# Calculating Error in Multiple Independent Variables

• IZlo0110
In summary: This will give you an algebraic expression for the error in the function due to the uncertainty in x.
IZlo0110

## Homework Statement

Okay so I was given this question to start calculating error on other problems.

Consider the equation F(x,y,z) = x^4+2y^3+5yz+5. Let the uncertainty in x be represented by the variable dx, the uncertainty in y be represented by the variable dy, and the uncertainty in z be represented by the variable dz.

## Homework Equations

I was given this equation to work from Δfx(x,y,z) = abs(f(x+Δx,y,z)-f(x,y,z))

## The Attempt at a Solution

Okay so I started to use the equation I was given, only substituting dx for Δx. Then I "solved" for Δfx to get Δx*y*z. Only that is not the right answer.

I am not sure how to incorporate the equation and I am unclear on how to proceed with this problem.

You haven't actually stated the question, you've only given some the parameters for it. Is the question to find the error in F? And what is Δfx? Your notation isn't clear. You might find section 9 of this resource useful, particularly the bit near the end.

Okay I apologize, the actual question is to find an algebraic expression for the uncertainty in the function due to the uncertainty in x.

I think you are just making an algebraic error. If you expand out $(x+\Delta x)^4$, you will have many more terms left over after cancellation than just the one you gave. Without seeing more of your work, I can't say where you've gone wrong.

Also, are you supposed to assume the error is small, or arbitrary? It would be very helpful to see the entire question.

Okay so here is what I tried to do. Though something is still off and I am not sure what I am doing wrong.

Δfx = abs(f(x+Δx,y,z)-f(x,y,z))
Δfx = abs((x+Δx)yz - x^4-2y^3-5yz-5)
Δfx = abs(xyz+Δxyz-x^4-2y^3-5yz-5)

You've forgotten the power of four in the first term. It's not $(x+\Delta x)$, it's $(x+\Delta x)$.

Edit: and you've forgotten the remaining three terms in $f(x+\Delta x)$, the additional $+2y^3 + 5yz + 5$.

Edit2: And I'm not sure why you're multiplying the first term by $yz$.

From the above, it seems you're confused about what $f(x+\Delta x,y,z)$ means. It means that in your original formula, you need to replace everywhere you see $x$ with $x+\Delta x$.

## What is the definition of error in multiple independent variables?

The error in multiple independent variables is the difference between the actual values of the variables and the predicted values based on a mathematical model or equation. It is a measure of how well the model fits the data.

## How is error calculated in multiple independent variables?

Error in multiple independent variables is typically calculated using a statistical method called least squares. This involves finding the sum of the squared differences between the actual values and the predicted values, and then minimizing this sum to find the best fit for the model.

## What is the purpose of calculating error in multiple independent variables?

Calculating error in multiple independent variables allows scientists to evaluate the accuracy and reliability of their mathematical models. It can also help identify any outliers or unusual data points that may be affecting the model's predictions.

## Can error in multiple independent variables be eliminated completely?

No, it is not possible to completely eliminate error in multiple independent variables. This is because there will always be some variability in the data and no model can perfectly capture the complexity of real-world systems. However, by minimizing the error, scientists can improve the accuracy of their models.

## How is error in multiple independent variables used in scientific research?

Error in multiple independent variables is used in various fields of science, such as physics, chemistry, and biology, to evaluate the effectiveness of mathematical models and make predictions about real-world phenomena. It is also used to compare different models and determine which one best fits the data.

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