Lesson Note Further Mathematics SS2 Third Term

Further Mathematics Lessons SS2 – Edudelight.com

THIRD TERM

SUBJECT: FURTHER MATHEMATICS      

CLASS: SS 2

SCHEME OF WORK

WEEK	TOPIC
1.	Revision of Second Term Examination Questions.
2.	Projectile: Trajectory of Projectile, Greatest Height Reached, Time of Flight, Range, and Projectile along Inclined Plane.
3.	Binomial Expansion: Pascal  Triangle Binomial Theorem of Negative, Positive and Fractional Power
4.	Mechanics: Vectors in Two and Three Dimension. Scalar Product of Vectors in Three Dimension.
5	Vector or Cross Product on Three Dimension.Application of Cross Product Cross Product of Two Vectors.
6	Review of the Half Term Work.
7.	Integration: Indefinite Integrals Concept, Different Methods of Integration.e.g (Algebraic and Trigonometric Substitution by Parts and Partial Fractions.
8.	Integration Continued: Definite Integral, Area Under Curve.
9.	Integration Continued: Application of Integration to Kinematics Volumes of Solids at Revolution and Trapezium Rule.
10.	Correlation and Regression: Concept, Scatter Diagram, Regression Line, Coefficient of Regression, Rank Correlation and Product Moment Correlation Coefficient.
11.	Revision.

REFERENCES

Further Mathematics Project 2 and 3.

WEEK ONE

REVISION OF SECOND TERM EXAMINATION QUESTIONS.

WEEK TWO

TOPIC:PROJECTILES: MOTION UNDER GRAVITY IN TWO DIMENSION,DERIVATION AND APPLICATION OF EQUATIONS INVOLVING GREATEST HEIGHT, TIME OF FLIGHT AND RANGE                                                                                       

Motion Under Gravity in Two Dimensions:

If a particle is projected with an initial velocity u at angle  to the horizontal, the prativle will be resolved into vertical and horizontal components of the velocity.

Horizontal components: Vx = ucos

Horizontal distance: Sx = utcos

Vertical components: Vy = usin

Vertical distance: sy = utsin – ½ gt²

Magnitude of the velocity, v = v_x ² + v_y²

The acceleration due to gravity acts against the motion of the body, hence it is negative.

Example:

1. A particle is projected with an initial velocity of 46m/s at an angle of 55⁰ to the horizontal. After 3 seconds, find: (i) the vertical component of the velocity (ii) horizontal component (iii) vertical distance traveled. (iv) Magnitude of the velocity.

SOLUTION:

 = 55⁰ u=46m/s

 (i) Vy = usin– gt

              = 46 sin 55 – 10 x 3

             = 37.68 – 30

             = 7.68/s   

(ii) Vx = ucos

            = 46 cos 55

            =26.38m/s

(iii) sy = utsin –  (10×9)

           = 138sin55 – 5×9

           = 113.04 – 45

             = 68.04m

(iv) = vx² +vy²

7.68² + 26.38²      = 58.98 + 695.9

        = 27.48m/s

EVALUATION: A particles is fired with an initial speed of 40m/s at an angle of 30⁰ to the horizontal. Determine the vertical and horizontal components of the velocity after 2.5 seconds.

GREATEST HEIGHT REACHED:  when a projected particle reaches its greatest height, the vertical components become zero. Therefore;

RecallVy, = usin – gt

Squaring both sides, (Vy) ² = (usinn – gt)²

Vy^{2 =} u²sin² – 2gs_y

Since: vy =0, hence, 0 = u²sin² – 2gs_y

S_y =u²sin²

  2g

Therefore the greatest height reaches is represented by    H=u²sin²

2g

Time taken to reach the greatest height: The time taken to reach the maximum height I at the point when the vertical component is zero. Hence,

                                ,Vy = usin–gt

                                 0 = usin–gt

                                T=usin

                                g

Example:

1. A particle is projected with velocity 56m/s at an angle of 60⁰ from a point O on a horizontal plane. The particle moves freely under gravity and hits the plain again A. Calculate, correct to 3 significant figures: (a) the greatest height above OA attainedby the particle (b) the time taken by the particle to reach A from O.

Solution:

   U = 56m/s        = 60⁰

• Greatest height reached, h = U²sin²O

                                               2g

                                   h = 56² x (sin 60)²

                                           2 x 9.8

                                   h = 2352                      h = 120m.

                                          19.6

(a) Time taken to reach A from O;        t = usin

                                                           g

                                               t = 56 sin 60

                                                        9.8

                                               T = 4.9secs.

Evaluation: A project is fired with a velocity of 45m/s and at angle of elevation of 81⁰ to the horizontal. Find the time taken by the particle to reach its destination. (Take g = 10m/s²)

Time of flight: This is the time taken by a particle which is projected to return to its original point of projection. At this point the vertical distance becomes zero. Hemce,

   T = 2usin

g

Range:  This is the horizontal distance covered when the particle returns to its original point of projection. The range is equal to the product of the horizontal component and the time of flight.    

Hence,

                              R =ucosx2usin

                                                      g

                              R =u² x 2sincos                 (but; 2sincos = sin 2)

g

R = u² x 2sin

                              g

Maximum range:  A particle will cover a maximum range if it is projected at angle 45⁰ to the horizontal. That is;  = 45⁰. Thus sin2 =1

Hence, R_max= U²

                  g

Example: The vertical and horizontal components of the initial velocity of a projectile are 36m/s and 64m/s. find (i) initial velocity of the projectile (ii) the inclination to the horizontal at which the projectile was fired. (iii) the greatest height reached; (iv) the time of flight; (v) the horizontal range of the projectile.

Solution:

      V_y = 36m/s   V_x= 64m/s

• VV_x²  +V_y²

 U = 64^{2 +} 36²;   U = 73.43m/s

• Inclination to the horizontal; ( the angle of projection)

Vx = u cos

64 = 73.43 cos

 = cos^-1 (64/73.43);  = 29.4⁰

• Greatest height reached; h = U² sin

                                              2g

                     h = 73.43² x (sin 29.4)²

                  2 x10

                              h = 5391.96 x 0.2410

                                          20

                              h = 64.97m

• Time of flight: T = 2usin   

g

      T = 2x 73.43 x sin 29.4

10

                            T = 7.2 secs.

• Horizontal range:  R = u² sin2

g

                              R = 73.43² x sin (2×29.4)

      10

                              R = 461.2m

EVALUATION:A  particle is projected into the air with a speed of 50m/s at an inclination sin^-1(3/5). Find the: (greatest height reached by the particles; (ii) horizontal range; (iii) time of flight

Reading Assignment

New Further Maths Project 2 page 262 -270.

GENERAL EVALUATION

1) A particle is projected with an initial speed of 45m/s at an angle of 35 to the horizontal, find the time it takes for the particle to (i) reach the highest level (ii) return to its original level

2) A particle is projected horizontally with a velocity of 40m/s from the top of a tower 80.5m above the level ground  find how far from the bottom of the tower the particle when it hits the ground

3) A particle is projected into the air with a speed of 20m/s at an inclination 30 to the horizontal , find the (i) greatest height reached  (ii) horizontal range  (iii) time of flight

4) Show that a particle which is projected with a given velocity reaches its maximum range at an elevation of sin^-1 (2^1/2 /2)

WEEKEND ASSIGNMENT

The vertical and horizontal components of the initial velocity of a projectile are 36m/s and 64m/s respectively find the

1) greatest height reached  a) 32.4m  b) 97.2m  c) 64.8m  d) 16.2m

2) time of flight a) 7.2s   b) 3.6s  c) 1.8s  d) 14.4s

3) horizontal range  a) 23.04m  b) 46.08m  c) 11.5m  d) 92.16m

4) initial velocity of the projectile  a) 73.4m/s  b) 146.8m/s  c) 36.7m/s  d)18.4m/s

5) inclination to the horizontal  a) 19  b) 21  c) 29  d) 49

THEORY

1) Find the initial speed which a projectile must be subjected to give a maximum horizontal range of 490m

2) Prove that the maximum range on a horizontal plane of a particle fired with velocity V at an angle x to the horizontal is V² / g

WEEK THREE

TOPIC:BINOMIAL EXPANSION: PASCAL TRIANGLE, BINOMIAL THEOREM OF NEGATIVE, POSITVE AND FRACTIONAL POWER

PASCAL’S TRIANGLE

Consider the expressions of each of the following:

(x + y)⁰;  (x + y )¹; (x + y)²; (x + y)³; (x + y)⁴

(x + y)⁰ = 1

(x + y)¹ = 1x + 1y

(x + y)² = 1x² + 2xy + 1y²

(x + y)³= 1x³ + 3x²y + 3xy² + 1y³

(x + y)⁴ = 1x⁴ + 4x³y + 6x²y² + 4xy³ + 1x⁴

The coefficient of x and y can be displayed in an array as:

                                                1

                                    1                      1

            1                      2                      1

            1                      3                      3                      1

1                      4                      6                      4                      1

The array of coefficients displayed above is called Pascal’s triangle, and it is used in determining the co-efficients of the terms of the powers of a binomial expression

Coefficient of (x + y)⁰                                                             1

Coefficient of (x + y)¹                                                 1                      1

Coefficients of (x + y)²                                                1                      2                      1

Coefficients of (x + y)³                                    1                      3                      3                      1

Coefficients of (x + y)⁴                        1                      4                      6                      4                      1

Example 1

Using Pascal’s riangle, expand and simplify completely: (2x + 3y)⁴

Solution:

(2x + 3y)⁴ = (2x)⁴ + 4(2x)³ (3y) + 6(2x)²(3y)² + 4(2x)(3y)³ + (3y)⁴

= 16x⁴ + 96x³y + 216x²y² + 216xy³ + 81y⁴

Examples 2:

Using pascal’s triangle, the coefficients of (x + y)⁵are: 1,5,10,10,5,1.

Therefore (x – 2y)⁵      = x⁵ + 5x⁴(-2y) + 10x³(-2y)² + 10x²(-2y)³ + 5x(-2y)⁴ + (-2y)⁵

                                                = x⁵ – 10x⁴y + 40x³y² – 80x²y³ + 80xy⁴ – 32y⁵

Example 3

Using Pascal’s triangle, simplify, correct to 5 decimal places (1.01)4

Solution

We can write (1.01)⁴ = (1 + 0.01)⁴

(1 + 0.01)⁴ = 1 + 4(0.01) + 6(0.01)² + 4(0.01)³+(0.01)⁴

= 1 + 0.04 + 0.0006 + 0.000004 + 0.00000001

= 1.04060401

= 1.04060 (5 d.p)

The Binomial Expansion Formula

Consider the expansion of (x + y)⁵ again

(x + y)⁵ = (x + y)(x + y)(x + y)(x + y)(x + y)

The first term is obtained by multiplying the xs in the five brackets. there is only one way to doing this

(x + y)ⁿ = xn + nx^{n – 1}y +    xⁿ – ²y² +     x^n-3y³ + ….

x^n-ry^r + …. yⁿ

It can be shown that the binomial expansion formula holds for positive, negative, integral or any rational value of n, provided there is a restriction on the values of x and y in the expansion of (x + y)ⁿ

We shall however consider only the binomial expansion formula for a positive integral n

Example 4:

1. Write down the binomial expansion of  ⁶ simplifying all the terms
2. Use the expansion in (a) to evaluate (1.0025)⁶ correct to five significant figures.

Solution

⁶  = 1 + ⁶C₁ ¹    +   ⁶C₂²

    +   ⁶C₃  ³    +    ⁶C₄ 

    +   ⁶C₅  ⁵    +    ⁶C₆  ⁶

⁶=  1 +   x   +  ³   +   x

⁴    +    x  ⁵   +  ⁶

= 1 + x   +   x²+   x³  +x⁴  +   x⁵ +   x⁶

• (1.0025)⁶   =  (1 + 0.0025)⁶

=  )⁶

=  )⁶

Put x  =

x =   x 4   = =  0.01

therefore   (1.0025)⁶   =  1  +    (0.01)  +   (0.01)²  + (0.01)  +(0.01)⁴ + …

= 1 + 0.015 + 0.00009375 + 0.0000003125

= 1.0150940625

= 1.0151 (5 s.f.)

EVALUATION

Expand ( 2 + 4x )⁴ simplifying the terms

Example 5
(a) Using the binomial theorem, obtain the expansion of (1 + 3x)⁶ + (1 – 3x)⁶  simplifying all the terms

(b)Use the above result to calculate the value of (1.03)⁶ + (0.97)⁶, correct to five decimal places

Solution:

(1 + 3x)⁶ = 1 + ⁶C₁ (3x) + ⁶C₂ (3x)² + ⁶C₃ (3x)³ _ ⁶C₄ (3x)⁴ _ ⁶C₅ (3x)⁵ + ⁶C₆ (3x)⁶          ….. (1)

(1 – 3x)⁶ = 1 – ⁶C₁ (3x) + ⁶C₂ (3x)² – ⁶C₃ (3x)³ + ⁶C₄ (3x)⁴ _ ⁶C₅ (3x)⁵ + ⁶C₆ (3x)⁶          ….. (2)

Adding (1) and (2)

(1 + 3x)⁶ +(1 – 3x)⁶  = 2 + 2 x  ⁶C₂ (3x)²  + 2 x ⁶C₄ (3x)⁴  + 2 x ⁶C₆ (3x)⁶

= 2 + 2 x   9x² + 2 x   x 81x⁴  + 2  x 729x⁶

= 2 + 270x² + 2430x⁴ + 1458x⁶

(1.03)⁶    =   (1 + 0.03)⁶

(0.97)⁶    =   (1 – 0.03)⁶

Put 1 + 0.03 = 1 + 3x

Therefore    3x   =   0.03

Therefore     x    =   0.01

Hence

(1.03)⁶ + (0.97)⁶   = 2 + 270(0.01)² + 2430(0.01)⁴ + 1458(0.01)

= 2 + 0.027 + 0.0000243 + 2.0270243

= 2.02702    (5 d.p)

Example 6

• Using the binomial theorem, expamd (1 + 2x)⁵, simplifying all the terms
• Use your expansion to calculate the value of 1.02⁵, correct to six significant figures

If the first three terms of the expansion of (1 + px)ⁿ in ascending powers of x are 1 + 20x + 160x,

Find the values of n and p

Solution:

• (1 + 2x)⁵ = 1 . ⁵C₁(2x) + ⁵C₂(2x)² + ⁵C₃(2x)² + ⁵C₄(2x)⁴ + ⁵C₅(2x)⁵

= 1 + 5.(2x) +   . 4x² +   . 8x³  +  . 16x   + 32x⁵

= 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵

• (1.02) = (1 + 0.02)

Put 1 + 0.02 = 1 + 2x

Therefore 2x = 0.02

x = 0.01

Hence:

(1.02)⁵ = 1 + 10(0.01) + 40(0.01)² + 80(0.01)³ + 80(0.01)⁴ + 32(0.01)⁵

= 1 + 0.1 + 0.004 + 0.0008 + 0.00000008

= 1.10408 (6.s.f.)

6.3 The Binomial Theorem for any index

The Binomial expansion formula is also applicable to any index n, where n can be positive or negative integer or even a fraction

If /x/  1, then:

(1 + x)ⁿ = 1 + nx +   +   +  x⁴  + … where n may be a negative integer or a fraction.

Example 7

Use the Binomial expansion formula to obtain the first five terms of the expansion of  (1 +   x)^-2

Solution:

(1 +   x)^-2  = 1 + (-2) ( x) + ( ( x)² +  ( x)³ +  ( x)⁴ +  ….

= 1 – x + 3. ²  – 4.³  + 5.⁴

• (1 + px)ⁿ = 1 + 20x + 160x² + …

(1 + px)ⁿ = 1 + ⁿc₁ (px) + ⁿc₁ (px)² + …

            = 1 + npx   +   p²x² …

            = 1 + 20x + 160x² + …

By equating coefficients

np = 20                                                …        (1)

p²  = 160                                   …        (2)

From (1) p =                         …        (3)

Therefore p² =                     …        (4)

Substituting (4) into (2)

  x    = 160

x 200   =  160

There 200(n – 1) = 160n

200n – 200           = 160n

200n – 160n         = 200

                40n         = 200

                     n         = 5

From (3)p =    = 4

Hence, n = 5, p = 4

Example 8

Obtain the first four terms of the explanation of (2 +   x)⁸in ascending powers of x. hence, find the value of (2.005)⁸, correct to five significant figures.

Solution:

(2 +   x)⁸= 2⁸(1+   x)⁸  

= 2⁸[1 +⁸C₁(  x)  + ⁸C₂ (  )²  +  ⁸C₃ (  )³   + … ]

=   2⁸[1 +8(  x)  +  (  )²  +  (  )³   + …]

=   2⁸[1 + 2X +  X²+ X³  + …]

Write 2.0.005

Put 2 +  x    =  2 + 0.005

Therefore   x    =   0.005

Therfore  x   =  0.005 x 2

                       = 0.01

Hence,

(2.005)⁸   =   2⁸[1 +2(0.01)  + (0.01)²+  (0.01)³ ]

(2.005)⁸   = 2⁸ + 2⁹(0.01) + 2⁶.7(0.01)² + 2⁵ x 7(0.01)³ + …

= 256 + 5.12 + 0.0448 + 0.000224

               = 261.165025

               = 261.17 (5 s.f.)

GENERAL EVALUATION

1) Write down and simplify all the terms of the binomial expansion of ( 1 – x )⁶ . Use the expansion to evaluate  0.997⁶  correct to 4 dp

2) Write down the expansion of  ( 1 + ¼ x ) ⁵ simplifying all its coefficients

3) Use the binomial theorem to expand  ( 2 – ¼ x)⁵ and simplify all the terms

4) Deduce  the expansion of   ( 1 – x +x² )⁶  in ascending powers of x

Reading Assignment

New Further Maths Project 2  page 73 – 78

WEEKEND ASSIGNMENT

If the first three terms of the expansion of ( 1 + px )ⁿ in ascending powers of  x are   1 + 20v + 160x find  the value of

1)  n  a) 2  b) 3  c) 4  d) 5

2) p   a) 2  b) 3  c) 4  d) 5

3) In the expansion of  ( 2x + 3y )⁴  what is the coefficient of  y⁴   a) 16  b) 81  c) 216  d) 96

4) How many terms are in the expansion  of  ( 1 – 4x ) ⁵  a) 3  b) 5  c) 6  d) 8

5) What is the third term in the expansion of  ( 1 – 3x )⁶ in ascending powers of x  a) 18  b) -540  c) 135  d) 729

THEORY

1) Using binomial theorem, write down and simplify the first seven terms of the expansion of  ( 1 + 2x )¹⁰ in ascending powers of x

2) Expand  ( 2 + x )⁵ ( 1 – 2x ) ⁶ as far as the term in x³   . Evaluate  ( 1.999 )⁵ ( 1.002 )⁶

WEEK FOUR

TOPIC:MECHANICS (VECTOR GEOMETRY)

SCALAR OR DOT PRODUCT OF TWO VECTORS

The scalar or dot product of two vectors a and b is written as a.b and pronounced as (a dot b). Therefore, a.b =|a| |b| cos dot is defined as a.b =            a b cos  where  is the angle between vectors a and b

If a = a₁ I + a₂j and b = b₁ I b2j

Thus a .b = (a)1bi ii + ab2j I 1 + 2 bi I h +a2 b2 j

Recall that I and j are mutually perpendicular unit vector hence

i.i = |x| cos 0 =1

i.j = |x| cos 90 =0

j.i = |x| cos 90 =0

j.i =|x| cos 0 =1

Hence, a.b =a₁b₁ + a₂ b₂

Examples

1. Find the scale product of the following vectors 9i -2j + k and I – 3j -4k

Solution:

            A=(9i- 2j +k) and b= (i-3j -4k)

            a.b = (9i-2j +k) (i-3j-4k)

            =9 (1) -2(-3) + 1(-4)=9+6-4a.b =11

2.         Let a = 3i+2j, b = -4i+2j and c = i+4j, calculate a.b, a.c and a. (b+c)

Solution:

            I a.b = (3i + 2j ) (-4i+2j) = 3 (-4) +2(2)

            = -12+4

            =-8

II a.c = (3i+2j) (I +4j)

            = 3 (1) + 2 (4)

            = 3+8   = 11

III a.(b+c)

Find (b+c) = -4i + 2j +i +4j

                                    =-3i +6j

1. (b+c) = (3i+2j) (-3i +6j)

            =3(-3) + 2(6)

            = -9+12 = 3.

PERPENDICULARITY OF VECTORS:

If two vectors P and q are in perpendicular directions, thus p.q =0

 Example 1: show that the vectors p = 3i+ 2j and q= -2i + 3j are perpendicular.

Solution:

P:q = (3i+2j) (-2i +3j)

=3(-2) + 2(3)

=-6+ =0

Since p.q=0, then the vectors p and q are perpendicular.

2. If p= 4i + kj and q=2i – 3j are perpendicular, find the value of k, where k is a scalar..

Solution:

p.q=0

(4i+kj)(2i-3j)=0

4(2) + k(-3)=0

-3k=-8

K=8/3.

EQUAL VECTORS: Vectors p ad q are equal if p is equal to q.

Example: find the value of the scalar K for which the vectors 2ki + 3j and 8i+kj

Solution:

2ki +3j = 8i +kj

Hence, 2ki =8i,                                    3j=

2k = 8                                      12=3k

K=8/2                                      k=12/3

K=4                                         k=4

EVALUATION

1. The vectors AB and C are -2i+6j-3k and -2i-3j+6k respectively. Find the scalar product AB.AC
2. Find the value of the scalar A for which the pairs of vectors 5i +3j and 2i-4Aj are perpendicular.

ANGLES BETWEEN TWO VECTORS

 Is the angle between two vectors and from dot product where a.b=|a|b| cos . Hence, Cos

Where = Magnitude of vector a= ²₁  +²₂

                 |b| =Magnitude of vector b=²₁  +²₂

Example:

Find the angle between the vectors pp=2i – 2j + k and q=12i +4j – 3k

Solution:

Cos =

p.q= (2i-2j+k)(12i+4j-3k)

            =2(12) -2(4)_1(3)

            = 24-8-3

            =13

|p|=² + (-2)² + 1²=+4+1 ==3

|q|= ² + 4² + (-3) = 144+16+9= 169 =13

Cos

Cos =1/3.                =Cos^-1 (1/3)

DIRECTION COSINES A VECTOR:

The direction is specified by the angles which the vector makes wit x and y axes. If we represent these angles by  and  respectively then,

Cos   =       Cos  =

Example: find the direction cosine of the vector 4i + 3J – 11k

Solution:

Let a = 4i + 3J-11k

            |a|= ²            + y² +z² = |=² + ² + (-11)² =

Direction cosine, Cos  Cos =Cos=

EVALUATION

1. Find the angle between the vectors 2i + 3j +6k and 3i+4j+12k
2. Find the direction cosine of vector a = 10i- j+2k

EVALUATION: find the projection of the vector a on the vector b if a=5i-4j+2k and b=6i – j +3k

GENERAL REVISION EVALUATION

1. Given that a=4i – 2j +k, b=2i – j +3k and c=5i +2k find (i) (a+b)c          (ii) a=c+b.c
2. If  a = 4i – 2j +k, b=6i +5j find (i) the unit vector  in direction of b. (ii) the projection of a on b (iii) the unit ve4xtor in the direction of a (iv) the projection of b on a.

READING ASSIGNMENT: Read vector Geometry, Further Mathematics project II page 236-240

WEEKEND ASSIGNMENT

1. If = 3i + 4j and b=gi +2k are perpendicular, what is the value of g?    A.-4     B.3    C.-8/3
2. Find the value of the scalar k for which the vectors ki + 8j and 3i +  are equal.                                          A. 3     B.6       C.9
3. Find the projection of the vector a on the vector if a = 4i + 6j and b=3i-2j.                                                        A.-3\ 52   B.5\ 13     C. 0  
4. Calculate the angle between a = -4i +2j and b =I -3j. A.45⁰    B.60⁰    C.135⁰
5. Find the scalar product of vectors – 2i-3j and 4i +5j? A.-23    B.23     C.7

THEORY

1a) Given that  a = 4i – 5j + 2k  and  b =  -7i + 3j – 6k  find the scalar product of a and b  (b)  find the direction  cosine  2a + 3b

2 ) Find the angle between  p = 6i + 2j – 4k    and   q = 9i + 5j

WEEK FIVE

TOPIC : MECHANICS ; VECTORS OR CROSS PRODUCT ON TWO OR THREE DIMENSION , CROSS PRODUCT OF TWO VECTORS AND APPLICATION OF CROSS PRODUCT

Vector Product of two vectors

Given two vectors andwhose directions are inclined at an angle, their vector productis defined as a vector whose magnitude issinand whose directions is perpendicular to both and and also being positive relative to a rotation from themvector and also being positive relative to a rotation from the vectorto the re

ctor.

            The vector product ofand b is designated

b

Thus:

             =  x  =|| || sin . whereis a unit vector perpendicular to the plane of  and .

 Properties of vector Product

•  x = |b||a| sin(-) 0 <<

                        = – |a||b| sin (-)

                        = –  b

            Thus the vector product of two vectors is not commutative .

• (k) x                                   = x (k )

                                                = k (x)

                                                = k |||| sin )

Where k is a scalar.

• x ( + c)                    = x  + x c

Distribute law

• x       = x =             x

x                    =                    = – x , x k =  =

• x

             x  =                         – x

•             |a x b| = area of parallelogram with sides 

and .

•             If  x = 0 and  and b are non zero vectors, then a and b are parallel
•  If      a = a₁ i+ a₂i + a₃k

            b = b₁ + b₂i +b₃k then

            i      j     k

a       ba₁      a₂     a₃

                 b₁    b₂    b₃

We shall make use of the following important result in determinant of order 2 x 2 and order 3 x 3 defined respectively as follows.

a     b

c     d    = ad – bc

 

a     b     c                  e     f       – b   d   f     + c               d       e

d     e     f         = a     h     I              g   I             g       h 

g     h     i

The expansion of the determinant of order 3 x 3 is along the first row.

            Notwithstanding it can be along any other row or any column.

Example 1

Find the vector of a andb where:

a = 4I  – 3j       + 2k, b = i + 2 j – 5k

Solution

a = 4 I – 3 j + 2 k

b = I + 2j – 5k

 

a x b =   i              j      k

                    4      -3     2

                    1       2     -5

 

=i    -3  2            4        2          4          -3

                                  –j                 +k              

              2        -5           1        -5         1          2 

= I (15 -4) –j (-20 -2) + k (8 + 3)

= 11i + 20j + 11k

If  p     =2i – 3j + 4k

     q     =5i – 4j – 3k

Find :

• p x q;
• |p x q|

Solution

• p x q =    i           j        k

                     2   -3       4

                     5    4       -3

=i    -3  4            2        4          2          -3

            –j               +k              

            4          -3         5         -3         5          4 

= i (9 – 8) –j (- 6 -20) + k (8 + 15)

=i + 26j + 23 k

• |p x q|   =          |I + 26j + 23k|

                        =         

                        =         

`                       =         

                        =         

                        =         

Example 

Show that (a x b)² = a²b² – (a.b)²

Solution

(a x b)² =          (absin)²

                        =          a² b² sin²

                        =          a² b² (1 – cos² )

                        =          a² b² – a² b² cos²

                        =          a² b² – (a.b)²

Hence

(a x b)² =          a² b² – (a.b)²

EVALUATION

 Given that  p = 2i + 3j +4k  and  q= 5i – 6j +7k   find   ; (1)  p x q  ( 2)  (p + q ) . ( p-q)

Application of vector product

Area of a parallelogram
Example 

Show that the area of parallelogram with sides a andb is.

Solution

Area of parallelogram

OAC B=           h/b

                        =/a/ sin /b/

                        =/a/ /b/sin

                        =/a x b/

Area of angle

Example 

A

Show that the area of a triangle with sides a and b is  |a x b|

Solution

Area of            =          OAB =  |b| x h

                        =            |b||a| Sin

                        =            |a||b| Sin

                        =            (a x b)

Example

The adjacent sides of a parallelogram are

 = 2 i – j – 6k and  = i + 3 j – k . Find

the area of the parallelogram.

Solution

AB       =          2 i – j – 6k

AC                   =          i + 3 j – k

Area of parallelogram =           |AB x AC|

                                                =           x

i                 j       k

 2              -1     -6

    1           3      -1       

x

= i        -1         -6   –j    2         -6   +k  2          -1

            3          -1            1       -1           1        3         

= I (1 + 18) –j (-2 + 6) + k (6 + 1)

= 19 I – 4 j + 7 k

|AB x AC|                     =          |19i  – 4j + 7k|

                                    =         

                                    =         

                                    =         

Hence

Area of parallelogram =  sq. Units

GENERAL EVALUATION

1) Find the vector product of  a= 4i -3j +4k and  b = -I + 2j +7k

2) Given that p = 7i + 2j + k  and q = 3i – 2j + 4k find ; (i) p x q  (ii) | p x q | (iii) the unit vector perpendicular to both p and q

3) Find the sine of the angle between the vectors :  a = I – j + k  and  b = 8i + 2j + 3k

4) The adjacent sides of a parallelogram  are PQ= 4i + 3j + k  and PR = -5i + 2j +3k find the area of the parallelogram

5) The position vectors OA, OB and OC are  2i – 3j + 4k , 6i + 4j -8k and 3i + 2j + 5k respectively  find  (i) vector AB  (ii) vector BA  (iii) vector BC (iv)  AB x BC

Reading Assignment: New Further Maths Project 2 page 216 – 222

WEEKEND ASSIGNMENT

Given that  a = I + 2j + k  and   b = 2i +3j- 5k

1) find ( a x b ) . a    a)  0  b) 1  c) 2  d) 3

2) find ( a x b ) . b     a) 1  b) 2  c) 0  d) 3

Given that  p = I + 5j + 6k    and q = – 2i + j + 3k

3)  find p x q    a) 15i +11j -11k  b) 11i – 15j + 11k   c)  11i – 11j + 15k  d) 11i- 15j -11k

4) find q x p    a)  -11i  + 15j – 11k  b) 11i –  15j + 11k    c) 15i – 11j-11k  d)  15i+11j+11k

5) Given that  a = i – j+ 3k  and   b = 6i + 2j – 2k     find   ( a + b ) . ( a x b )     a)  1  b) 0   c) 2  d) 3

THEORY

1)  AB = 4i +3j+5k  and AC= 2i-3j+k  are two sides of a triangle  ABC , find the area of the triangle

2) PQ = 2i+5j+3k  and  PR = 3i-3j + k  are two adjacent sides  of  a parallelogram, find the area of the parallelogram.

WEEK SIX

REVIEW OF HALF TERM WORK

WEEK SEVEN

TOPIC: INTEGRATION

Integration: This is defined as anti- differentiation. Suppose, y = x³ + 2x, the first derivative is

 3x² + 2. (dy/dx = 3x² + 2)

then the anti – derivative of  3x² + 2 = x³ + 2x

 Thus, integration is the reverse process of differentiation and denoted by the symbol ∫.

If  dy/dx = xⁿ , then  ∫ dy/dx = xⁿ⁺¹   + C

n + 1           (n ≠ -1)

where c is the arbitrary constant.

INDEFINITE INTEGRAL CONCEPTS

General Concept

Example:  Evaluate the following integrals:
       1. ∫x² dx      2. ∫x^5/2 dx     3.∫ 4/ x5 dx      4.∫ √x⁸      5.∫ (7x⁴ + 2) dx 6.∫(x⁵ + 2x⁴ – x³ + 6) dx

Solution;

1. x³ + C        2.x^{5/2 + 1} =   2x^7/2  + c           3. ∫ 4x^-5 dx =  4x^-5+1=  – 4^{x – 4} + C

    3                        5/2 + 1         7                                           -5 + 1

4.  ∫  x⁴ = x⁴⁺¹ = x⁵ + C       5.  7x^{4 +1} + 2x⁰⁺¹  =7x⁵ + 2x + C

                                                  4 + 1                      5

6. x⁶ + 2x⁵ – x⁴+ 6x + C

    6        5      4

NB: Integral of a constant is not zero but the variable in the question.

Evaluation: Evaluate the following integrals; 1. ∫ (12x³ – x⁶ +1/x²) dx     2. ∫x²(3x² + 4x) dx

Trigonometric integral:

The trigonometric integrals can be summarized in the following table. Remember that this is the reverse process of differentiation.

    F(x)                                    ∫ f(x)dx

                           Sin x                          – cos x + c

                           Cos x                           sin x + c

                           Sec²x                            tan x + c

                         Cosec²x                       – cot x + c

                        Sec x tan x                       sec x + c

e^xe^x + c

                           1/x                                ln x + c

Example: Evaluate each of the following integrals.

1. ∫ sin x – 5 cos x)dx                2. ∫ (5sinx + 3x²)dx

Solution:

1.  . ∫ sin x – 5 cos x )dx      =     ∫sin x dx – ∫5 cos x dx

=  -cos x – 5 ( sin x ) + c

                                             = – cos x – 5sin x + c

2.∫ (5sinx + 3x² )dx            =   ∫5 sin x dx + ∫3×2 dx

                                              = -5cos x + x³ + c

3.  ∫  e^2x dx                           = e^2x/2 + c

Evaluation: Evaluate the integrals:

 1. ∫ (3 cos x + 2 sin x) dx              2. ∫ e^{2×2 + 5x} dx

INTEGRATION BY ALGEBRAIC SUBSTITUTION

Sometimes integral are not given in the standard form, such integral are then reduced to standard form format before evaluation by algebraic substitution.

Suppose, an integral is given in the form ∫ f(ax + b)ⁿ dx

Then, the algebraic substitution is to represent the function in the bracket by any letter.

Let              u = (ax + b)    du/dx = a,      dx = du/ a

                   ∫uⁿ dx = ∫ uⁿ du/a

                            = 1/a ∫ uⁿ du

Example:

Evaluate the following integrals.

1.  ∫ (2x² – 5x )⁴ dx       2.  ∫        7           dx           3. ∫ xcos 2x² dx   4. ∫ x² √(x³ + 5)dx

                                               (5x – 4 )⁵

Solution:

1. .∫ (2x² – 5x )⁴ dx        let u = 2x² – 5x ,  du/dx = 4x – 5 ,  dx = du/4x -5

       .  ∫ u 4 du/ 4x -5  =           u⁵+   c

                                        5(4x – 5)

= (2x² – 5x )⁵+  c

20x – 25

2. ∫        7           dx     let u = ( 5x – 4)    , du/dx = 5, dx = du/5

        (5x – 4 )⁵

       ∫ 7 u ^-5 du/5   = 7u^-4+  c

  5 x – 4

=  7(5x – 4 )^-4

                                     -20

       3. ∫ x cos 2x² dx     let u = 2x², du/dx = 4x,  dx = du/4x

then; ∫x cos 2x² dx =                    ∫x cos u du/4x  = 1 x ( sin u )  =   1 sin 2x²  +   c

                                                                                         4                      4 

       4. ∫ x² √(x³ + 5)dx     let u = x³ + 5,  du/dx = 3x²,   dx = du/3x²

∫ x² √(x³ + 5)dx     = ∫x² u^1/2 du/3x²  =  1 x u^3/2    = 2 (x³ + 5)^3/2+ c

 3/2 x 3            9

   Evaluation:

 Evaluate the following integrals:

        1. ∫(5x – 7 )^7/2dx    2. ∫cos 9x dx      3. ∫ xcos 2x dx.

       INTEGRATION BY PARTS

This technique is uniquely useful in evaluating integrals that are not in the standard form. Such integrals cannot be solved by algebraic substitution.

From the product rule of differentiation, it can be generalized thus;

                               ∫ vdu = uv – ∫ udv.

Example: Evaluate the following integral by parts.

1. ∫ 2x sin x dx               2. ∫ e^2xcos 2x dx

solution:

 1. ∫ 2x sin x dx               , let v = 2x, dv/dx = 2, dv = 2dx

                                             ∫du = sin x dx

                                    u =  -cos x

                        ∫ vdu = uv – ∫ udv.

                  ∫ x² sin x = – x²cos x  –  ∫- cos x x 2 dx

                              = – x²cosx  + 2∫ cos x

                              = – x²cos x + 2 sin x + c

2. ∫ x²e^x  dx    , let v = x², dv/dx = 2x, dv = 2xdx

du = e^x     u = e^x dx

∫ vdu = uv – ∫udv

              ∫ x²e^x   = e^x x² – ∫ e^x  2xdx

                          = x² e^x–  2∫e^x x dx                                         

the integral part in the RHS will have to be evaluated using integration by parts;

thus, v =  x,  dv/dx = 1,  dv = dx   , du =e^x,   u = e^x

∫ vdu = uv – ∫udv

  ∫ x e^x  =  e^x . x – ∫e^xdx

                    = e^x.x – e^x

finally,  ∫x²e^x  = x² e^x   –  2(e^x.x – e^x)

                          = x²e^x – 2xe^x + 2e^x  + c

Evaluation:

Evaluate 1.    ∫x²cos x dx    2. ∫x³ e^-x dx

INTEGRATION BY PARTIAL FRACTION

Sometimes rational functions are not expressed in the proper standard form; such function can be evaluated by transforming them into standard form through partial fractions. The knowledge of partial fractions is needed here to evaluate the functions.

Example; Integrate each of the following with respect to x;

1         2x + 32          x + 8

     (2x + 1) (x – 1)                                     ( x² + 3x + 2)

solution:

1.  resolve into partial fraction;           2x + 3       =      A       +     B        

                                                    (2x + 1)(x – 1)      (2x + 1)    (x – 1)

                                               2x + 3 = A(x-1) + B(2x + 1)

when x = 1,   2(1) + 3 = B(2 + 1)

                                    5       =   3B,        B= 5/3

when x = -1/2,  2(-1/2) + 3 = A(-1/2 – 1 )

                                            2      = – 3/2 A                      A = – 4/ 3

;           2x + 3       = ∫      -4         +  ∫    5       dx

     (2x + 1)(x – 1)     3( 2x + 1)          3(x – 1)  

                             = -4  ln (2x + 1)   + 5 ln (x – 1)

                                   2 x     3                               3

                             = – 2/3 ln (2x + 1)  +  5/3 ln ( x – 1) + c

2.x   +   8    =     x  + 8         = A       +  B

   (x² + 3x + 2)      (x + 1)(x + 2)            x + 1        x + 2

                                     x + 8 = A(x+2) + B(x+1)

when x = -2,

                  – 2 + 8 = B(-2+1) 

                         6 = -B,    B = – 6

when, x = -1,    – 1 + 8 = A(-1 + 2)

                                7 = A.

thus,   x   +   8   =∫  7       +  ∫     – 6

                                             x + 1         x + 2

                                     = 7ln (x+ 1) – 6 ln(x+2) + c

Evaluation

 Integrate by partial fraction.

     1.       4x + 3                      2.            1     

         (x – 3)(x+2)                    (x²+ 3x + 2)

GENERAL EVALUATION/REVISIONAL QUESTIONS

1.Find the derivatives of the following with respect to x;

(a)        y = (15 + 5x)(1 + 2x)               (b)        y = (1 + 2x)¹²                       (c) y = 3x² (3 – 2x + 4x²) ^½

2. Given that the gradient function of a curve is 8x – 2, find the equation of the curve at point (2, 4)

3. Find ∫(x² + 1)(x³ – 2)dx 

4. Find ∫x² e^2xdx                    

Reading Assignment: Read Integration, Page 31 – 46  Further Mathematics project III.

WEEKEND ASSINGMENT

1. Evaluate ∫ (x⁵ + 3)dx    . A. x⁶/6 + 3x  + c      B.   x⁵/6 + c     C. x⁶/6  + c

      2. Evaluate ∫ cos 7x dx       A. 7sin 7x      B. 1/7 sin 7x + c     C.  7sin 7x + c

3.  Integrate the function; (3x + 5)⁵wrt x   .A 12(2x+3)⁶ + c   B. (2x +3)⁶+ c      C.  (3x + 5)⁶ + c

                                                                                                            12                         18

4. Find ∫(x+1)(x²– 2)dx  A. x⁴ + x³ – x² – 2x + c    B. x⁴ – x³ + x² + c     C. x + x⁴ – x³ – x² + c

  4     3                                         3     4

5. Integrate 1/x⁵ wrt x.   A.   x ⁶ + c    B .x^-4 + c      C.    x^-4+ c

 6                – 4                    5

Theory:

1. Find ∫x sin2xdx                     2. Evaluate ∫    dx

x ( x + 2)

WEEK EIGHT

TOPIC : INTEGRATION [INDEFINITE  INTEGRALDEFINITE INTEGRAL AND AREA UNDER CURVE]

The process of reversing differentiation is called Integration. If dy/dx = 3x², then y could be x³, as the derivative of x³is 3x².

We say that x³ is an integral of 3x² with respect to x. The symbol for integration sign is given by ∫  . The expression to be integrated is put between the  ∫ sign and dx.

∫  3×2 dx could be x³

Since differentiating any constant gives zero, the following also have derivation 3x².

X³ + 2,  x³ + 4.5,  x³ – 17etc

In general, any function of the form x3 + c, where c is the constant has derivative of 3x²

Hence, ∫  3x² dx = x³ + C. C is called constant of integration. Because we do not know the actual or definite value of C, this is called INDEFINITE INTEGRAL.

Let y = xⁿ⁺¹/ n+1 ,  Differentiating dy/dx = (n+1) xⁿ⁺¹/n+1 = xⁿ

Reversing this,

∫  xⁿ dx = xⁿ⁺¹/n+1 + C

∫  kxn dx = K ^xn+1/n+1 + C

To Integrate a sum, integrate each term as this is similar to differentiating a sum.

Examples:

Evaluate (i)  ∫(2x³ + 3x² – 4) dx

            2/4x⁴ + 3/3x³ – 4x + C

            ½ x2 + x3 – 4x + C

Evaluate ∫(4t³ + 2t² + ½ t²) dt

4/4 t4 + 2/3 t3 + ½ × ½ t² + C

t⁴ + 2t3 + ¼ t2 + C.

Integration of basic Trigonometric functions

Consider the table below for differentiating trigonometric  functions.

Y	sinKx	Cos Kx	Tan Kx
dy/dx	kcoskx	-ksinkx	k/(coskx)2

Consider a similar table as the one given above

Y	Sin x	Cos x	Tan x
	Cosx+C	Sinx +C	Tanx +C

Y	sinkx	coskx	Tankx
∫ydx	-1/k coskx + C	1/k sinkx + C	1/k tankx + C

Example

∫(cos2x + sin3x) dx

∫cos2x dx + ∫ sin3x dx

½ sin2x + -1/3 cos3x

½ sin2x – 1/3 cos3x + K.

Sometimes,the value of the constant C can be found, if extra information is given.

If dy/dx = x2+2x-3, find y in terms of x given that x = 1, y =4

 Y =∫(x2 +2x-3)dx

Y =x3/3 + 2x/2 – 3x + C

Y = x3/3 +x2 -3x + C

Putting x=1,and y = 4

4 = 1/3 + 1 – 3 + C. hence; C = 5 2/

;. 1/3×3 – x2 – 3x + 5 2/3.

If dy/dx = 2Cos3x, find y given that y =2 and x = 1/6∏

∫2Cos3x = 2/3Sin3x + C

2 = 2/3Sin3(∏/6) + C = 2/3 Sin∏/2 + C

Multiplying by ½ ∏ (180/∏) = 90⁰

Sin ½ ∏ – Sin 90⁰ = 1

2 = 2/3 ×1 + C

C = 4/3

Y = 2/3Sin3x + 4/3 .

Evaluation:

Evaluate these indefinite integrals

1. ∫(y2 – 7y)dy
2. ∫(3×2 – 2x – 1)dx
3. ∫(Cos4x)dx
4. ∫(3cos2x + 4sin3x)dx
5. If dy/dx = 4×2 + 1 and y = 2 when x =3, find y in terms of x
6. If dy/dx = 2Sin1/3x and y = 4 when x =∏m, find y

DEFINITE INTEGRALS

In this part, the constant is removed. If a definite integration is performed, the function is evaluated between the values called limits. Upper and lower ie

Example: Evaluate

 =x³ + C     ,  (3³ + c) – (2³ + c)

(27 + C) – (8 + C)

27 + C -8 – C

19.

 = 19.

 = { ½ Sin2}

(½ sin× 0)

½ sin

 = ½

Area under curve using Definite Integral

Given in the diagram, the area between the curve and the x axis from x = a and to x = b. The area is given by  

 Area =

The area can be explained as: Area = ∫ y   ×   dx = Sum of height of rectangle  × width of rectangle

y

a                                b  

Ex 1: Find the Area between the curve y = x3 – x and the x axis when x =2 and x = 4.

y

0           2              4                       x

{ ¼ (4)⁴ – ½(4)2} – { ¼ X 2⁴ – ½ (2)²} 

64 – 8 -4 + 2 = 54 UNITS.

Ex 2 : Find the area in the diagram shown below

y = 4 – x²

                                                                                       4

           -2                                                         0                                                              2

∫(4 – x2 )dx 

{4x – x3/3}₂².

4(2) – 2³/3  – 4(-2) – (-2)³/3

(8- 8/3) – (-8 + 8/3)

8 – 8/3 + 8 – 8/3

32/3.square unitrs

Evaluation:

Find the area between the values as shown below

y

                                                                               y = 3x²

 0     2         4                       x

Find the area between the curve y = x² + 3 At the xs axis and when x = -1 and x = 3.

Evaluation

1. x² and the line y = 2.

General Evaluation:

1. ∫(3x – 1)(x + 2) dx
2. ∫5cos4x (dx)

Reading Assignment :Solve the evaluation questions given above

Weekend Assignment:

1. Evaluate  A. 2/3 B. -2/3 C. -6 2/3 D. 6 2/3
2. Evaluate  A. 4 B. 2 C. 4/3 D. 1/3
3. Evaluate  A. – ½ B. 1 C. -1 D. 0
4. Find the area enclosed by by the curve y = x² , X = 0 and X = 3 A. 9 B. 7 C. 5/2 D. 5
5. Given  y = 3x -2, x=3, x=4. Find the area under the curve A. 4/3 B. 17/2 C. 6 D. 3

Theory

1. Find the area enclosed between the curve y =x2 + x -2 and the x axis
2. Find the area enclosed by the curve y = x2 – 3x + 3 and the y = 1.

WEEK NINE

TOPIC : APPLICATION OF INTEGRATION II : SOLID REVOLUTION AND TRAPEZOIDAL RULE

A solid whichb has a central axis of symmetry is a solid of revolution.Forexample, a cone, a cylinder , a vase etc.

 

y

Consider the area under a portion AB of the curve y = f(x) revolved about the x axis through four right angle or 3600, each point of the curve describes a circle centered on the x axis. A solid revolution can be thought of as created in this way with the circular plane ends cutting the x – axis at x = a and x = b.

Let v be the volume of the solid for x = a up to an arbitrary value of x between a and b. Given abincreament dx in x , and y takes an increamentdy and v increases by dv.

The figure shows a section through the x axis, from this it is seen that the slice dv of bthickness dx is enclosed between two cylinders of outer radius y +dy and inner radius y .

Then ,πy2dx< dv < π (y + dy)2dx;

With appropriate modification, if the curve is falling at this point.

π y2 < dv/dx < π (y + dy)2

if dx          0, dy        0 as dv/dx           dv/dx

: . dv/dx = πy2 or V =

Where y = f(x) and v = volume of solid revolution of the curve where y = f(x) is rotation completely and x – axis between limits x = a and x =b.

Examples;The portion ofthe curve y = x² between x = 0 and x = 2 is rotated complrtely around the x axis, find  the volume of the solid generated?

V =

=

V =

V =  =

Put x = 2 and x = 0 then substitute into the expression above

V =

THE TRAPEZOIDAL RULE

There are many definite integrals which can’t be evaluated and thus required advance techniques e.g

etc

We can find an approximate value for such  integralsbyb finding the area approximately. There are many methods methods of doing this and such methods include the Trapezium rule.

         Y

                                        Y = f(x)

 y₁ y₂y₃ y_n-1­y_n

x₁        x₂         x₃          x_n-1x_n

 = ½ (y₁ +y₂)h + ½ (y₂ +y₃)h + ½ (y_n-1 +y_n)h

½ h(y1+2y2+2y3+……….+2yn-1 + yn)

½ (width of each trap. ) × (first ordinate + last ordinate )+ 2( sum of all other ord.)

F(x)dx = ½ h{y1 + yn} +2{y2 +y3 + …Yn}.

Example

Find the approximate value of  at interval 0.5

X	1	1.5	2	2.5	3.0
Y = 1/x	1	0.67	0.5	0.4	0.33

Applying the rule;

{ ½ . ½ { (1 +0.33) + 2 ( 0.67 + 0.5 + 0.4)}

¼ {(1.33) +2(1.57)}

¼ (4.47) = 1.12.                                                                      

 1    0.67   0.5     0.4      0.33

Ex (2). Make a table of value of y  for which y =  for which x =2 t0 x = 3 at interval of 0.2.

X	2	2.2	2.4	2.6	2.8	3
X2-1	3	3.84	4.76	5.76	6.84	8
	1.732	1.956	2.182	2.4	2.615	2.828
	0.5754	0.5703	0.4583	0.4167	0.3824	0.3536

Using the rule.

= ½ *0.2 (0.5774 +0.3536 + 3.5354)

0.44664 *2

0.89 correct to 2 dp

APPLICATION OF INTEGRATION TO KINEMATICS

If the ve;locityis  given as a function of time, the displacement is the integral of the velocity function with respect to the time .

ds/dt = f(t)

then S =

= f(t) + C

Similarly, if the acceleration is a function of time, the velocity is the integral of the acceleration function.

Ex. A particle is projected in a straight line from O until a speed of 6m/s is attained. At time t secs.Later,its acceleration is (1 + 2t) m/s² for the value of t = 4. Calculate for the particle (i) its velocity (ii) its distance from O

dv/dt = 1+ 2t

v =  = t + t² + C

when t = 0

v = 6m/sand c = 6.

V = (t² + t + 6) m/s

When t = 4, v = 16 + 4 + 6 = 26m/s.

(ii) distance (s) = ds/dt = t2 + t + 6

S =

S = {t3/3 + 16/2 6t}⁴

  = 160/3

m.

Evaluation

1. Find the area enclosed byb y = x2 – x -2 and the x axis

2. find  the area under the curve y = x2/3 between x = 2 and x = k is 8 times the area under the same curve between x = 1 and x =2, hence find the value of k.

GENERAL EVALUATION

1. A particle moves in astraight line from O until the initial velocity was 2m/s. its acceleration is given by (2t -3)m/s2. Calc. (i) its velocity after 3 secs. (ii) the distance from O when it is momentarily at rest.
2. Find the volume of solid revolution when a is  the region bounded by the cuerve y = 2x. and the ordinate at x = 2,and x = 4 and the x axis is revolved by 2π.

Reading Assignment :F/Matrhs Project, pg 47 – 63

WEEKEND ASSIGNMENT

1. Integrate 2√x A.  B.4x^3/2  + C C.  + C  D.

2. Integrate   A.  + C B. x^3/2 + C C. x² + C D.  + C

3. The gradient of a curve is 6x + 2 and it passes through the point (1,3), find its equation A. 3x² – 2x + 2 B. 3x² -2x -2 + 2x + 2 C. 3x² – 2x + 2 D. 3x² – 2x -2

4. Evaluate  A.  –  + x2 – 2x + C B. x3/3 – x3/2 + x2 + 2x +C C. x3/3 + x2/2 +x3-2x + C D. x4/4+x3/3-x2+2x+C

5. Eval.  A. 9 +C B. 8/3 + C C. 24 D. 18 + C

THEORY

1. Evaluate
2. Using trapezoidal rule, with ordinate x = -3,-2,-1,0,1,2,3 and 4. Calc correct to 3 dp an approximate value of

WEEK TEN

TOPIC: REGRESSION LINE AND CORRELATION COEFFICIENT

SCATTER DIAGRAM

Definition: a scatter diagram is a graphic display of bivariate data. A bivariate data involves two variables

TYPES OF SCATTER DIAGRAM:

Linear positive correlation.

A positive correlation between two variables x any y means that in general, increase in x is accompanied by increase in y. The regression line has a positive slope.

X

Linear  negative  correlation

A negative correlation between x and y means that an increase in x is accompanied by a decrease in y, negative correlation has a negative slope.

x

Zero Correlation:

There  is no apparent association between x and y.

y

Non Linear Correlation:

Most of the points lie on or near a curve which is parabolic in shape. The parabolic curve is called a regression curve.

x

REGRESSION LINE   OR    LINE OF BEST FIT   OR   THE LEAST SQUARES LINE

There are two variables where one is dependent and the other is independent variable. The regression line can be fit using scatter  diagram method and the least squares method.

LEAST SQUARES METHOD: If x is independent variable and y dependent variable, that is y on x. then :The equation of the regression line is written as y = ax + b

Where a is the slope and b is the y – intercept. Given two sets of variables x and y it can be deduced that

a  = n   ∑ xy  –   ∑ x  ∑ y

                                  ∑ x² –  (  ∑ x)²

b = y a – ax

Where  x   =      ∑ x 

n

y =       ∑ y

n

Example: use the least square method to fit a regression line of y on x for the following data

X	3	5	6	9	11	14	15	18
Y	2	3	5	7	10	12	13	17

 Find value of y when x = 8

SOLUTION:

X	y	Xy	x2
3	2	6	9
5	3	15	25
6	5	30	36
9	7	63	81
11	10	110	121
14	12	168	196
15	13	195	225
18	17	306	324
∑ x  = 81	∑  y = 69	∑ xy = 893	∑ x2= 1017

 a = n ∑ xy –   ∑x ∑ y=      8 (893) –  81x 69

n∑(x² ) – ( ∑x)²                8 (1017) – (81)²

                a =   7144 – 5589      = 1555

                        8136 – 6561          1575

                                                a = 0. 9873

x  = ∑ x    =   81  =   10.125

n            8  

y =     ∑ y    = 69   = 8. 625

n          8

b = y – ax

                         b = 8.625 — 0.9873 (10.125)

                            = 8.625 – 9.996

                          b = -1.37

                          y = ax + b

                           y = 0.9873x – 1.37    (regression line of y on x )

                 When x = 8

                           y = 0. 9873 (8) – 1.37

                            y = 6.5284 ~ 6. 5

EVALUATION

Use the least square method to fit a regression line of y on x for the following data

X	1	4	5	7	8	10	12	16	19	20
Y	2	3	4	5	7	8	10	15	20	18

 Use the line obtained to find the value of y when x = 9

CORRELATION COEFFICIENT

DEFINITION:

The correlation coefficient determines the amount or degree of linear relationship between two variables. The correlation coefficient is represented by r

The characteristics of r are as follows:

1. The value of r is the same irrespective of the variable labelled x or y.
2. the value of r satisfies the inequality   -1< x < + 1
3. if r is close to +1, the variables are highly positively correlated. If r is close to -1 then, x and y are highly negatively correlated. If r is close to zero, the correlation between x and y is very low. There is no correlation between x and y when r = 0

 There are two methods of obtaining the correlation coefficient.

1. Pearson’s  coefficient of correlation or product moment correlation coefficient
2. Rank correlation coefficient.

RANK CORRELATION COEFFICIENT: It is also known as Spearman’s rank correlation coefficient and defined as :

r_k = 1  –    6 ∑ D²

n(n² -1)

As the name implies, the variables (if not ranked) can be ranked in ascending order or descending order. Where there are ties, the average is used as the rank.

Where D is the difference between the pairs of variables and n is the number of variables. D = Rx – Ry

Example:

The table below gives the examination marks of 10 students in mathematics and history.

Maths	51	25	33	55	65	38	35	53	61	44
History	20	65	25	36	51	50	77	31	60	5

A         Calculate the rank correlation coefficient

b)         Comment briefly on your result

SOLUTION:

MATHS (x)	HISTORY (y)	Rx	Ry	D	D2
51	20	5	9	-4	16
25	65	10	2	8	64
33	25	9	8	1	1
55	36	3	6	-3	9
65	51	1	4	-3	9
38	50	7	5	2	4
35	77	8	1	7	49
53	31	4	7	-3	9
61	60	2	3	-1	1
44	5	6	10	-4	16

                                                                                                                                     ∑D² = 178

r_k =     1-  6 ∑D²

                        n(n² – 1)

=1 – 6  x 178

        10 (10² – 1)

   1 –  1068/990

= 1-1.178= -0.078

There is a very low negative correlation between the marks obtained in mathematics and history.

EVALUATION:

The table below shows the marks obtained by ten students in both theory (x) and practical (y) examination.

X	50	70	85	35	60	65	75	40	45	80
Y	45	55	75	40	50	60	70	35	30	65

 Calculate the rank correlation coefficient between x and y comment on your result.

PEARSON’S CORRELATION COEFFICIENT: It is fully called Pearson’s product moment correlation coefficient. It is simple to calculate and it does not recognise any of the variables as independent or dependent. It is obtained using the formula below.

                                             r =            n ∑ xy – ∑x∑ y

                                                 √ [n∑(x² ) – (∑x)² ][n∑(y²) – (∑y)²

Example:        

            Calculate the product moment correlation coefficient for the following data        

X	2	4	7	9	11
Y	1	2	3	7	9

Comment on your result.

SOLUTION:

X	Y	XY	X2	Y2
2	1	2	4	1
4	2	8	16	4
7	3	21	49	9
9	7	63	81	49
11	9	99	121	81
∑x =   33	∑y =     22	∑xy =  193	∑x2 =    271	∑y2 =   144

√[5(271) – ( 33)²][5(144) – ( 22)²]

   r =   965 – 726

                                    √266 x 236  

                              r =    239

                                      250.55

                              r = 0.9539.        r = 0.95    (approximately to 2 s.f)

Comment: The relationship between x and y is highly positive.

EVALUATION: The following data are the marks obtained by five students in statistics (X) and mathematics(Y). Calculate the product moment correlation coefficient and comment on your result.

X	33	36	42	52	40
Y	42	46	38	62	52

GENERAL EVALUATION/REVISIONAL QUESTIONS

1. If Cos A = 24/25 and Sin B= 3/5, where A is acute and B is obtuse, find without using tables, the values of (a) Sin 2A (b) Cos 2B (c) Sin (A-B)
2. Use the addition formula to find the values of the following

(a)Sin 75⁰ (b) cos 75⁰ (c) tan 45⁰

• Calculate the Product moment correlation coefficient and the Spearman’s rank correlation coefficient.

X	50	45	43	30	30	43	23	43	25
Y	12	13.5	14	11	12	15	13.5	12	14

READING ASSIGNMENT: Read correlation and regression.Page313–320. Further Mathematics project 2.

WEEKEND ASSIGNMENT

Use the table below to answer questions 1 and 2.

Height	160	161	162	163	164	165
No of students	4	6	3	7	8	2

1. The mean of the distribution is

(a)  4875.1 cm ( b)  4001.2   (c) 3571.0cm  (d) 162.2 cm (e)  129.2cm

2.         The median of the distribution is

            (a)    160    (b)   162    (c)  163  (d)  164   (e)  165

3.         Calculate the standard deviation of 3,4, 5,6,7,8,9

             (a)   2      (b)     2.4     (c)     3.6    (d)   4.0        (e)    4.2

4.         Calculate the mean deviation of 6 ,  8 , 4  , 0  ,  4

            (a)  4 .0   (b) 3.6   (c)   3.0   (d)  2. 8   (e)  2 . 1

5.         The table below shows the rank Rx and Ry of marks scored by 10 candidates in an oral and

written tests respectively. Calculate the spearman’s rank correlation  coefficient   of   the   data.

Rx	1	2	3	4	5	6	7	8	9	10
Ry	2	3	4	1	6	5	8	7	10	9

(a)51/55    b) 6/55  c)49/55   d)54/55  e) 61/55

THEORY

1          The distribution of marks scored in statistics and mathematics by ten students is given in the table below:

Maths(x	11	20	23	42	48	50	57	64	80	90
Stat(y)	26	23	35	46	44	50	50	58	68	70

1. Plot a scatter diagram for the distribution
2. Draw an eye- fitted line of  best fit
3. Use your line to estimate the students marks in statistics  if  his mark in maths is  40

2.         The table below gives the marks obtained by members of  a class in maths and                                           physics examination

STUDENTS	A	B	C	D	E	F	G	H	I	J
Maths	85	75	59	43	74	69	62	80	54	63
Physic	92	72	62	48	85	73	46	74	58	50

1. Calculate the product moment correlation coefficient.
2. Comment on your result.