Using differentiation to find maximum length problems

In summary: The sum of those is \sqrt{x^2+ 1}. What do you want to maximize?In summary, the problem asks to find the greatest possible value of the sum of the lengths of AB and AC, given that the line segment AB lies on a diameter of a circle of radius 1 and the angle BAC is a right angle. One approach to solving this problem is to set up a coordinate system with the origin at the center of the circle, A at (-1, 0), and the y-axis parallel to BC. This allows us to express AB and BC in terms of the parameter x, and the sum of their lengths can be simplified to \sqrt{x^2+ 1}. To maximize this
  • #1
lmstaples
31
0

Homework Statement



The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle.

Find the greatest possible value of the sum of the lengths of AB and AC.


Homework Equations





The Attempt at a Solution



I have no idea what parameters to use or how to set up the problem.

Any help would be much appreciated.
 

Attachments

  • image.jpg
    image.jpg
    37.5 KB · Views: 423
Physics news on Phys.org
  • #2
lmstaples said:

Homework Statement



The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle.

Find the greatest possible value of the sum of the lengths of AB and AC.


Homework Equations





The Attempt at a Solution



I have no idea what parameters to use or how to set up the problem.

Any help would be much appreciated.

Why not try picking the parameter to be the x coordinate of A?
 
  • #3
check out the attachment, does that seem right?
 

Attachments

  • image (1).jpg
    image (1).jpg
    26 KB · Views: 487
  • #4
lmstaples said:
check out the attachment, does that seem right?

Well, you ignored my advice on which parameter to use, so apparently you weren't all that confused to begin with. Yes, it looks ok.
 
  • #5
Haha sorry, I had so many ideas of ways to do it and that's the only one I managed to fully work through :)
 
  • #6
Here (I think) is what Dick was suggesting: set up a coordinate system so the origin is at the center of the circle, A is at (-1, 0), and the y-axis is parallel to BC. Then AB has length x+ 1 and BC has length [itex]\sqrt{1- x^2}[/itex].
 

1. What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function at any given point. It involves finding the derivative of a function, which is a measure of how the function changes with respect to its input.

2. How is differentiation used to find maximum length problems?

Differentiation can be used to find the maximum length of a curve or surface by finding the point at which the derivative of the function is equal to 0. This point, known as the critical point, is where the slope of the curve or surface is flat and can indicate the maximum or minimum point.

3. What is the role of the first and second derivatives in finding maximum length problems?

The first derivative represents the rate of change of the function, while the second derivative represents the rate of change of the first derivative. In finding maximum length problems, the first derivative is set to 0 to find the critical points, and the second derivative is used to determine if these points are maximum or minimum points.

4. Can differentiation be used for any type of function to find maximum length problems?

Yes, differentiation can be used for any type of continuous and differentiable function to find maximum length problems. However, the process may become more complex for more complicated functions, and some functions may require the use of advanced differentiation techniques.

5. Are there any limitations to using differentiation to find maximum length problems?

While differentiation is a useful tool for finding maximum length problems, it is important to note that it may not always provide the most accurate or practical solutions. In some cases, other mathematical methods or real-world constraints may need to be considered in addition to differentiation to find the most optimal solution.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
792
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Precalculus Mathematics Homework Help
Replies
27
Views
1K
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
303
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
999
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
771
Back
Top