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 tangent which is parallel to the y–axis. Show that the y-coordinate of the point of contact of the tangent with C must satisfy 63216 4 0y y k   . Hence show that k  1 54 . Find the possible values of k in the case where the line x = –6 is a tangent to C. 6. A curve is defined by the parametric equations, x = t2, y = t3. Prove that the equation of the tangent at the point with parameter t is 32 3 0y tx t   . (i) This tangent passes through a fixed point (X, Y). Give a brief argument to show that there cannot be more than 3 tangents passing through (X, Y). (ii) The tangent at the point where t = 2 meets the curve again at the point where t = u. Find the value of u. 7. A curve is given parametrically by the equations 12 1, 21x t y t    , where t . (i) Find x y d d in terms of t. Deduce that the curve shows a decreasing function. (ii) Show that there is no tangent to the curve parallel to the chord joining the points where 1t and 5 8t  . (iii) The curve passes through the point Q when 0.5t . Find the coordinates of the point at which the normal to the curve at Q meets the curve again. (iv) The origin is denoted by the point O. The normal to the curve at Q intersects the x-axis at the point P. Find the exact area of the triangle OPQ.

National Junior College Mathematics Department 2016 Differentiation Applications Page 4 of 8 8. A metal tank with square base is expanding due to heating. After t seconds, the tank has dimensions x cm by x cm by 10 x cm. Given that the area of the horizontal cross - section is increasing at 0.032 cm2 s1 when x = 8, find, at this instant, (i) the rate of increase of the side of the cross-section, and (ii) the rate of increase of the volume. [2006/SRJC/P1] 9. The diagram below shows an isosceles triangle ABC with fixed lengths AB and AC of 10 cm each. A is a variable point which is at a height h cm directly above the point O while B and C are variable points which move horizontally along the line l. Given that A descends vertically towards the point O such that the area of triangle ABC is decreasing at a constant rate of 0.7 cm 2/s, determine at the instant when A is 6 cm above O, (i) the rate of change of  , (ii) the rate at which C is moving away from O. 10. A piece of wire of length 8 cm is c ut into 2 pieces, one of length x cm, the other (8 ) x cm. The piece of length x cm is bent to form a circle with circumference x cm. The other piece is bent to form a square with perimeter (8 ) x cm. Show that, a s x varies, the sum of the areas enclosed by these 2 pieces of wire is a minimum when the radius of the circle is 4 4  cm. 11. The perimeter of an isosceles triangle of base length x cm and two sides of equal length y cm each is 12 cm. Show that its area can be expressed as 36 62 x x . Hence show that the area is maximum when the triangle is equilateral. [2008/SAJC/Promo] A B C l h 10 cm 10 cm O 

National Junior College Mathematics Department 2016 Differentiation Applications Page 5 of 8 12. An art sculptor has conceptualised a glass sculpture design as shown below. The sculpture consists of a right conical shell of height 3 metres and base radius r metres. Inside the cone holds an inverted solid glass cone of height h metres and base radius x metres (see above diagram). Let V denotes the volume of the inverted cone. (i) Show that 3 2212 3 3 9 hV r h h     . (ii) The sculptor decides to cast the inverted cone in gold to increase the value of his sculpture. Find the height h of the inverted cone that would maximise the value of the sculpture, justifying your answer. 13. A curve C has parametric equations 3sinx  , 23sin cosy  , π0 2 . (i) Show that d 2cot tand y x  . (ii) Show that C has a turning point when tan k  , where k is an inte ger to be determined. Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum. The line with equation y ax , where a is a positive constant, meets C at the origin and at the point P. (iii) Show that 3tan a  at P. Find the exact value of a such that the line passes through the maximum point of C. [N2015/I/11] h 3 u n r x

National Junior College Mathematics Department 2016 Differentiation Applications Page 6 of 8 Further Practice Questions 1. A curve C, given by 44yxy e x , cuts the x-axis at point P. The line L is the tangent to C at P. Show that the equation of L is 5 4 4yx . (a) Find the exact coordinates of the point R on L such that the distance OR is the shortest, where O is the origin. (b) A point Q moves along L with its x-coordinate decreasing at a rate of 4 units/s. Find the rate of change of the area of triangle OPQ. [2012/AJC/Promo] 2