Differential Equation: Tangents & Normals to y=x^2

In summary, the differential equation satisfied by the equation of the family of tangents to y=x^2 is y' = 2x, and the differential equation satisfied by the equation of the family of normals to y=x^2 is y' = -1/2x. Both of these equations can be derived by using the fact that the slope of a tangent line is equal to the derivative of the function at the point of tangency, and the slope of a normal line is equal to -1/m, where m is the slope of the tangent line.
  • #1
rozin
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0
Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.
 
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  • #2
I presume you know that a tangent line to a graph has slope equal to the derivative of the function at the point of tangency. Surely you know that the derivative of [tex]y= x^2[/tex], at x= a, is y'= 2a. So what is the equation of the tangent line there? What differential equation does every such tangent line satisfy?

Do you know that a line "normal" to a graph is perpendicular to the tangent line at that point? And that the slope of a line, perpendicular to a line with slope "m", is -1/m?
 

FAQ: Differential Equation: Tangents & Normals to y=x^2

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a function at a given point.

What is the tangent line to y=x^2 at a specific point?

The tangent line to y=x^2 at a specific point is a line that touches the curve y=x^2 at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of y=x^2 at that point.

How do you find the equation of the tangent line to y=x^2 at a specific point?

To find the equation of the tangent line to y=x^2 at a specific point, you need to find the slope of the curve at that point using the derivative of y=x^2. Then, you can use the point-slope formula to find the equation of the tangent line.

What is the normal line to y=x^2 at a specific point?

The normal line to y=x^2 at a specific point is a line that is perpendicular to the tangent line at that point. It represents the rate of change of the tangent line at that point.

How do you find the equation of the normal line to y=x^2 at a specific point?

To find the equation of the normal line to y=x^2 at a specific point, you can first find the slope of the tangent line at that point using the derivative of y=x^2. Then, you can use the negative reciprocal of this slope to find the slope of the normal line. Finally, you can use the point-slope formula to find the equation of the normal line.

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