Ceramah Add Math (4 of 4)

Slide 1: 

Additional Mathematics Part 4 / 4

9. DIFFERENTIATION : : 

9. DIFFERENTIATION : Given that , find F4

Differentiation : The second derivative: 

Differentiation : The second derivative Given that f(x) = x 3 + x 2 – 4x + 5 , find the value of f ” (1) ‏ f’ (x) = 3x 2 + 2x – 4 f” (x) = 6x + 2 f” ( 1 ) = 8 F4

Differentiation : The second derivative: 

Differentiation : The second derivative Given that , find the value of g ” (1) . g’ (x) = 10x (x 2 + 1) 4 F4 g’’ (x) = 40x (x 2 + 1) 3 . 2x Ya ke ??

Slide 5: 

g’ (x) = 10x (x 2 + 1) 4 F4- 9 Given that , find the value of g ” (-1) . g’’ (x) = 10x . 4(x 2 + 1) 3 .2x +(x 2 +1) 4 . 10 g’’ ( -1 ) = 10( -1 ) . 4[( -1 ) 2 + 1] 3 +[( -1 ) 2 +1) 4 . 10 = 800 Mid-year, Paper 2

Differentiation : Small increments: 

Given that y = 2x 3 – x 2 + 4, find the value of at the point (2, 16). Hence, find the small increment in x which causes y to increase from 16 to 16.05. Differentiation : Small increments K1 K1 N1 = 6x 2 – 2x = 20 , x = 2 F4

10. Progressions : A.P & G.P : 

10. Progressions : A.P & G.P A.P. : a, a+ d , a+2 d , a+3 d , …….. Most important is “ d ” F5 G.P. : a, a r , a r 2 , a r 3 , …….. Most important is “ r ” !!

Progressions : G.P - Recurring Decimals : 

Progressions : G.P - Recurring Decimals SPM 2004, P1, Q12 Express the recurring decimal 0.969696 … as a fraction in the simplest form. x = 0. 96 96 96 … (1) ‏ 100x = 96. 96 96 ….. (2) ‏ (2) – (1) 99x = 96 x = = F5

Back to basic… …: 

Usual Answer : S 10 – S 5 = ……. ??? Correct Answer : S 10 – S 4 Progressions Given that S n = 5n – n 2 , find the sum from the 5 th to the 10 th terms of the progression. Back to basic… … Ans :-54 F5

11. Linear Law: 

Y X 1. Table for data X and Y 2. Correct axes and scale used 3. Plot all points correctly 4. Line of best fit 5. Use of Y-intercept to determine value of constant 6. Use of gradient to determine another constant 1 1-2 1 1 2-4 11. Linear Law F5

Linear Law: 

Y X Bear in mind that …...... 1. Scale must be uniform 2. Scale of both axes may defer : FOLLOW given instructions ! 3. Horizontal axis should start from 0 ! 4. Plot ……… against ………. Linear Law Vertical Axis Horizontal Axis F5

Linear law: 

0 2 4 6 8 10 12 x 0.5 1.0 1.5 2.5 2.5 3.0 3.5 4.5 Y x x x x x x Linear law F5 Read this value !!!!!

12. INTEGRATION : 

= = = = 12. INTEGRATION F5

INTEGRATION: 

INTEGRATION SPM 2003, P2, Q3(a) 3 marks Given that = 2 x + 2 and y = 6 when x = – 1 , find y in terms of x. Answer: = 2x + 2 y = = x 2 + 2x + c x = -1, y = 6: 6 = 1 + 2 + c c = 3 Hence y = x 2 + 2x + 3 F5

INTEGRATION: 

INTEGRATION SPM 2004, K2, S3(a) 3 marks The gradient function of a curve which passes through A(1, -12) is 3x 2 – 6 . Find the equation of the curve. Answer: = 3x 2 – 6 y = = x 3 – 6x + c x = 1, y = – 12 : – 12 = 1 – 6 + c c = – 7 Hence y = x 3 – 6 x – 7 Gradient Function F5

Vectors : Unit Vectors: 

A B Given that OA = 2 i + j and OB = 6 i + 4 j , find the unit vector in the direction of AB AB = OB - OA = ( 6 i + 4 j ) – ( 2 i + j ) = 4 i + 3 j l AB l = = 5 Unit vector in the direction of AB = Vectors : Unit Vectors K1 N1 K1 F5

Parallel vectors: 

Parallel vectors Given that a and b are parallel vectors, with a = (m-4) i +2 j and b = -2 i + m j . Find the the value of m. a = k b (m-4) i + 2 j = k (-2i + mj) m- 4 = -2 k m k = 2 1 2 a = b m = 2 K1 N1 K1 F5

Prove that tan2 x – sin2 x = tan2 x sin2 x: 

Prove that tan 2 x – sin 2 x = tan 2 x sin 2 x sin 2 x tan 2 x – sin 2 x = K1 N1 K1 5 TRIGONOMETRIC FUNCTIONS F5

Slide 19: 

Solve the equation 2 cos 2x + 3 sin x - 2 = 0 sin x ( -4 sin x + 3 ) = 0 sin x = 0 , 2( 1 - 2sin 2 x ) + 3 sin x - 2 = 0 -4 sin 2 x + 3 sin x = 0 sin x = x = 0 0 , 180 0 , 360 0 x = 48.59 0 , 131.41 0 K1 N1 K1 N1 5 TRIGONOMETRIC FUNCTIONS F5

Slide 20: 

5 TRIGONOMETRIC FUNCTIONS (Graphs) ‏ (Usually Paper 2, Question 4 or 5) - WAJIB ! F5 1. Sketch given graph : (4 marks) ‏ (2003) y = 2 cos x , (2004) y = cos 2x for (2005) y = cos 2x , (2006) y = – 2 cos x ,

Slide 21: 

Find the number of four digit numbers exceeding 3000 which can be formed from the numbers 2, 3, 6, 8, 9 if each number is allowed to be used once only. No. of ways = 4 . 4. 3. 2 = 96 3, 6, 8, 9 F5 PERMUTATIONS AND COMBINATIONS

Slide 22: 

Vowels : E, A, I C onsonants : B, S, T, R Arrangements : C V C V C V C No. of ways = 4 ! 3 ! = 144 Find the number of ways the word BESTARI can be arranged so that the vowels and consonants alternate with each other [ 3 marks ] F5

Slide 23: 

Two unbiased dice are tossed. Find the probability that the sum of the two numbers obtained is more than 4. Dice B, y 4 1 5 6 2 3 Dice A, x 2 3 4 5 1 6 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X n(S) = 6 x 6 = 36 Constraint : x + y > 4 Draw the line x + y = 4 We need : x + y > 4 P( x + y > 4) = 1 – = F5

The Binomial Distribution: 

The Binomial Distribution r = 0, 1, 2, 3, …..n n = Total number of trials q = probability of ‘failure’ p = Probability of ‘success’ r = No. of ‘successes’ p + q = 1 F5 PROBABILITY DISTRIBUTIONS Mean = np Variance = npq

The NORMAL Distribution: 

The NORMAL Distribution F5 PROBABILITY DISTRIBUTIONS Candidates must be able to … determine the Z -score Z = use the SNDT to find the values (probabilities) z f ( z ) 0 0.5 0

Slide 26: 

z f(z) 0 1.5 z z f(z) 0 -1.5 1 = – 1 f(z) 0 1 – T5

Index Numbers: 

Index Numbers Index Number = Composite Index = Problems of index numbers involving two or more basic years. F4

Solution of Triangles: 

Solution of Triangles The Sine Rule The Cosine Rule Area of Triangles Problems in 3-Dimensions. Ambiguity cases (More than ONE answer)‏

Motion in a Straight Line: 

Motion in a Straight Line Initial displacement, velocity, acceleration... Particle returns to starting point O... Particle has maximum / minimum velocity.. Particle achieves maximum displacement... Particle returns to O / changes direction... Particle moves with constant velocity ...

Motion in a Straight Line: 

Motion in a Straight Line Question involving motion of TWO particles. ... When both of them collide / meet ??? … how do we khow both particles are of the same direction at time t ??? The distance travelled in the nth second. The range of time at which the particle returns …. The range of time when the particle moves with negative displacement Speed which is increasing Negative velocity Deceleration / retardation

Linear Programming: 

Linear Programming To answer this question, CANDIDATES must be able to ..... form inequalities from given mathematical information draw the related straight lines using suitable scales on both axes recognise and shade the region representing the inequalities solve maximising or minimising problems from the objective function (minimum cost, maximum profit ....) ‏

Linear Programming: 

Linear Programming y ≤ 2x 12. The ratio of the quantity of Q ( y ) to the quantity of P ( x ) should not exceed 2 : 1 x ≥ y + 10 11. x must exceed y by at least 10 y - 2x >10 13. The number of units of model B ( y ) exceeds twice the number of units of model A ( x ) by 10 or more. x + y > 40 10. The sum of x and y must exceed 40 x + y ≥ 50 9. The sum of x and y is not less than 50 3x - 2y ≥ 18 8. The minimum value of 3x – 2y is 18 x + 2y ≤ 60 7. The maximum value of x+ 2y is 60 y ≥ 35 6. The minimum value of y is 35 x ≤ 100 5. The maximum value of x is 100 y ≥ 2x 4. The value of y is at least twice the value of x x ≤ y 3. x is not more than y x ≤ 80 2. x is not more than 80 x ≥ 10 1. x is at least 10 Ketaksamaan Maklumat

Selamat maju jaya !: 

Selamat maju jaya !