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

Click For Summary
SUMMARY

The discussion focuses on finding the differential equations for the family of tangents and normals to the curve defined by the equation y = x^2. The slope of the tangent line at any point x = a is determined by the derivative, y' = 2a, leading to the equation of the tangent line as y - a^2 = 2a(x - a). For normals, which are perpendicular to tangents, the slope is -1/(2a), resulting in the equation y - a^2 = -1/(2a)(x - a). These equations represent the respective differential equations satisfied by tangents and normals to the parabola.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives and slopes.
  • Familiarity with the equation of a line in slope-intercept form.
  • Knowledge of the geometric interpretation of tangents and normals to curves.
  • Basic understanding of differential equations and their applications.
NEXT STEPS
  • Study the derivation of tangent lines for other polynomial functions.
  • Explore the concept of normals in greater depth, particularly for non-linear curves.
  • Learn about higher-order derivatives and their implications for curve analysis.
  • Investigate the applications of differential equations in physics and engineering contexts.
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in the geometric properties of functions and their derivatives.

rozin
Messages
1
Reaction score
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.
 
Physics news on Phys.org
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 y= x^2, 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?
 

Similar threads

Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Expressing a differential equation into a different format
  • · Replies 1 ·
Replies
1
Views
2K
Graduate A question about an 'extremal' surface
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Integral-form change of variable in differential equation
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad General solution vs particular solution
  • · Replies 52 ·
2
Replies
52
Views
8K
Undergrad Can I Differentiate Both Sides to Solve y in y^2=ln(1-y^2)?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad On sub and super solutions: Teschl and others
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Direction Fields and Isoclines
  • · Replies 1 ·
Replies
1
Views
729
Undergrad Partial Differential Equation solved using Products
  • · Replies 2 ·
Replies
2
Views
2K