Explanation Given information: The point on the parabola y = x 2 is ( x , y ) , that has a minimal distance from the point ( 16 , 1 2 ) as shown in the below figure. Explanation: Use the distance formula to find the distance between the two points ( x , y ) and ( 16 , 1 2 ) in the given figure. Distance Formula: Let the distance between point ( x , y ) and ( x 1 , y 1 ) is d = ( x − x 1 ) 2 + ( y − y 1 ) 2 By using the above formula, The distance between ( 16 , 1 2 ) and ( x , y ) is d d = ( x − 16 ) 2 + ( y − 1 2 ) 2 So, d is an objective function that is to be minimized in two variables x and y is, d = f ( x , y ) = ( x − 16 ) 2 + ( y − 1 2 ) 2 subject to the constraint y = x 2 Use Lagrange multipliers and constrained optimization method: Replace f ( x , y ) by an auxiliary function of three variables F ( x , y , λ ) as, F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Substitute d = f ( x , y ) = ( x − 16 ) 2 + ( y − 1 2 ) 2 and g ( x , y ) = y − x 2 in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . ⇒ F ( x , y , λ ) = ( x − 16 ) 2 + ( y − 1 2 ) 2 + λ ( y − x 2 ) Find partial derivatives of F ( x , y , λ ) = ( x − 16 ) 2 + ( y − 1 2 ) 2 + λ ( y − x 2 ) and set them equal to zero. The partial derivative of F ( x , y , λ ) = ( x − 16 ) 2 + ( y − 1 2 ) 2 + λ ( y − x 2 ) with respect to x is ∂ F ∂ x = x − 16 ( x − 16 ) 2 + ( y − 1 2 ) 2 − 2 x λ = 0 …

15th Edition

ISBN: 9780137638826

Author: Goldstein

Publisher: PEARSON

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Question

Chapter 7.4, Problem 22E

To determine

The point on the parabola y=x2, when distance is minimum from a point

(16,12) to parabola

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Chapter 7.4, Problem 21E

Chapter 7.4, Problem 23E

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Chapter 7 Solutions

CALCULUS+ITS APPL.,BRIEF-MYLAB MATH

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The point on the parabola y = x 2 , when distance is minimum from a point ( 16 , 1 2 ) to parabola

Solution Summary: The author explains the distance formula used to determine the point on the parabola y=x2.

The point on the parabola y=x2, when distance is minimum from a point

(16,12) to parabola

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Chapter 7 Solutions

The point on the parabola y = x 2 , when distance is minimum from a point ( 16 , 1 2 ) to parabola

Solution Summary: The author explains the distance formula used to determine the point on the parabola y=x2.

The point on the parabola y=x2, when distance is minimum from a point (16,12) to parabola

Want to see the full answer?

Chapter 7 Solutions

The point on the parabola y=x2, when distance is minimum from a point

(16,12) to parabola