This is a broad chapter containing three fundamental topics. They are Differentials, Errors and Approximations. Here, students obtain a basic knowledge of these topics and also get a chance to practice the problems on all these topics.
Firstly, students study how to define differentials and understand their meaning. In calculus, differential denotes the principal part of the change that occurs in a function, y=f (x) with respect to changes in the independent variable. So, the differential dy can be defined as,
dy=f' x dx
where, f' (x) is considered the derivative of f with respect to x, and dx is an additional real variable, which means dy is a function of x, and dx signifies a change in the value of x. So, the notation can be changed as,
After learning how to use differentials, students are exposed to a topic called Errors. Error is nothing but a mistake that happens while doing a process. In calculus, errors are of different types. The following are discussed in this chapter: Absolute error, Relative error, Percentage error.
Absolute error is defined as the amount of error in the practical measurement. It is also said to be the difference between the measured value and true value. Relative error, while using it as a measurement of precision, is considered as the ratio of the absolute error of a measurement to the measurement which is being taken. It can be denoted in percentage and has no specific units. Percentage error is defined as the difference between the estimated value and the actual value compared to the actual value and is expressed in percentage. The below mentioned is the formula to calculate percent error,
T = True or Actual value
E = Estimated value
Other than all this, students also study the geometrical meaning of differentials, algorithms or steps involved in finding the geometrical meaning of differentials and learning how to find the approximate value using differentials. Lastly, they get a chance to study some of the important results of differentials.