3 handouts section3-11
0 likes226 views
This document discusses linear approximations and differentials. It introduces: 1) Finding the linear approximation of non-linear functions by using the equation of the tangent line. 2) Finding the differential dy of a function y=f(x), which represents the change in y for an infinitesimal change dx in x. 3) Using differentials to approximate small changes or errors in a function when there is a small change in the input variable.
3 handouts section3-11
- 1. Section 3.11 Linear approximations and differentials Learning outcomes After completing this section, you will inshaAllah be able to 1. find linear approximation of non-linear functions 2. find differential of a function 3. use differentials to approximate small changes or errors 13.11
- 2. Linear approximation Why do we need linear approximations? • See class explanation How to approximate? • Given a function ( )y f x= . • Equation of its tangent line at ( ), ( )a f a is ( ) ( )( ) or ( ) ( )( ) y f a f a x a y f a f a x a ′− = − ′= + − 23.11 Main idea of linear approximation “Near point x=a, the tangent line and the function f(x) have approximately same graph” Linear Approximation of f(x) For values of x near x=a ( ) ( ) ( )( )f x f a f a x a′≈ + − . See class explanation See examples 1, 2 done in class
- 3. Differential of a function Finding change y∆ in ( )y f x= corresponding to change x dx∆ = in x Using differentials to approximate small change y∆ in the function Note: For small x dx∆ = we have y dy∆ ≈ . Since finding dy is easy, it is a good idea to use dy to approximately find y∆ . 33.11 See class explanation Given a function ( )y f x= . The differential ‘dy’ of y is given by ( )dy f x dx′= where dx denotes change in x See example 3 done in class See class explanation to see difference between dyand y∆ Given a function ( )y f x= . If x changes from x to x+dx then ( ) ( )y f x x f x∆ = + ∆ − (*) See example 4 done in class See example 5 done in class
- 4. Application of differentials in estimating the errors When you make measurements “are you always exact?” • Suppose we make a small error x∆ in measuring x . • This will obviously lead to an error y∆ in ( )y f x= . • As seen above, for a small change x∆ , we have y dy∆ ≈ . End of 3.11 43.11 So we can use dyto estimate error y∆ in ( )y f x= See example 6 done in class We will find following types of errors • dy: Error in y • dy y : Relative Error in y • Relative error expressed as percentage: Percentage error Which of these give better information? See class explanation