05_2016_-_2017_H2_Maths_Differentiation_Applications_Tutorial

National Junior College Mathematics Department 2016 Differentiation Applications Page 1 of 8 National Junior College 2016 – 2017 H2 Mathematics Differentiation Applications Tutorial Basic Mastery Questions Geometrical Results of the Gradient Function 1. The diagram below shows the graph of f ( )yx . Sketch, on a separate diagram, the graph of f ( )yx  . Tangents and Normals 2. The equation of a curve is 222 3 5x xy y   . Find the equation of the tangent and normal to the curve at the point (4, 3). 3. A curve has equation 22 8 4 6 4 0.x x y y xy      (i) Find an expression for d d y x in terms of x and y. (ii) Find the coordinates of the point(s) on the curve at which the tangent is parallel to the x–axis. 4. It is given that a is a positive constant. A curve has parametric equations 5 sec ,xa  3 tan ,ya  where 22   . Find the coordinates of the point on the curve at which the normal is parallel to y = x. Practical Problems involving Differentiation 5. A metal sheet of area 100 cm 2 is used to manufacture a closed cylinder can. Find, to two significant figures, the largest possible volume of the can. 6. A viscous liquid is poured onto the top of a table and a circular patch is formed that increases in area at a constant rate of 215 cm s4   . Find the rate at which the radius r is increasing at the instant when r = 20 cm. y x 2y −2 2 3 −3 −3 f ( )yx O

National Junior College Mathematics Department 2016 Differentiation Applications Page 2 of 8 Practice Questions 1. The diagram below shows the sketch of the graph of f ( )yx , where 2f ( ) . ( 2) xx xx   (i) Give the equations of the asymptotes. (ii) Using an algebraic method, find the exact range of f. (iii) Sketch the graph of f ( ),yx  giving the coordinates of stationary point(s). (iv) Deduce the range of values of x for which f ( ) 0.x  2. The diagram below shows the graph of f ( )yx  . Deduce the range of values of x where the graph of ()fyx is concave upwards. 3. The diagram shows the graph of f ( )yx . On a separate clearly labelled diagram, sketch the graph of y = f (x). [08/HCI /JC1/Promo] (–1, 0) y = x (1, 2) y = 1 2 x y O y x −2 2 f ( )yx O (–1, 0) y = x (1, 2) y = 1 2 x y O

National Junior College Mathematics Department 2016 Differentiation Applications Page 3 of 8 4. The diagram shows a sketch of the curve ex xy . (i) Find, by differentiation, the exact maximum value of y. (ii) Hence show that ln 1xx for all positive values of x. (iii) Determine the range of values of x for which the graph of ex xy is concave upwards. [08/DHS/JC2/Prelim] 5. The equation of a curve C is 33 2x xy y k   , where k is a constant. Find d d y x in terms of x and y. It is given that C has a