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Non Differentiable Graphs of Derivatives

Non Differentiable Graphs of Derivatives
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  • 0:10 The Normal Line
  • 1:06 Equation of a Normal Line
  • 2:19 First Example
  • 4:07 Second Example:…
  • 7:22 Lesson Summary
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Lesson Transcript
Instructor: Heather Higinbotham
When I walk along a curve, I stand normal to it. That is, I stand perpendicular to the tangent. Learn how to calculate where I'm standing in this lesson.

The Normal Line

The normal line is at a 90-degree angle from the original line
Normal Line

In general, when I'm walking along some curve my feet are traveling in the line that's tangent to the curve, and my head is sticking straight up away from the curve. If I draw a line from my head to my feet, that line is perpendicular to the tangent at that point on the curve. The line of my body, from my head to my feet, is called the normal line. If you have some original line, your tangent in this case, then the line normal to it is at a 90-degree angle from the original line. It's perpendicular to the original line.

Anytime you have a line that is perpendicular to another line, the slopes are related by a pretty nifty formula. If you have your original line with a slope of m, then the slope of the normal line will be -1/m. This is the negative reciprocal.

Equation of a Normal Line

How can we find out the equation for the line of my body at any point in time when I'm walking along a curve? We're going to follow a five-step formula for finding normals.

The graph corresponding to example #1
Normal Example 1

First, we're going to find the point on the curve where my feet are. So given some x value, I'm going to plug in x to a function to get y. Now I've got the point on the curve where my feet are. Next, I need to figure out where the tangent is. What tangent am I walking along? I'm going to first calculate the derivative of my function, so I'm going to find y`. Once I've calculated the derivative, I'm going to find the slope of the tangent at the point where my feet are. So this is going to tell you which direction I'm walking. Once I've found the slope of the tangent, I'm going to use the negative reciprocal to find the slope of the normal. This is the direction that my head is compared to my feet. Once I have the slope of the normal, I'm going to use point-slope form to find an equation for my head and feet. This is going to be y = y sub 1 + the slope of the normal, n(x - x sub 1), where x sub 1 and y sub 1 are the positions of my feet.

First Example

Let's do an example. Let's say I'm walking (rather upside-down, I might add) on the curve y=x^2. In particular, I'm walking, and I've just gotten to x=1, so the first step is find out exactly where I am. At x=1, y=x^2, so y=1. My feet are going to be at the point (1,1). Then I'm going to find the derivative of my y=x^2. This is my second step in finding the normal: y` equals d/dx(x^2). I'm going to use the power rule here. I find that y`=2x. Now I need to find the slope of the tangent at the point 1,1. To do this, I'm going to plug in my point (x=1, y=1) into the derivative. When x=1, y`=2x, so y`= 2 * 1. The slope of the tangent at the point (1,1) is equal to 2. What about the slope of the normal? That's going to be -1 divided by the slope of the tangent: -1/2. Now I have the slope of the normal, and I have a point for the normal, so I'm going to use point-slope form to find the equation for the normal line: y=y sub 1 + n(x - x sub 1), so y = 1 - 1/2(x - 1) because my point is (1,1) and the slope is -1/2. That's the equation of this line that is perpendicular to the function at the point (1,1). My head is downward. Great.

Finding the equation for the normal at the point x=1 in example #2
Normal Example 2

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