RI Applications of Differerentiation Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 8 (Summary and Tutorial) Topic: Applications of Differentiation Summary for Applications of Differentiation 1 Strictly Increasing/Strictly Decreasing & Concavity IF Then f is over the interval (,)ab f( ) 0x  Strictly increasing f( ) 0x  Strictly decreasing f() 0x  Concave up f() 0x  Concave down To show “ 0 ” or “ 0 ”, we can complete the square of the expression if it is quadratic otherwise we can try to “build up the expression” from the given interval. For example to show  1f2 0x x   for 1 2x . We can rewrite the expression:  21f xx x   So the strategy here is to try to examine the signs the numerator and denominator from the given condition of 1 2x . 2 Methods for determining the nature of a stationary point Method 1: Second Derivative Test IF at xk Then  , f( )kk is f() 0x  a minimum turning point f() 0x  a maximum turning point Note that if f() 0x  then there is no conclusion about the nature of the stationary point. The first derivative test must then be used. If 1 2x , then the denominator must be “ 0 ” If 1 2x , then 21x which implies that 21 0x Therefore if the numerator is “ 0 ” and the denominator is also “ 0 ”, we can conclude that for 1 2x , f0 x 

Method 2: First Derivative Test To determine the nature of the stationary point at  , f( )kk , we need to check the signs of f( )x for x k and x k . Important note: you need to fully factorise f( )x where possible BEFORE discussing the signs of f( )x . Example: 53f( ) 2 5x xx . It is not sufficient to just differentiate and get 42f( ) 1 0 1 5x xx  . You need to factorise further to get 2 33f( ) 1 0 22xx x x        , BEFORE discussing stationary points at (0,0) , 33,322   and 33,322  making reference to the table below. x kk k kk k kk k f( )x ve 0 ve ve 0 ve ve 0 ve or ve 0 ve Nature of Stationary Point Maximum Minimum Stationary point of inflexion 3 TANGENTS AND NORMALS To find the equations of the tangent or normal at a point  , f( )kk on a curve, we need: Equation of a (straight) line f( ) ( )yk m x k   with gradient m passing through  , f( )kk .  Obtain m by finding f( )k which will the gradient of the tangent at  , f( )kk .  If the question needs equation of the normal at  , f( )kk , gradient of the normal at  , f( )kk is 1 f( )k  . When gradient at a point on a curve f( )x is parallel to the x-axis f( ) 0x  (Usually mean that the numerator of f( ) 0x  ) y-axis f( )x is undefined. (Usually mean that the denominator of f( ) 0x  )

4 MAXIMIZATION AND MINIMIZATION Some guidelines to solve problems involving maximization and minimization :  Denote each changing quantity by a variable.  Write d