What point on the parabola y = 3 -x is closest to the point (3,3)? Let D be the distance between the two points. What is the objective function in terms of the x-coordinate? D=D (Type an expression.) The interval of interest of the function is. (Simplify your answer. Type your answer in interval notation.) The point on the parabola y = 3-x² that is closest to the point (3,3) is. (Type an ordered pair. Round to two decimal places as needed.)
What point on the parabola y = 3 -x is closest to the point (3,3)? Let D be the distance between the two points. What is the objective function in terms of the x-coordinate? D=D (Type an expression.) The interval of interest of the function is. (Simplify your answer. Type your answer in interval notation.) The point on the parabola y = 3-x² that is closest to the point (3,3) is. (Type an ordered pair. Round to two decimal places as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
What point on the parabola \( y = 3 - x^2 \) is closest to the point \( (3,3) \)?
**Objective Function:**
Let \( D \) be the distance between the two points. What is the objective function in terms of the x-coordinate?
\[ D = \] \(\_\_\_\_\_\)
*(Type an expression.)*
**Interval of Interest:**
The interval of interest of the function is \(\_\_\_\_\_\).
*(Simplify your answer. Type your answer in interval notation.)*
**Closest Point:**
The point on the parabola \( y = 3 - x^2 \) that is closest to the point \( (3,3) \) is \(\_\_\_\_\_\).
*(Type an ordered pair. Round to two decimal places as needed.)*
**Instructions:**
- Fill in the blanks with the appropriate mathematical expressions and solutions.
- For the distance \( D \), use the distance formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Determine the interval where the solution is applicable by considering the points on the parabola.
- Use calculus or another optimization method to find the point that minimizes \( D \).
**Notes:**
- Ensure calculations maintain accuracy up to two decimal places where applicable.
- Consider using the vertex form of the parabola for easier manipulation, if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa66e66dd-9c0e-423a-baa5-9e5f9c2a5700%2F825eb176-cdcd-4381-a16a-9d04307a8246%2Fizgrp4n_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
What point on the parabola \( y = 3 - x^2 \) is closest to the point \( (3,3) \)?
**Objective Function:**
Let \( D \) be the distance between the two points. What is the objective function in terms of the x-coordinate?
\[ D = \] \(\_\_\_\_\_\)
*(Type an expression.)*
**Interval of Interest:**
The interval of interest of the function is \(\_\_\_\_\_\).
*(Simplify your answer. Type your answer in interval notation.)*
**Closest Point:**
The point on the parabola \( y = 3 - x^2 \) that is closest to the point \( (3,3) \) is \(\_\_\_\_\_\).
*(Type an ordered pair. Round to two decimal places as needed.)*
**Instructions:**
- Fill in the blanks with the appropriate mathematical expressions and solutions.
- For the distance \( D \), use the distance formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Determine the interval where the solution is applicable by considering the points on the parabola.
- Use calculus or another optimization method to find the point that minimizes \( D \).
**Notes:**
- Ensure calculations maintain accuracy up to two decimal places where applicable.
- Consider using the vertex form of the parabola for easier manipulation, if needed.
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