 The Tangent Line Problem.notebook
6
April 09, 2013
The Tangent Line Problem The Tangent Line Problem
The Tangent Line ProblemFind the locations where the tangent line is horizontal, given the following conditions.
The Tangent Line Problem
The Tangent Line Problem The Tangent Line ProblemFind the locations where the tangent line is horizontal, given the following conditions.
PreCalc CH 12.3 The Tangent Line Problem.notebook
7
April 09, 2013
The Tangent Line Problem The Tangent Line Problem