 Hello and welcome to the session. In this session we will discuss approximations. Here we will learn how we use differentials to approximate values of certain quantities. Let y be equal to fx. Suppose del x is a small increment in x del y be increment in y corresponding to the increment in x that is we have del y is equal to f of x plus del x minus f of x. Then differential of x denoted by dx is defined by dx equal to del x and differential of y denoted by dy is defined by dy equal to f dash x dx or this could be written as dy is equal to dy by dx into del x. Also dy is a good approximation of del y when dx equal to del x is relatively small. We denote this by dy approximately equal to del y. So we note that the differential of the dependent variable that is the variable y is not equal to the increment of the variable that is dy is not equal to del y whereas differential of the independent variable that is x is equal to the increment of the variable that is dx is equal to del x. Let's try and find out the approximate value of the square root 26. For this let y be equal to square root x then we take x equal to 25 and del x equal to 1. We have del y is equal to square root x plus del x minus square root x that is this becomes equal to square root 25 plus 1 minus square root 25 that is we have del y is equal to square root 26 minus square root 25 is 5. So from here we get square root 26 is equal to del y plus 5. Now dy is approximately equal to del y and dy is given by dy upon dx into del x. We have y equal to square root x so from here we get dy by dx equal to 1 upon 2 square root x that is equal to 1 upon 2 square root 25 since x was equal to 25 and that becomes equal to 1 upon 2 into 5 that is 10. So we have dy upon dx is equal to 1 upon 10. Thus here we get dy is equal to 1 upon 10 into del x that is equal to 1 so we have dy is equal to 0.1 and we know dy is approximately equal to del y so from here we get root 26 is equal to del y or dy that is 0.1 plus 5 that is equal to 5.1 so we get square root 26 is equal to 5.1. So this is how we use differentials to find approximate values of certain quantities. This completes the session. Hope you have understood the concept of approximations.