Differentiating $f(F,\theta)$: Why the Answers Differ?

• agnimusayoti
In summary, the conversation discusses two different methods for calculating the relative error in a given equation. The first method uses a derivative of the equation while the second method uses logarithms. The question asks why the answers using these two methods are different. It is then pointed out that the derivative in the first method is missing a negative sign, which changes the result. The discussion then turns to the concept of correlated errors and how they can affect the calculation of relative error. Ultimately, it is concluded that both methods can be used to calculate the relative error, but it is important to pay attention to signs and potential correlations between errors.
agnimusayoti
Homework Statement
A force of 500 N is measured with a possible error of 1 N. Its component in a
direction 60◦ away from its line of action is required, where the angle is subject to an
error of 0.5◦. What is (approximately) the largest possible error in the component?
Relevant Equations
For ##f(F,\theta) \rightarrow dF= \frac{\partial f}{\partial F} dF +\frac {\partial f}{\partial \theta} d\theta ##
From the question,
$$f(F,\theta)=F \cos \theta$$

1. If I use:
$$df=dF \cos{\theta} -F \sin {\theta} d\theta$$
$$df=dF \cos{\theta} -F \sin {\theta} d\theta \frac {\pi}{180^\circ}=5.28 N$$

2. But, if I take logarithm to both side:
$$ln f=ln F+ln \cos{\theta}$$
differentiate both sides:
$$\frac{df}{f}=\frac{dF}{F} + \frac{\sin\theta}{\cos \theta} d\theta$$
Using radian, it gives ##df=4.28 N##My question is, why the answers are different? Thanks.

Last edited by a moderator:
Signs matter.

Delta2, fresh_42 and DaveE

I think it should also be added that there is no reason to assume that the errors are correlated.

Yeah the second should be negative ##-\tan \theta d\theta##. But then for the worst case or should be positive, because we can take d\theta ia negative then ##-\tan \theta d\theta## becomes positive.

With this condition (worst case), so the first one becomes positive and brings ##df=5.28 N##What do you mean by the errors are correlated? Could you please explain itu further? Because the book show to get relative error by the second method. But i just wonder why can't be calculated by 1st method.

agnimusayoti said:
Because the book show to get relative error by the second method. But i just wonder why can't be calculated by 1st method.
You can. If you divide your expression for ##df## you got using the first method by ##f##, you get the expression you found for ##df/f## using the second method.

agnimusayoti, joshyewa and Delta2
Uh youre right. Thanks!

But again what is the meaning by the errors are correlated?

1. What is the main purpose of differentiating $f(F,\theta)$?

The main purpose of differentiating $f(F,\theta)$ is to determine how the output of the function changes with respect to its input variables, F and $\theta$. This allows us to understand the relationship between the variables and how small changes in one variable affect the output.

2. Why do the answers differ when differentiating $f(F,\theta)$?

The answers differ because the function $f(F,\theta)$ may be a complex mathematical expression with multiple variables and operations. Differentiation involves applying various rules and techniques to find the derivative of the function, which can lead to different answers depending on the method used.

3. What are the common methods used for differentiating $f(F,\theta)$?

The most common methods for differentiating $f(F,\theta)$ include the power rule, product rule, quotient rule, and chain rule. These rules are used to differentiate different types of functions and can be combined to find the derivative of more complex functions.

4. How does differentiating $f(F,\theta)$ help in solving problems?

Differentiating $f(F,\theta)$ helps in solving problems by providing information about the rate of change of the function and the relationship between its input and output variables. This information can be used to optimize the function or solve real-world problems involving rates of change, such as in physics, economics, and engineering.

5. Can differentiating $f(F,\theta)$ be applied to any type of function?

Yes, differentiating $f(F,\theta)$ can be applied to any type of function as long as it is a continuous and differentiable function. This means that the function must be defined for all values of its input variables and have a well-defined derivative at each point.

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