Optimization: A Distance Example
Main Concept

An optimization problem involves finding the best solution from all feasible solutions. One is usually solving for the largest or smallest value of a function, such as the shortest distance or the largest volume.
A minimum or maximum of a continuous function over a range must occur either at one of the endpoints of the range, or at a point where the derivative of the function is 0 (and thus the tangent line is horizontal). These are called critical points.

Steps


1.

Identify what value is to be maximized or minimized.

2.

Define the constraints.

3.

Draw a sketch or a diagram of the problem.

4.

Identify the quantity that can be adjusted, called the variable, and give it a name, such as x.

5.

Write down a function expressing the value to be optimized in terms of x.

6.

Differentiate the equation with respect to x.

7.

Set the equation to 0 and solve for .

8.

Check the value of the function at the end points.





Problem: Every morning Tom leaves his house, gets water from the river, and takes it to the farm. What is the shortest possible path that Tom has to walk?
Let x be the distance downstream from the house at the point where Tom gets water from the river.
Adjust the value of using the slider to find value that minimizes the distance traveled.

Numerical solution



,



Total distance :








Calculate first derivative:




Set it to 0:




Solve for an x value that minimizes T:
































Choose the positive root:








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