Ri Review The Use of Graphing Calculators Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 16 Topic: Review the Use of Graphing Calculators In this revision, we will review the use of graphing calculators for the following topics. There may be tricks that you may not be aware of. 1: Graphing – Functions 2: Graphing – Parametric Equations 3: Equations & Inequalities 4: Complex Numbers 5: Discrete Random Variables 1: Graphing – Functions Skills Required: By using the FUNCTION mode, 1. Sketch the graphs of functions and composite functions [ o, p, q]. 2. Determine the y-value for a given x-value [/ value]. 3. Determine the x-intercepts [/ zero]. 4. Determine the coordinates of the minimum & maximum points [/ minimum/maximum] 5. Determine the points of intersection of 2 graphs [ / intersect]. 6. Determine the gradient of the function [ / d d y x OR nDeriv]. 7. Determine the area bounded by the function with the x -axis [/ ʃ f(x) dx OR fnInt]. 8. Sketch piecewise functions (OS 5.3.0 and 5.3.1) [» B: piecewise] Exercise 1 (a) Functions f and g are defined as follows: 2 21f( ) 2 xxx x −−= + , for ,x∈  2x≠− , ( )g( ) 3 e ln 1 xxx x −=−− − + , for 1x>− . Sketch, on separate diagrams, showing clearly any asymptotes, intersections with the axes and coordinates of stationary points, the graphs of (i) f( )yx= , (ii) g( )yx= [Note for Teachers: May need to guide students for (ii)]. The function h is defined as h: f ( ), for , 2.xx xx ∈ >−  (iii) Find the equation of the tangent to gh( )yx= at the point where x = 0. (iv) Find the area bounded by the graph of gh( )yx= with the x-axis.

(v) Solve the inequality gh( ) 2ln( 1)xx<+ . (b) Sketch the curve 1 101 , , 1 , 0 ,( 1) xy xx x xx −= + ∈ ≠− ≠+  stating clearly equations of any asymptotes, intersections with the axes and coordinates of turning points. (c) Sketch the graph of f( )yx= where 7 12 2 sin cos 1 for 0, f( ) ( 2)1 for 0 2.4 xx x x x x π+ − − ≤< = −−− ≤≤ State clearly the coordinates of the endpoints. Answers to Exercise 1(a) (i) Use 2 1 21 2 xxY x −−= + to sketch f( )yx= . Asymptotes: y = x – 4, x = –2 Intercepts: (–0.414, 0), (2.41, 0), (0, –0.5) Stationary Points: (0.646, –0.708), (–4.65, –11.3) (ii) Use ( )2 3 e ln 1xYx x −=−− − + to sketch g( )yx= . [Note for Teachers: GC will not show the complete graph below (regardless of zoom)] Asymptote: x = –1 Intercepts: (–0.999, 0), (4.76, 0), (0, –4) Stationary Point: (–0.659, –4.52) (iii) Use 3 21 ()Y YY= to sketch gh( )yx= Use [/ value] to obtain y = –4.46 at x = 0 OR Type 3Y (0) on the home screen. Use [/ d d y x ] to obtain d d y x = –0.487 at x = 0 OR Type 3 X0 d (Y )dX = on the home screen. Equation of tangent at x = 0 is y = – 0.487x – 4.46 (iv) Use [/ zero] to obtain x = –1.3044 and x = 8.0638 when y = 0. Use [/ f ( ) dxx∫ ] to obtain area = 26.3 units2 OR Type 8.0638 31.3044 (Y ) dX−∫ on the home screen. (v) Sketch 3 21 ()Y YY= and 2 ln( 1)yx= + Use [/ intersect] to obtain intersections at x = –0.786 and x = 14.5 gh( ) 2ln( 1) –0.786 14.5xx x< +⇒ << x y 0 2 4 6 8 10 12 14 16 18 0 5 10 15

Answers to Exercise 1(b) & 1(c) 1(b) Intercepts: (0.113, 0) and (8.89, 0) 1(c) Use »→B: piecewise to sketch graph Endpoints: ( –1.83, –2.22) and (2, –1) 2: Graphing – Parametric Equations Skills Required: By using the PARAMETRIC mode, 1. Sketch the curve for the given domain [ o, p]. 2. Determine the coordinates of a point on the curve [ r, / value]. 3. Determine the gradient of the curve at a point [/ d d y x OR nDeriv]. 4. Obtain the value of an integral with parametric equations [fnInt]. Exercise 2 (a) A curve has parametric equations sin 2 sinx θθ= + , cosy θ= , where 20 3 πθ≤≤ . (i) Find the equation of the tangent to the curve at the point where 2 πθ = . (ii) Sketch the curve, giving the coordinates of any intercepts. (b) A curve has parametric equations 2cosxt= , 3sinyt= , for 0. 2t π≤≤ (i) Sketch the curve, giving the coordinates of any intercepts. (ii) Find the area bounded by the curve and the axes. Answers to Exercise 2(a) (i) Use 1TX and 1TY to sketch the curve. Use [/ value] to obtain 1x= and 0y = at 2 πθ = . Use [/ d d y x ] to obtain d1 d2 y x = at 2 πθ = . [OR Type 1T 2X ()π , 1T 2Y() π , etc.] Equation of tangent at 2 πθ = is 1 ( 1)2yx= − . (ii) Intercepts: (1, 0), (0, 1), (0, –0.5) Answers to Exercise 2(b) O y x x = –1 x = 0 y=1 (−0.232,−17.6) (0.432, – 4.37)

(i) Use 2TX and 2TY to sketch the curve. Intercepts: (0, 1) and (1, 0) (ii) Type 0 2T 2T 2 TT d(Y (X ) ) dTdTπ = ×∫ or 2 2T 2T0 TT d(X (Y ) ) dTdT π = ×∫ on the home screen to get required area = 0.4 units2. 3: Equations & Inequalities Skills Required: 1. Solve equations & inequalities. • By using Œ→ PlySmlt2→ SIMULTANEOUS EQN SOLVER, 2. Solve system of linear equations. Exercise 3 (a) Find the smallest positive integer n such that ( ) ( )10000 1 0.9 25 3n nn−< + . (b) Find, for 22 x−≤ ≤ , the set of values of x such that 3 2e sin 0.053 x xx xx−++ < . (c) Use your calculator to find a vector equation of the line of intersection between ( )1 : 32 4ri j kπ ⋅++ = and ( )2 :4 rijkπ ⋅ −− = . (d) A finite arithmetic progression 123 2, , ,..., nuuu u has first term a and common difference d. It is given that 1 4027 4025 n n u u + = , 2 36228nnuu+= and the average of all the odd-numbered terms is 12 075. By formulating three equations involving a, d and nd, find the values of a, d and n. Answers to Exercise 3 (a) ( )1 10000 1 0.9 nY = − and ( )2 25 3Y nn= + Least n = 17 (b) 3 2 1 e sin 3 x xY x xx = −++  and 2 0.05Y = The set of values of x is ( 1.17,1.02)− . (c) 1 2 : 324 :4 xyz xyz π π ++= −−= An equation of the intersecting line is (d)

41 03 04 r t      = +−       , t∈  . 2 4027 2 0 2 2 3 36228 + 12075 a d nd a d nd a d nd − += − += −= From the GC, a = 3, d = 6 and nd = 12078 2013n⇒= 4: Complex Numbers Skills Required: 1. Perform operations on complex numbers. • By using Œ→ PlySmlt2→ POLYNOMIAL ROOT FINDER, 2. Find the roots of a polynomial equation with real coefficients. • By using »→ CMPLX, 3. Find the conjugate, modulus and argument of a complex number. 4. Convert a complex number from the Cartesian form to the Polar form and vice versa. Exercise 4 (a) Evaluate the following: (i) ( ) ( ) 3 5 3 4i 1i + − , (ii) 5 12i− . [(i) 161 73 i88+ ; (ii) 3 2i− ] (b) Find the roots of the following equations: (i) 2 ( 1 4i) ( 5 i) 0zz+−+ +−+ = , (ii) 43 2 8 16 0zzz− +−= . (c) Find the conjugate, modulus and argument of (i) (3 i)(i 3)−− , (ii) (5 i) (3 2i) − + . [(i) 8 6i , 10 , 2.50−− ; (ii) 1 i , 2 , 4 π+− ] (d) Express the following complex numbers in cartesian form. (i) i42e π , (ii) i32e π . [(i) 1i+ ; (ii) 1 3i+ ] (e) Express the following complex numbers in polar form. ( i) (2 2i) (i 1) + − , (ii) 13 i (i 3)22 −− −  . [(i) i22e π− ; (ii) i62e π ] Answers to Exercise 4(b) (a)(i) Note: GCs with older OS cannot perform division in Math Print, i.e. fraction. (b)(i) [Use Quadratic Formula & GC] 2 3i or 1 iz=− −− (b)(ii) [Use APPS] 2, 1 3iz= ±± 5: Discrete Random Variables Skills Required: Use of …, summation, 0.

Exercise 5 (a) Find the mean and standard deviation of the random variable X with the following probability distribution: X 4− 3− 2− 1− 0 1 2 3 ( )P Xx= 0.0 4 0.1 6 0.2 4 0.