Differentials

Differentials, small changes that make big differences

Recall that the derivative has several notations.
.
We call Leibniz notation, do you know who Leibniz was?
Gottfried Wilhelm Leibniz

In differential calculus it wasn't very important which notation we used, but in integral calculus it is very important to use Leibniz notation.
Now we will consider that dy can be separated fromdx, dy is the differential of y and dx is the differential of x. This means that the following operation is possible:
If , then

Example: Find the differential for each of the following functions.
a)
This is a derivative.
This is the same derivative using Leibniz Notation.
This is a differential. Leibniz Notation is convenient because we can separate dy and dx.

b)Derivative
Differential

c)
Derivative using Leibniz Notation
Differential

What is the difference between these three problems?


What are differentials and what are they for?
We have used Leibniz notation to denote the derivative of y with respect to x, but we have regarded it as a single entity and not a ratio. Now we will give dy and dx separate meanings. Thesequantities are called differentials and they are useful in finding approximate values of functions.

Approximate values of functions
Let's begin with a very simple example, suppose that we want to calculate the area of a square. As you all know, this is not very difficult, all you need is to measure one of the sides of the square. If we call this measure x, then we can compute the area with theformula:
A(x) = x2
Suppose that the measure of the side is 12 cm, compute the area of the square.

A = ___________

Ok, but now let's suppose that instead of measuring 12 cm, which is the correct measurement of the side, we make a mistake and measure 12.2 cm instead. If we use this measure to compute the area of the square, what is the answer now?

A = __________


We call the error inmeasurement of the side a change in x or delta x (x).
In this case, x is 0.2 cm.
The error in computing the area of the square is called delta A (A). This error is the difference between the answer obtained with an error in measurement and the correct answer.
In this case 148.84 cm2 - 144 cm2 = 4.84 cm2.
Conclusion: Small errors in measurements can become big errors, when using thisinformation in calculations.
Another way of expressing the error in the area is using percents. The error in computing the area: 4.84 cm2 doesn’t tell us if the error is big or small. The only way to decide how big the error in the area really is is to compare it with the correct area and find the percentage error.
Correct area ---- 100%
Error in the area ---- ?

To calculate the percentage errordivide A over the correct area (the area that you computed with the correct x measurement), then multiply times 100.
In this case the answer is (4.84/144) * 100= 3.36%.
Sometimes we need to find the relative error; this is the ratio of A over the correct area.
In this case 4.84/144 = 0.0336.
I prefer the percentage error, but it is common to use the relative error. You can see that theyare similar, in the percentage error you multiply times 100, in the relative error you don't.

Stop for a moment and consider what you know up to now. Write the meaning of the following concepts:
x __________________________________________________
A _________________________________________________
Relative Error _________________________________________
Percentage Error___________________________________________

Now find the error in computing the area of the square and the percentage error using different values of x and x.
x= 6cm.
x = 0.2 cm.

A_________________________________________________
Relative Error _________________________________________
Percentage Error ____________________________________________

What relation does this have with...