MHB Finding the point on a parabola closest to a given point

  • Thread starter Thread starter leprofece
  • Start date Start date
  • Tags Tags
    Parabola Point
Click For Summary
The discussion focuses on finding the coordinates on the parabola defined by x² = y that are closest to the point P(3,0). The approach involves setting up the distance formula and deriving it to find critical points. The calculations lead to the equation 4y² - 13y + 9 = 0, which is solved to find y = 1 and subsequently x = 1. The closest point on the parabola to P is determined to be (1,1). The solution process demonstrates the application of calculus in optimizing distances to curves.
leprofece
Messages
239
Reaction score
0
Find the curve coordinates nearest to P
a)x2 = y P(3,0)

ok to the boox exercise model I must do so

(x-3)2+ (y)2 = D

then x2-6x +9+y2 = D

introducing y into x
y-6sqrt(y) +9 + (y)2 = D

derivating
1-3/(sqrt(y) +0+2y

mcm = sqrt(y) -3 +2y = 0
solving
sqrt(y) = -2y+3
Squaring
y = 4y2-12y +9
4y2-13y +9 = 0
x = 1
y = 1
 
Physics news on Phys.org
leprofece said:
Find the curve coordinates nearest to P
a)x2 = y P(3,0)

ok to the boox exercise model I must do so

(x-3)2+ (y)2 = D

then x2-6x +9+y2 = D

introducing y into x
y-6sqrt(y) +9 + (y)2 = D

derivating
1-3/(sqrt(y) +0+2y

mcm = sqrt(y) -3 +2y = 0
solving
sqrt(y) = -2y+3
Squaring
y = 4y2-12y +9
4y2-13y +9 = 0
x = 1
y = 1

Looks like you already figured this one out. :)
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

Similar threads

MHB How to Find the Second Derivative with Given Equation at a Specific Point?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Finding the point on a parabola closest to the origin
  • · Replies 4 ·
Replies
4
Views
4K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
MHB -apc.3.3.01 Find the equation of the curve that passes through the point (1,2)
  • · Replies 11 ·
Replies
11
Views
3K
MHB 219 AP Calculus Exam Inverse function
  • · Replies 1 ·
Replies
1
Views
2K
MHB Critical Points & Extrema of Multivariable Function
  • · Replies 1 ·
Replies
1
Views
2K
Volume between a region (above x-axis and below parabola) and a surface
  • · Replies 4 ·
Replies
4
Views
2K
MHB 219 GRE what is the value of g'(1)
  • · Replies 3 ·
Replies
3
Views
1K
MHB Implicit function theorem for f(x,y) = x^2+y^2-1
  • · Replies 1 ·
Replies
1
Views
2K
MHB Finding the Area of a Figure Given by an Equation
  • · Replies 1 ·
Replies
1
Views
1K