Total differential to calculate approximately the largest error

In summary, the largest possible error in determining the area of a right triangle with cathetuses measuring 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement, is approximately 0.7 cm. This is calculated using the total differential to account for errors in both the length of the sides.
  • #1
Ereisorhet
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I have the following problem:
Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement.

I did this:
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  • #2
Ereisorhet said:
I have the following problem:
Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement.

I did this:

Greetings hero, and welcome to MHB!

Your end result is correct.

What you wrote to get there, is not quite right though. For starters there is no actual reference to the total differential.
Let me clean it up a bit.

Let $A$ be the area of the triangle, and let $a$ and $b$ be the lengths of the sides.
The total differential is then $dA$.
And:
\begin{aligned}
A&=\frac 12 ab \\
dA&=d\left(\frac 12 ab\right) = \frac 12 b\,da + \frac 12 a\,db \\
\text{largest error} &\approx \left| \frac 12 b\,da \right| + \left|\frac 12 a\,db\right| = \left| \frac 12\cdot 6\cdot 0.1 \right| + \left| \frac 12\cdot 8\cdot 0.1 \right| = 0.3 + 0.4 = 0.7
\end{aligned}
 

FAQ: Total differential to calculate approximately the largest error

1. What is a total differential?

A total differential is a mathematical concept used to calculate the approximate change in a function's output based on small changes in its input variables. It takes into account all the partial derivatives of the function with respect to each input variable.

2. How is a total differential used to calculate the largest error?

To calculate the largest error, we use the total differential to approximate the change in the function's output due to small changes in the input variables. This approximation can then be used to estimate the maximum possible error in the function's output.

3. What is the difference between total differential and partial differential?

The total differential takes into account all the partial derivatives of a function, while the partial differential only considers the derivative with respect to one variable. In other words, the total differential gives a more comprehensive understanding of how the function changes with respect to all its input variables.

4. Can the total differential be used for any type of function?

Yes, the total differential can be applied to any differentiable function. It is a general concept that can be used to approximate the change in any type of function's output based on small changes in its input variables.

5. How is the total differential related to the concept of linear approximation?

The total differential is closely related to the concept of linear approximation, as it is used to approximate the change in a function's output using its first-order partial derivatives. This is similar to how linear approximation uses the tangent line to estimate the change in a function's output at a specific point.

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