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- Derivatives: Tangent and Normal Lines
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Series: Derivatives: Tangent and Normal Lines
Average Rate of Change
Finding the average rate of change of a function over an interval is the same thing as finding the slope of the function over that same interval. In other words, if you drew a straight line between the points on the function at the endpoints of...Show More
Finding the average rate of change of a function over an interval is the same thing as finding the slope of the function over that same interval. In other words, if you drew a straight line between the points on the function at the endpoints of the interval, the slope, or the average rate of change, would be how fast the line is rising or falling, compared with how fast it is moving to the left or to the right. Show Less
Normal Lines
Learn how to find the equation of the normal line at a given point. To find the equation of the normal line, you'll need to first calculate the derivative of the function, then plug the given point into the derivative to find the slope of the...Show More
Learn how to find the equation of the normal line at a given point. To find the equation of the normal line, you'll need to first calculate the derivative of the function, then plug the given point into the derivative to find the slope of the tangent line. Plug the slope and the given point into the point-slope formula for the equation of the line to find the equation of the tangent line. Then take the negative reciprocal of the slope of the tangent line to find the slope of the normal line, which is the line perpendicular to the tangent line. Finally, plug the new slope and the given point into the point-slope formula for the equation of the line to find the equation of the normal line. Show Less
Tangent Lines
When you're finding the equation of the tangent line, the most important thing to remember is that you're looking for the equation of a line, which means you'll want to use the point-slope formula for the equation of a line. In order to use the...Show More
When you're finding the equation of the tangent line, the most important thing to remember is that you're looking for the equation of a line, which means you'll want to use the point-slope formula for the equation of a line. In order to use the point-slope formula, you'll of course need a point and a slope. You've probably already been given a point, because you've been asked to find the equation of the tangent line at a particular point. Which means that all you need to do is find the slope. In order to find the slope of the tangent line at the given point, all you need to do is take the derivative of the original function, and then evaluate the derivative at the given point. Doing so will give you a constant value, and this is the value you'll use for the slope. Therefore, to write the equation of the tangent line, just take the slope and the point and plug them into the point-slope formula for the equation of the line. Simplify, and you've got the equation of the tangent line to the function at the given point. Show Less
Value That Makes Two Tangent Lines Parallel
In this video we'll learn how to find the value of 'a' that makes two tangent lines of a function parallel to one another, assuming that the tangent lines intersect the graph at the points x=a and x=a+1. Our approach will be to write equations...Show More
In this video we'll learn how to find the value of 'a' that makes two tangent lines of a function parallel to one another, assuming that the tangent lines intersect the graph at the points x=a and x=a+1. Our approach will be to write equations for the tangent lines at these points, then simplify them into slope-intercept form in order to identify the slopes of each of them. Since parallel lines have equal slopes, we'll set the slopes equal to one another to solve for the value that makes them parallel. Show Less
Values That Make the Function Differentiable
When we talk about differentiability, it’s important to know that a function can be differentiable in general, differentiable over a particular interval, or differentiable at a specific point. In order for the function to be differentiable in...Show More
When we talk about differentiability, it’s important to know that a function can be differentiable in general, differentiable over a particular interval, or differentiable at a specific point. In order for the function to be differentiable in general, it has to be differentiable at every single point in its domain. If there’s just a single point where the function isn’t differentiable, then we can’t call the entire curve differentiable. In that case, we could only say that the function is differentiable on intervals or at points that don’t include the points of non-differentiability. So how do we determine if a function is differentiable at any particular point? Well, a function is only differentiable if it’s continuous. So if there’s a discontinuity at a point, the function by definition isn’t differentiable at that point. This applies to point discontinuities, jump discontinuities, and infinite/asymptotic discontinuities. But there are also points where the function will be continuous, but still not differentiable. Remember, differentiability at a point means the derivative can be found there. If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined. That means we can’t find the derivative, which means the function is not differentiable there. In the same way, we can’t find the derivative of a function at a corner or cusp in the graph, because the slope isn’t defined there, since the slope to the left of the point is different than the slope to the right of the point. Therefore, a function isn’t differentiable at a corner, either. Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Barring those problems, a function will be differentiable everywhere in its domain. Show Less