Find the sum of a function, then minimize?

[mrjust]

I am in my first semester of calculus in college.

Find the value of m such that the sum of the squares of the vertical distances from each of the points (1,1), (2,2), and (3,2) to the line y=mx is minimized. Hint: Find the sum as a function of m ( no x in the expression) and then minimized it.



I am stuck and don't know how to get started. I know that I have to take the derivative of the equation that I find and minimize m, but I feel what's confusing me is translating the question into numbers. Can I get some hints to get me going? Thank you.

 

[Deleted member 4993]

I am in my first semester of calculus in college.

Find the value of m such that the sum of the squares of the vertical distances from each of the points (1,1), (2,2), and (3,2) to the line y=mx is minimized. Hint: Find the sum as a function of m ( no x in the expression) and then minimized it.

View attachment 2754

I am stuck and don't know how to get started. I know that I have to take the derivative of the equation that I find and minimize m, but I feel what's confusing me is translating the question into numbers. Can I get some hints to get me going? Thank you.

Let us calculate the vertical distance of the first point from the line.
The abscissa of the point is x = 1 → What is the y value of the point? What is the y value on the line at that x value ? What is the difference between these two (y) values? Now do this for all the three points and continue....

 

[JeffM]

I am in my first semester of calculus in college.

Find the value of m such that the sum of the squares of the vertical distances from each of the points (1,1), (2,2), and (3,2) to the line y=mx is minimized. Hint: Find the sum as a function of m ( no x in the expression) and then minimized it.

View attachment 2754

I am stuck and don't know how to get started. I know that I have to take the derivative of the equation that I find and minimize m, but I feel what's confusing me is translating the question into numbers. Can I get some hints to get me going? Thank you.

This is a GREAT problem, but I can see why it is hard for you. You see a function of y = mx, but that is NOT the function that you need to minimize. How to solve many problems becomes a very easy mechanical task once you translate the problem into mathematical language. Let's think about what the problem says about the function to be minimized. First, the problem says the function is a sum. How many summands in the sum? Second, the problem says that each term summed is itself a square. The square of what? How do you calculate what is to be squared? If those questions plus Subhotosh Khan's hint are not enough, come back and give us the answers to the questions above that you can answer, and someone will assist you further.

 

[mrjust]

Thanks for your help, so these is what I came up with:
(1,1) (2,2) (3,2)

y=m
y=2m
y=3m

(1,m), (2,2m), (3,3m) These are the points on the line.
so using d= | yline- ydata| I will get the distance between the points.

s(m)= (m - 1)^2 + (2m - 2)^2 + (3m - 2)^2
s(m)= m^2 - 2m + 1 + 4m^2 - 8m + 4 + 9m^2 - 12m + 4
s(m)=14m^2 - 22m + 9
Now I take the derivative
s'(m)=28m - 22
0=28m - 22
m=11/14

Is this correct?

 

[Deleted member 4993]

Thanks for your help, so these is what I came up with:
(1,1) (2,2) (3,2)


Is this correct?

As far as I can tell - your work is correct .

 

[mrjust]

Thank you guys very much for you help!