TMJC H2 Chapter 8 Applications of Differentiation Learning Package 2024

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Chapter 8 Applications of Differentiation TMJC 2024 Page 1 of 18 . Normal at A Tangent at A ( )fyx= x y 00( , )A x y R S P Q H2 Mathematics (9758) Chapter 8 Applications of Differentiation Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Differentiate an equation of a curve defined implicitly to find gradient of tangents to a curve  Differentiate parametric equations to find gradient of tangents to a curve  Using relationship 1 gradient of tangent − to find gradient of normal to a curve  Find equations of tangents and normal to a curve  Use Chain Rule d d d d d d y y x t x t  = to link up the derivatives and rate given  Solve local maxima and minima problems  Solve connected rates of change problems §1 Tangents and Normals Given A ( )00,xy is a point on a curve, then a) the line PQ touching the curve at A is called the tangent to the curve at A and b) the line RS perpendicular to the tangent at A is called the normal to the curve at A. Recall: Gradient of curve at A = Gradient of the tangent at A 0 d d xx y x = =

Chapter 8 Applications of Differentiation TMJC 2024 Page 2 of 18 The gradient of the tangent and normal at 00( , )xy are respectively given by: • Gradient of tangent, 0 d d xx ym x = = • Gradient of normal 1 m=− Recall: The product of gradients of perpendicular lines is −1 The equation of the tangent and normal at any point ( )00,xy on a curve f ( )yx= is given by: Equation of tangent ( )00y y m x x− = − where 0 d d xx ym x = = Equation of normal ( )00 1y y x x m− = − − Example 1 The equation of a curve is 2 51y x x= + − . Find the equations of the (i) tangent and (ii) normal at the point (2, 13). Solution: (i) Non-GC approach: d 25d y xx =+ 2 d 9d x y x = = Equation of tangent: ( )13 9 2yx− = − 95yx=− GC approach to get equation of tangent directly: Step 1 Press !. Enter the equation of the graph 2 51y x x= + − under 1y Press % to see the graph Step 2 To draw/ obtain the tangent press `p and select 5: Tangent( Check using GC Gradient of Tangent = d d y x

Chapter 8 Applications of Differentiation TMJC 2024 Page 3 of 18 Using GC, Equation of tangent: 95yx=− Step 3 At the graph screen, key in the x-coordinate of the point required e.g. 2 and press e