Minimizing as a function of variables

[Lady M]

Homework Statement

I am given an integral with the bound of -1 to 1, of e^x minus a string of polynomials , x^n, running from 0 to 4, all times constants b[n], running from 0 to 4, all of which is squared. (sorry I don't know how to type functions into this interface yet, but I will include a photo of the question below).

I am asked to "minimize it as a function of 5 variables". Then it wants me to compare the polynomial I get to the Taylor series of e^x to 4 terms, as well as another series approximation of e^x obtained earlier using the inner product of Legendre polynomials.

Relevant Equations

The problem is that I am not sure what's being asked here. Specifically, I am not sure what "minimize the function" means in this context.

I found this post here (https://www.physicsforums.com/threads/minimizing-a-integral.533829/) from a while back, which seems to be addressing a similar question. However, it looks like the "resolution" to the question came from using shifted Legendre polynomials, and then using that to determine the coefficients. Because my class has not done something like that yet, I doubt that's what she wanted us to do.

As promised, here is the original question, with the integral written in a more legible form.

 

[tnich]

Homework Statement: I am given an integral with the bound of -1 to 1, of e^x minus a string of polynomials , x^n, running from 0 to 4, all times constants b[n], running from 0 to 4, all of which is squared. (sorry I don't know how to type functions into this interface yet, but I will include a photo of the question below).

I am asked to "minimize it as a function of 5 variables". Then it wants me to compare the polynomial I get to the Taylor series of e^x to 4 terms, as well as another series approximation of e^x obtained earlier using the inner product of Legendre polynomials.
Homework Equations: The problem is that I am not sure what's being asked here. Specifically, I am not sure what "minimize the function" means in this context.

I found this post here (https://www.physicsforums.com/threads/minimizing-a-integral.533829/) from a while back, which seems to be addressing a similar question. However, it looks like the "resolution" to the question came from using shifted Legendre polynomials, and then using that to determine the coefficients. Because my class has not done something like that yet, I doubt that's what she wanted us to do.

As promised, here is the original question, with the integral written in a more legible form.

View attachment 252974

First, recognize that your integral is a function of the $b_i$s, $f(b_0, b_1, b_2, b_3, b_4)$. How would you find the minimum of a function of five variables?

 

[Lady M]

Perhaps I'm being naive, but I think I would take the div of the function with respect to each variable (in this case b₀,..., b₄) and set it equal to zero.

 

[tnich]

Perhaps I'm being naive, but I think I would take the div of the function with respect to each variable (in this case b₀,..., b₄) and set it equal to zero.

Almost. You don't need the div (I assume you mean $\nabla \cdot f$). Just take the derivative with respect to each variable and set each one equal to zero. That would be $\nabla f = 0$. In other words, find the point where the gradient of the function is zero.

 

[Lady M]

Okay, so would I do that with the entire function in the integral (e^x - (b₀+b₁ x + b₂ x² + b₃ x³ +b₄ x⁴))², or just e^x - (b₀+b₁ x + b₂ x² + b₃ x³ +b₄ x⁴)?

Assuming I take the derivative of the squared function (because otherwise the min value is just zero except for b₀ which is weird), I'm left with a system of 5 equations. What do I do with them? I'm not allowed to treat it like a system of equations and solve for each of the b's, am I?

 

[tnich]

Okay, so would I do that with the entire function in the integral (e^x - (b₀+b₁ x + b₂ x² + b₃ x³ +b₄ x⁴))², or just e^x - (b₀+b₁ x + b₂ x² + b₃ x³ +b₄ x⁴)?

Assuming I take the derivative of the squared function (because otherwise the min value is just zero except for b₀ which is weird), I'm left with a system of 5 equations. What do I do with them? I'm not allowed to treat it like a system of equations and solve for each of the b's, am I?

Your function is
$$f(b_0, b_1, b_2, b_3, b_4) = \int_{-1}^1(e^x-(b_0+b_1x+b_2x^2+b_3x^3+b_4x^4)^2 dx$$
So when you differentiate it with respect to $b_i$, you get
$$\frac d {db_i} f(b_0, b_1, b_2, b_3, b_4) = \frac d {db_i}\int_{-1}^1(e^x-(b_0+b_1x+b_2x^2+b_3x^3+b_4x^4)^2 dx$$
Fortunately, in this case you can interchange the differentiation and integration operators. The point is, you still need to do the integration, but once you complete the differentiation you have something that is easily integrable.
EDIT: And yes, once you have done the integration(s) you have a system of 5 equations in 5 unknowns to solve.

 

[WWGD]

Kind of weird: like doing polynomial regression for $e^x$ but ultimately you can throw in infinitely-many terms and your regression improves with each.