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7. RESOLUTION OF VECTORS 

If Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev be any two non-zero vectors in a plane with different directions and Resolution of Vectors Class 11 Notes | EduRev be another vector in the same plane. Resolution of Vectors Class 11 Notes | EduRev can be expressed as a sum of two vectors-one obtained by multiplying by a real number and the other obtained by multiplying by another real number.

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev (where l and m are real numbers)

We say that Resolution of Vectors Class 11 Notes | EduRev has been resolved into two component vectors namely

Resolution of Vectors Class 11 Notes | EduRev (where l and m are real number)

We say that Resolution of Vectors Class 11 Notes | EduRev has been resolved into two component vectors namely

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev respectively. Hence one can resolve a given vector into two component vectors along a set of two vectors - all the three lie in the same plane.

7.1 Resolution along rectangular component : 

It is convenient to resolve a general vector along axes of a rectangular coordinate system using vectors of unit magnitude, which we call as unit vectors. Resolution of Vectors Class 11 Notes | EduRev are unit along x, y and z-axis as shown in figure below :

Resolution of Vectors Class 11 Notes | EduRev

7.2 Resolution in two Dimensions 

Consider a vector Resolution of Vectors Class 11 Notes | EduRev that lies in xy plane as shown in figure,

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev ⇒ Resolution of Vectors Class 11 Notes | EduRev

The quantities Ax and Ay are called x-and y-components of the vector Resolution of Vectors Class 11 Notes | EduRev.

Ax is itself not a vector but Resolution of Vectors Class 11 Notes | EduRev is a vector and so it Resolution of Vectors Class 11 Notes | EduRev.

Ax = A cosθ and Ay = A sinθ

It's clear from above equation that a component of a vector can be positive, negative or zero depending on the value of q. A vector Resolution of Vectors Class 11 Notes | EduRev can be specified in a plane by two ways :

(a) its magnitude A and the direction q it makes with the x-axis; or

(b) its components Ax and Ay            A = Resolution of Vectors Class 11 Notes | EduRev, θ = Resolution of Vectors Class 11 Notes | EduRev

Note : If A = Ax ⇒ Ay = 0 and if A = Ay ⇒ Ax = 0 i.e., components of a vector perpendicular to itself is always zero. The rectangular components of each vector and those of the sum Resolution of Vectors Class 11 Notes | EduRev are shown in figure.

Resolution of Vectors Class 11 Notes | EduRev

We saw that

Resolution of Vectors Class 11 Notes | EduRev is equivalent to both

Cx = Ax + Bx

and Cy = Ay + By

Refer figure (b)

Resolution of Vectors Class 11 Notes | EduRev

Vector Resolution of Vectors Class 11 Notes | EduRev has been resolved in two axes x and y not perpendicular to each other. Applying sine law in the triangle shown, we have

Resolution of Vectors Class 11 Notes | EduRev

or Rx = Resolution of Vectors Class 11 Notes | EduRev and Ry = Resolution of Vectors Class 11 Notes | EduRev

If α+β = 90°, Rx = R sinβ and Ry = R sin

Ex.7 Resolve the vector Resolution of Vectors Class 11 Notes | EduRev along an perpendicular to the line which make angle 60° with x-axis. 

 

Sol.Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

so the component along line = |Ay cos 30° + Ax cos 60°| and perpendicular to line = |Ax sin 60° - Ay sin 30°|

Ex.8 Resolve a weight of 10 N in two directions which are parallel and perpendicular to a slope inclined at 30° to the horizontal 

Sol. Component perpendicular to the plane

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

= Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev N

and component parallel to the plane

W|| =W sin 30° = (10) Resolution of Vectors Class 11 Notes | EduRev = 5 N

Ex.9 Resolve horizontally and vertically a force F = 8 N which makes an angle of 45° with the horizontal. 

Sol. Horizontal component of is          

Resolution of Vectors Class 11 Notes | EduRev

FH = F cos 45° = (8) Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev                   

and vertical component of is

Fv = F sin 45° = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRevAns.

 8. PROCEDURE TO SOLVE THE VECTOR EQUATION

Resolution of Vectors Class 11 Notes | EduRev ...(1)

(a) There are 6 variables in this equation which are the following :

(1) Magnitude of Resolution of Vectors Class 11 Notes | EduRev and its direction

(2) Magnitude of Resolution of Vectors Class 11 Notes | EduRev and its direction

(3) Magnitude of Resolution of Vectors Class 11 Notes | EduRev and its direction.

(b) We can solve this equation if we know the value of 4 variables [Note : two of them must be directions]

(c) If we know the two direction of any two vectors then we will put them on the same side and other on the different side.

For example 

If we know the directions of Resolution of Vectors Class 11 Notes | EduRev and and Resolution of Vectors Class 11 Notes | EduRev direction is unknown then we make equation as follows:-

 Resolution of Vectors Class 11 Notes | EduRev

(d) Then we make vector diagram according to the equation and resolve the vectors to know the unknown values.

Ex.10 Find the net displacement of a particle from its starting point if it undergoes two successive displacements given by Resolution of Vectors Class 11 Notes | EduRev, 37° North of West, Resolution of Vectors Class 11 Notes | EduRev, 53° North of East 

Sol. Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev
Angle from west - east axis (x - axis)
Resolution of Vectors Class 11 Notes | EduRev

Ex.11 Find magnitude of Resolution of Vectors Class 11 Notes | EduRev and direction of Resolution of Vectors Class 11 Notes | EduRev . If Resolution of Vectors Class 11 Notes | EduRev makes angle 37° and Resolution of Vectors Class 11 Notes | EduRev makes 53° with x axis and Resolution of Vectors Class 11 Notes | EduRev has magnitude equal to 10 and Resolution of Vectors Class 11 Notes | EduRev has 5. (given Resolution of Vectors Class 11 Notes | EduRev) 

Sol.

Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev

B = 5 (magnitude can not be negative) & Angle made by A
Resolution of Vectors Class 11 Notes | EduRev

Ex.12 Find the magnitude of F1 and F2. If F1, F2 make angle 30° and 45° with F3 and magnitude of F3 is 10 N. (given Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev) 

Sol. 

Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev

9. SHORT-METHOD

Resolution of Vectors Class 11 Notes | EduRev

If their are two vectors Resolution of Vectors Class 11 Notes | EduRev and their resultant make an angle α with Resolution of Vectors Class 11 Notes | EduRev. then A sin α = β sin β

Means component of Resolution of Vectors Class 11 Notes | EduRev perpendicular to resultant is equal in magnitude to the component of Resolution of Vectors Class 11 Notes | EduRev perpendicular to resultant.

Ex.13 If two vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev make angle 30° and 45° with their resultant and Resolution of Vectors Class 11 Notes | EduRev has magnitude equal to 10, then find magnitude of Resolution of Vectors Class 11 Notes | EduRev

Sol. B sin 60° = A sin 30°   

Resolution of Vectors Class 11 Notes | EduRev                                

⇒ 10 sin 60° = A sin 30°

⇒ A = Resolution of Vectors Class 11 Notes | EduRev

Ex.14 If Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev have angle between them equals to 60° and their resultant make, angle 45° with Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev have magnitude equal to 10. Then Find magnitude of Resolution of Vectors Class 11 Notes | EduRev. 

Sol. here a = 45° and b = 60° -45° = 15°

so A sinα = B sinβ

Resolution of Vectors Class 11 Notes | EduRev

10 sin 45° = B sin 45°

So B = Resolution of Vectors Class 11 Notes | EduRev

= Resolution of Vectors Class 11 Notes | EduRev

10. ADDITION AND SUBTRACTION IN COMPONENT FORM :

Suppose there are two vectors in component form. Then the addition and subtraction between these two are

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Also if we are having a third vector present in component form and this vector is added or subtracted from the addition or subtraction of above two vectors then

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Note : Modulus of vector A is given by

Resolution of Vectors Class 11 Notes | EduRev

Ex.15 Obtain the magnitude of Resolution of Vectors Class 11 Notes | EduRev if 

Resolution of Vectors Class 11 Notes | EduRev      and         Resolution of Vectors Class 11 Notes | EduRev 

Sol. Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev Magnitude of Resolution of Vectors Class 11 Notes | EduRev

= Resolution of Vectors Class 11 Notes | EduRevAns. 

Ex.16 Find Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev if Resolution of Vectors Class 11 Notes | EduRev make angle 37° with positive x-axis and Resolution of Vectors Class 11 Notes | EduRev make angle 53° with negative x-axis as shown and magnitude of Resolution of Vectors Class 11 Notes | EduRev is 5 and of B is 10.

Sol. Resolution of Vectors Class 11 Notes | EduRev

for Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev    +   Resolution of Vectors Class 11 Notes | EduRev    =     Resolution of Vectors Class 11 Notes | EduRev

so the magnitude of resultant will be = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev

and have angle θ = Resolution of Vectors Class 11 Notes | EduRev from negative x - axis towards up

for Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

So the magnitude of resultant will be

= Resolution of Vectors Class 11 Notes | EduRev

and have angle Resolution of Vectors Class 11 Notes | EduRev from positive x-axis towards down.

11. MULTIPLICATION OF VECTORS (The Scalar and vector products) : 

11.1 Scalar Product 

The scalar product or dot product of any two vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev, denoted as Resolution of Vectors Class 11 Notes | EduRev. Resolution of Vectors Class 11 Notes | EduRev (read Resolution of Vectors Class 11 Notes | EduRev dot Resolution of Vectors Class 11 Notes | EduRev) is defined as the product of their magnitude with cosine of angle between them.

Resolution of Vectors Class 11 Notes | EduRev

Thus,

Resolution of Vectors Class 11 Notes | EduRev (here θ is the angle between the vectors)

Properties : 

  • It is always a scalar which is positive if angle between the vectors is acute (i.e.< 90°) and negative if angle between them is obtuse (i.e., 90° < q £ 180°)
  • It is commutative i.e. Resolution of Vectors Class 11 Notes | EduRev
  • It is distributive, i.e. Resolution of Vectors Class 11 Notes | EduRev
  • As by definition Resolution of Vectors Class 11 Notes | EduRev.Resolution of Vectors Class 11 Notes | EduRev = AB cosθ . The angle between the vectors θ = Resolution of Vectors Class 11 Notes | EduRev
  • Resolution of Vectors Class 11 Notes | EduRev

Geometrically, B cosθ is the projection of Resolution of Vectors Class 11 Notes | EduRev onto Resolution of Vectors Class 11 Notes | EduRev and vice versa

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Component of Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev = B cosθ = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev (Projection of Resolution of Vectors Class 11 Notes | EduRev on Resolution of Vectors Class 11 Notes | EduRev)

Component of Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev = A cosθ = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev (Projection of Resolution of Vectors Class 11 Notes | EduRev on Resolution of Vectors Class 11 Notes | EduRev)

Resolution of Vectors Class 11 Notes | EduRev

  • Scalar product of two vectors will be maximum when cosθ = max = 1, i.e., θ = 0°,

             i.e., vectors are parallel ⇒ Resolution of Vectors Class 11 Notes | EduRev

  • If the scalar product of two non-zero vectors vanishes then the vectors are perpendicular.
  • The scalar product of a vector by itself is termed as self dot product and is given by

           Resolution of Vectors Class 11 Notes | EduRev = AA cosθ = A2Resolution of Vectors Class 11 Notes | EduRev

  • In case of unit vector Resolution of Vectors Class 11 Notes | EduRev,  
    Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev
    In case of orthogonal unit vectors,Resolution of Vectors Class 11 Notes | EduRev
    Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev                                                                                                     

 Ex.17 If the vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev are perpendicular to each other. Find the value of a? 

Sol. If vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev are perpendicular

Resolution of Vectors Class 11 Notes | EduRev     ⇒   Resolution of Vectors Class 11 Notes | EduRev

⇒ a2 -2a -3 = 0      ⇒   a2 -3a a -3 = 0

⇒ a(a -3) +1 (a -3 )    ⇒      a = -1, 3

Ex.18 Find the component of Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev ?

Sol. Component of Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev is given by Resolution of Vectors Class 11 Notes | EduRev hence required component

= Resolution of Vectors Class 11 Notes | EduRev

Ex.19 Find angle between Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev ? 

Sol. We have cosθ = Resolution of Vectors Class 11 Notes | EduRev

cosθ = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev                    θ = cos-1Resolution of Vectors Class 11 Notes | EduRev

Ex.20 (i) For what value of m the vector Resolution of Vectors Class 11 Notes | EduRev is perpendicular to Resolution of Vectors Class 11 Notes | EduRev

(ii) Find the component of vector Resolution of Vectors Class 11 Notes | EduRev along the direction of Resolution of Vectors Class 11 Notes | EduRev ? 

Sol. Resolution of Vectors Class 11 Notes | EduRev

(i) m = -10 (ii) Resolution of Vectors Class 11 Notes | EduRev

Important Note : 

Components of b along and perpendicular to a.

Let Resolution of Vectors Class 11 Notes | EduRev . Resolution of Vectors Class 11 Notes | EduRev represent two (non-zero) given vectors a, b respectively. Draw BM perpendicular to Resolution of Vectors Class 11 Notes | EduRev

From ΔOMB, Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev
Thus Resolution of Vectors Class 11 Notes | EduRev are components of b along a and perpendicular to a.

Now
Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Hence, components of b along a perpendicular to a are.
(a . b/ |a|2) a and b - (a . b / |a|2) a respectively.

Ex.21 The velocity of a particle is given by Resolution of Vectors Class 11 Notes | EduRev. Find the vector component of its velocity parallel to the line Resolution of Vectors Class 11 Notes | EduRev.

Sol. Component of Resolution of Vectors Class 11 Notes | EduRev along Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

11.2 Vector product 

The vector product or cross product of any two vectors and Resolution of Vectors Class 11 Notes | EduRev, denoted as

Resolution of Vectors Class 11 Notes | EduRev (read Resolution of Vectors Class 11 Notes | EduRev cross Resolution of Vectors Class 11 Notes | EduRev) is defined as :

Resolution of Vectors Class 11 Notes | EduRev

Here θ is the angle between the vectors and the direction Resolution of Vectors Class 11 Notes | EduRev is given by the right - hand - thumb rule.

Right - Hand - Thumb Rule : 

To find the direction of Resolution of Vectors Class 11 Notes | EduRev, draw the two vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev with both the tails coinciding. Now place your stretched right palm perpendicular to the plane of Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev in such a way that the fingers are along the vector Resolution of Vectors Class 11 Notes | EduRev and when the fingers are closed they go towards Resolution of Vectors Class 11 Notes | EduRev. The direction of the thumb gives the direction of Resolution of Vectors Class 11 Notes | EduRev.  

Resolution of Vectors Class 11 Notes | EduRev

Properties : 

  • Vector product of two vectors is always a vector perpendicular to the plane containing the two vectors i.e. orthogonal to both the vectors and , though the vectors and Resolution of Vectors Class 11 Notes | EduRev may or may not be orthogonal.
  • Vector product of two vectors is not commutative i.e. Resolution of Vectors Class 11 Notes | EduRev But Resolution of Vectors Class 11 Notes | EduRev
  • The vector product is distributive when the order of the vectors is strictly maintained i.e.Resolution of Vectors Class 11 Notes | EduRev
  • The magnitude of vector product of two vectors will be maximum when sinθ = max = 1. i.e. θ = 90°                                                                                                                 Resolution of Vectors Class 11 Notes | EduRev
  • The magnitude of vector product of two non-zero vectors will be minimum when |sinθ| = minimum = 0, i.e., θ = 0° or 180° and Resolution of Vectors Class 11 Notes | EduRev i.e., if the vector product of two non-zero vectors vanishes, the vectors are collinear.
  • The self cross product i.e. product of a vector by itself vanishes i.e. is a null vector.Resolution of Vectors Class 11 Notes | EduRev
  • In case of unit vector Resolution of Vectors Class 11 Notes | EduRev, Resolution of Vectors Class 11 Notes | EduRev ⇒ Resolution of Vectors Class 11 Notes | EduRev
  • In case of orthogonal unit vectors Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev in accordance with right-hand-thumb-rule,
    Resolution of Vectors Class 11 Notes | EduRev
  • In terms of components. Resolution of Vectors Class 11 Notes | EduRev
    Resolution of Vectors Class 11 Notes | EduRev
     

Ex.22 Resolution of Vectors Class 11 Notes | EduRev is East wards and Resolution of Vectors Class 11 Notes | EduRev is downwards. Find the direction of Resolution of Vectors Class 11 Notes | EduRev × Resolution of Vectors Class 11 Notes | EduRev ? 

Sol. Applying right hand thumb rule we find that Resolution of Vectors Class 11 Notes | EduRev is along North.

Ex.23 If Resolution of Vectors Class 11 Notes | EduRev, find angle between Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev 

Sol. Resolution of Vectors Class 11 Notes | EduRev AB cosθ = AB sinθ            tanθ = 1                ⇒ θ = 45°

Ex.24 Resolution of Vectors Class 11 Notes | EduRev ⇒ Resolution of Vectors Class 11 Notes | EduRev here Resolution of Vectors Class 11 Notes | EduRev is perpendicular to both Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev 

Ex.25 Find Resolution of Vectors Class 11 Notes | EduRev if Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev

Sol. Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev

Ex.26 (i) Resolution of Vectors Class 11 Notes | EduRev is North-East and Resolution of Vectors Class 11 Notes | EduRev is down wards, find the direction of Resolution of Vectors Class 11 Notes | EduRev

(ii) Find Resolution of Vectors Class 11 Notes | EduRev × Resolution of Vectors Class 11 Notes | EduRev if Resolution of Vectors Class 11 Notes | EduRev and Resolution of Vectors Class 11 Notes | EduRev 

Ans. (i) North - West.                 (ii) Resolution of Vectors Class 11 Notes | EduRev

12. POSITION VECTOR : 

Position vector for a point is vector for which tail is origin & head is the given point itself.

Position vector of a point defines the position of the point w.r.t. the origin.

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

13. DISPLACEMENT VECTOR :

Resolution of Vectors Class 11 Notes | EduRev
Change in position vector of particle is known as displacement vector.

Resolution of Vectors Class 11 Notes | EduRev
Thus we can represent a vector in space starting from (x , yj & ending at

Resolution of Vectors Class 11 Notes | EduRev

CALCULUS 

14. Constants : They are fixed real number which value does not change

Ex. 3, e, a, -1, etc.

15. Variable : 

Something that is likely to vary, something that is subject to variation.

or

A quantity that can assume any of a set of value.

Types of variables. 

(i) Independent variables : Independent variables is typically the variable being manipulated or changed

(ii) dependent variables : The dependent variables is the object result of the independent variable being manipulated.

Ex. y = x2

here y is dependent variable and x is independent variable

16. FUNCTION : 

Function is a rule of relationship between two variables in which one is assumed to be dependent and the other independent variable.

The temperatures at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). Here elevation above sea level is the independent & temperature is the dependent variable.

The interest paid on a cash investment depends on the length of time the investment is held. Here time is the independent and interest is the dependent variable.

In each case, the value of one variable quantity (dependent variable), which we might call y, depends on the value of another variable quantity (independent variable), which we might call x. Since the value of y is completely determined by the value of x, we say that y is a function of x and represent it mathematically as y = f(x).

Resolution of Vectors Class 11 Notes | EduRev

all possible values of independent variables (x) are called domain of function.

all possible values of dependent variable (y) are called Range of function.

Think of function f as a kind machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain (figure).

When we study circles, we usually call the area A and the radius r. Since area depends on radius, we say that A is a function of r, A = f(r). The equation A = πr2 is a rule that tells how to calculate a unique (single) output value of A for each possible input value of the radius r.

A = f(x) = πr2. (Here the rule of relationship which describes the function may be described as square & multiply by π)

if    r = 1   A = π

if    r = 2   A = 4π

if    r = 3    A = 9π

The set of all possible input values for the radius is called the domain of the function. The set of all output values of the area is the range of the function.

We usually denote functions in one of the two ways :

1. By giving a formula such as y = x2 that uses a dependent variable y to denote the value of the function.

2. By giving a formula such as f(x) =x2 that defines a functions symbols f to name the function.

Strictly speaking, we should call the function f and not f(x).

y = sin x. Here the function is y since, x is the independent variable.

Ex.27 The volume V of ball (solid sphere) of radius r is given by the function V(r) = Resolution of Vectors Class 11 Notes | EduRev

The volume of a ball of radius 3m is? 

Sol. V(3) = Resolution of Vectors Class 11 Notes | EduRev = 36 pm3.

Ex.28 Suppose that the function F is defined for all real numbers r by the formula. 

F(r) = 2 (r -1) +3. 

Evaluate F at the input values 0, 2 x 2, and F(2). 

Sol. In each case we substitute the given input value for r into the formula for F:

F(0) = 2(0 -1) + 3 = -2 + 3 = 1

F(2) = 2(2 -1) + 3 = 2 + 3 =5

F(x + 2) = 2 (x + 2 -1) + 3 = 2x + 5

F(F(2)) = F(5) = 2(5 -1) 3 = 11

 

Ex. 29 function f(x) is defined as f(x) = x2 + 3, Find f(0), f(l), f(x>), f(x + 1) and f(f(l))

Sol.  f(0) = 02 + 3  = 3
f(1)  =  l2 + 3 = 4
f(x2) =  (x2)2 +3  = x4 + 4
f(x +1)  =  (x + 1)2  + 3   = x2 + 2x + 4
= f(4)  = 42+3  = 19

17. Differentiation

Finite difference : 

The finite difference between two values of a physical is represented by Δ notation.

For example :

Difference in two values of y is written as Δy as given in the table below.

y2

100

100

100

y1

50

99

99.5

Δy = y2 - y1

50

1

0.5


Infinitely small difference : 

The infinitely small difference means very-very small difference. And this difference is represented by 'd' notation instead of 'D'.

For example infinitely small difference in the values of y is written as 'dy'

if y2 = 100 and y1 = 99.9999999999999.....

then dy = 0.00000000000000..........00001

Definition of differentiation 

Another name of differentiation is derivative. Suppose y is a function of x or y = f(x)

Differentiation of y with respect to x is denoted by symbols f' (x)

where f'(x) = Resolution of Vectors Class 11 Notes | EduRev; dx is very small change in x and dy is corresponding very small change in y.

Notation : There are many ways to denote the derivative of function y = f(x), the most common notations are these :
 

y

"y prime"

Nice and brief and does not name the independent variable

dy/dx

" dy by dx"

Names the variables and uses d for derisive

df/dx

" df by dx"

Emphasizes the function's name

Resolution of Vectors Class 11 Notes | EduRev

” d by dx of f"

Emphasizes the idea that differentiation is an operation performed on f.

Dxf

" dx of f"

A common operator notation

Resolution of Vectors Class 11 Notes | EduRev

” y dot"

One of Newton's notations, now common for time derivative i.e. dy/dt

 

Average rates of change : 

Given an arbitrary function y = f(x) we calculate the average rate of change of y with respect to x over the interval (x, x +Δx) by dividing the change in value of y, i.e., Dy = f(x+Δx) -f(x), by length of interval Δx over which the change occurred.

The average rate of change of y with respect to x over the interval [x, x+Δx]

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Geometrically

Resolution of Vectors Class 11 Notes | EduRev = tanθ = Slope of the line PQ

In triangle QPR tanθ = Resolution of Vectors Class 11 Notes | EduRev

therefore we can say that average rate of change of y with respect to x is equal to slope of the line joining P & Q.

The derivative of a function 

We know that Average rate of change of y w.r.t x is -

Resolution of Vectors Class 11 Notes | EduRev

If the limit of this ratio exists as Δx → 0, then it is called the derivative of given function f(x) and is denoted as

Resolution of Vectors Class 11 Notes | EduRev

18. GEOMETRICAL MEANING OF DIFFERENTIATION : 

The geometrical meaning of differentiation is very much useful in the analysis of graphs in physics. To understand the geometrical meaning of derivatives we should have knowledge of secant and tangent to a curve.

Secant and Tangent to a Curve 

Secant : - A secant to a curve is a straight line, which intersects the curve at any two points.

Resolution of Vectors Class 11 Notes | EduRev

Tangent : 

A tangent is a straight line, which touches the curve a particular point. Tangent is limiting case of secant which intersects the curve at two overlapping points.

Resolution of Vectors Class 11 Notes | EduRev

In the figure - 1 shown, if value of Δx has gradually reduced then the point Q will move nearer to the point P. If the process is continuously repeated (Figure-2) value of Δx will be infinitely small and secant PQ to the given curve will become a tangent at point P.

Therefore

Resolution of Vectors Class 11 Notes | EduRev

we can say that differentiation of y with respect to x, i.e. Resolution of Vectors Class 11 Notes | EduRev is equal to slope of the tangent at point P (x,y)

or tanθ = Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

(From fig-1 the average rate change of y from x to x+Δx is identical with the slope of secant PQ)

Rule No. 1 Derivative Of A Constant

The first rule of differentiation is that the derivative of every constant function is zero.

If c is constant, then Resolution of Vectors Class 11 Notes | EduRev

Ex.30Resolution of Vectors Class 11 Notes | EduRev, Resolution of Vectors Class 11 Notes | EduRev, Resolution of Vectors Class 11 Notes | EduRev

Rule No.2 Power Rule

If n is a real number, then Resolution of Vectors Class 11 Notes | EduRev

To apply the power Rule, we subtract 1 from the original exponent (n) and multiply the result by n.

Ex.31Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev

Function defined for x > 0 derivative defined only for x > 0
Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Function defined for x > 0 derivative not defined at x = 0

Rule No.3 The Constant Multiple Rule

If u is a differentiable function of x, and c is a constant, then Resolution of Vectors Class 11 Notes | EduRev

In particular, if n is a positive integer, then Resolution of Vectors Class 11 Notes | EduRev

Ex.34 The derivative formula 

Resolution of Vectors Class 11 Notes | EduRev

says that if we rescale the graph of y = x2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3. 

Ex.35 A useful special case

The derivative of the negative of a differentiable function is the negative of the function's derivative. Rule 3 with c = -1 gives.

Resolution of Vectors Class 11 Notes | EduRev

Rule No.4 The Sum Rule

The derivative of the sum of two differentiable functions is the sum of their derivatives.

If u and v are differentiable functions of x, then their sum u+v is differentiable at every point where u and v are both differentiable functions in their derivatives.

Resolution of Vectors Class 11 Notes | EduRev

The sum Rule also extends to sums of more than two functions, as long as there are only finite functions in the sum. If u1, u2, ........ un is differentiable at x, then so if u1+u2 ....... +un, then

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev

 Notice that we can differentiate any polynomial term by term, the way we differentiated the polynomials in above example.

Rule No. 5 The Product Rule 

If u and v are differentiable at x, then if their product uv is considered, then Resolution of Vectors Class 11 Notes | EduRev.

The derivative of the product uv is u times the derivative of v plus v times the derivative of u. In prime notation

(uv)' = uv' + vu'.

While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product of two functions is not the product of their derivatives. For instance,

Resolution of Vectors Class 11 Notes | EduRev    while Resolution of Vectors Class 11 Notes | EduRev, which is wrong

Ex.37 Find the derivatives of y = (x2+1) (x3+3)  

Sol. Using the product Rule with u = x2+1 and v = x3+3, we find

Resolution of Vectors Class 11 Notes | EduRev = (x2+1) (3x2) + (x3+3) (2x)

= 3x4 + 3x2 + 2x4 + 6x = 5x4 + 3x2 + 6x

Example can be done as well (perhaps better) by multiplying out the original expression for y and differentiating the resulting polynomial. We now check :

y = (x2 + 1) (x3 + 3) = x5 + x3 + 3x2 + 3

 Resolution of Vectors Class 11 Notes | EduRev = 5x4 + 3x2 + 6x

This is in agreement with our first calculation.

There are times, however, when the product Rule must be used. In the following examples. We have only numerical values to work with.

Ex.38 Let y = uv be the product of the functions u and v. Find y'(2) if u(2) = 3, u'(2) = -4, v(2) = 1, and v'(2) = 2. 

Sol.

From the Product Rule, in the form y' = (uv)' = uv' + vu',
we have y'(2) = u(2) v'(2) + v(2) u'(2)
= (3) (2) + (1) (-4) = 6-4 = 2

Rule No.6 The Quotient Rule

If u and v are differentiable at x, and v(x) ¹ 0, then the quotient u/v is differentiable at x,

and Resolution of Vectors Class 11 Notes | EduRev

Just as the derivative of the product of two differentiable functions is not the product of their derivatives, the derivative of the quotient of two functions is not the quotient of their derivatives.

Ex.39 Find the derivative of Resolution of Vectors Class 11 Notes | EduRev 

Sol. We apply the Quotient Rule with u = t2 -1 and v = t2 1

Resolution of Vectors Class 11 Notes | EduRev           Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Rule No. 7 Derivative Of Sine Function

Resolution of Vectors Class 11 Notes | EduRev

Ex.40 Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev 

Rules No.8 Derivative Of Cosine Function

Resolution of Vectors Class 11 Notes | EduRev

Ex.41 (a) y = 5x + cos x           Sum Rule

Resolution of Vectors Class 11 Notes | EduRev    Product Rule
 = sin x(— sin x) + cos x (cos x)
 = cos2 x - sin2 x - cos 2x

Rule No. 9 Derivatives Of Other Trigonometric Functions 

Because sin x and cos x are differentiable functions of x, the related functions

Resolution of Vectors Class 11 Notes | EduRev ;    Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev ;    Resolution of Vectors Class 11 Notes | EduRev

are differentiable at every value of x at which they are defined. There derivatives, Calculated from the Quotient Rule, are given by the following formulas.

Resolution of Vectors Class 11 Notes | EduRev ;   Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev ;    Resolution of Vectors Class 11 Notes | EduRev

Ex.42 Find dy / dx if y = tan x. 

Sol.Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Ex. 43 
Resolution of Vectors Class 11 Notes | EduRev

Rule No. 10 Derivative Of Logarithm And Exponential Functions

Resolution of Vectors Class 11 Notes | EduRev ,                        Resolution of Vectors Class 11 Notes | EduRev

Ex.44 y = ex . loge (x)

Resolution of Vectors Class 11 Notes | EduRev   ⇒   Resolution of Vectors Class 11 Notes | EduRev

Rule No. 11 Chain Rule Or `Outside Inside' Rule

Resolution of Vectors Class 11 Notes | EduRev

It sometimes helps to think about the Chain Rule the following way. If y = f(g(x)),

Resolution of Vectors Class 11 Notes | EduRev = f'[g(x)] . g'(x)

In words : To find dy/dx, differentiate the "outside" function f and leave the "inside" g(x) alone; then multiply by the derivative of the inside.

We now know how to differentiate sin x and x2 -4, but how do we differentiate a composite like sin(x2 -4)?

The answer is, with the Chain Rule, which says that the derivative of the composite of two differentiable functions is the product of their derivatives evaluated at appropriate points. The Chain Rule is probably the most widely used differentiation rule in mathematics. This section describes the rule and how to use it. We begin with examples.

Ex.45 The function y = 6x -10 = 2(3x -5) is the composite of the functions y = 2u and u = 3x -5. How are the derivatives of these three functions related ?

Sol. We have Resolution of Vectors Class 11 Notes | EduRev, Resolution of Vectors Class 11 Notes | EduRev, Resolution of Vectors Class 11 Notes | EduRev

Since 6 = 2 × 3 Resolution of Vectors Class 11 Notes | EduRev

Is it an accident that Resolution of Vectors Class 11 Notes | EduRev ?

If we think of the derivative as a rate of change, our intuitions allows us to see that this relationship is reasonable. For y = f(u) and u = g(x), if y changes twice as fast as u and u changes three times as fast as x, then we expect y to change six times as fast as x.

Ex.46 Let us try this again on another function. 

y = 9x4 +6x2 +1 = (3x2 +1)2

is the composite y = u2 and u = 3x2 + 1. Calculating derivatives. We see that

Resolution of Vectors Class 11 Notes | EduRev = 2 (3x2 + 1). 6x = 36x3 + 12 x

and Resolution of Vectors Class 11 Notes | EduRev = 36 x3 + 12 x

Once again, Resolution of Vectors Class 11 Notes | EduRev

The derivative of the composite function f(g(x)) at x is the derivative of f at g(x) times the derivative of g at x.

Ex.47 Find the derivation of Resolution of Vectors Class 11 Notes | EduRev 

Sol. Here y = f(g(x)), where f(u) = Resolution of Vectors Class 11 Notes | EduRev and u = g(x) = x2 + 1. Since the derivatives of f and g are

f' (u) = Resolution of Vectors Class 11 Notes | EduRev and g'(x) = 2x,

the Chain Rule gives

Resolution of Vectors Class 11 Notes | EduRev = f' (g(x)).g'(x) = Resolution of Vectors Class 11 Notes | EduRev.g'(x) = Resolution of Vectors Class 11 Notes | EduRev. (2x) = Resolution of Vectors Class 11 Notes | EduRev

Ex.48

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Ex. 49   Resolution of Vectors Class 11 Notes | EduRev u = 1 - x2 and n = 1/4
   (Function defined) on [-1, 1]

  Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

Rule No. 12 Power Chain Rule 

If Resolution of Vectors Class 11 Notes | EduRev

Ex.50 Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev = -1 (3x -2)-2Resolution of Vectors Class 11 Notes | EduRev

= -1 (3x -2)-2 (3) = -Resolution of Vectors Class 11 Notes | EduRev

In part (d) we could also have found the derivation with the Quotient Rule.

Ex.51 (a)Resolution of Vectors Class 11 Notes | EduRev

Sol. Here u = Ax B, Resolution of Vectors Class 11 Notes | EduRev

(b)Resolution of Vectors Class 11 Notes | EduRev
(c)Resolution of Vectors Class 11 Notes | EduRevlog(Ax B) = Resolution of Vectors Class 11 Notes | EduRev.A

(d)Resolution of Vectors Class 11 Notes | EduRevtan (Ax + B) = sec2 (Ax + B).A
(e)Resolution of Vectors Class 11 Notes | EduRev

Note : These results are important

19. DOUBLE DIFFERENTIATION 

If f is differentiable function, then its derivative f' is also a function, so f' may have a derivative of its own, denoted by Resolution of Vectors Class 11 Notes | EduRev. This new function f'' is called the second derivative of because it is the derivative of the derivative of f. Using Leibniz notation, we write the second derivative of y = f(x) as

Resolution of Vectors Class 11 Notes | EduRev

Another notation is f''(x) = D2 f(x).

Ex.52 If(x) = x cos x, find f'' (x)

Sol. Using the Product Rule, we have f'(x)   Resolution of Vectors Class 11 Notes | EduRev

To find f" (x) we differentiate f'(x)

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

= - x cos x - sinx - sinx = - x cos x - 2 sin x

20. Application of derivative Differentiation as a rate of change

Resolution of Vectors Class 11 Notes | EduRev is rate of change of 'y' with respect to 'x' :

For examples :

(i) v = Resolution of Vectors Class 11 Notes | EduRev this means velocity 'v' is rate of change of displacement 'x' with respect to time 't'

(ii) a = Resolution of Vectors Class 11 Notes | EduRev this means acceleration 'a' is rate of change of velocity 'v' with respect to time 't'.

(iii) Resolution of Vectors Class 11 Notes | EduRev this means force 'F' is rate of change of momentum 'p' with respect to time 't'.

(iv) Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev this means torque 't' is rate of change of angular momentum 'L' with respect to time 't'

(v) Power = Resolution of Vectors Class 11 Notes | EduRev this means power 'P' is rate of change of work 'W' with respect to time 't'

Ex.53 The area A of a circle is related to its diameter by the equation Resolution of Vectors Class 11 Notes | EduRev.

How fast is the area changing with respect to the diameter when the diameter is 10 m ?

Sol. The (instantaneous) rate of change of the area with respect to the diameter is

Resolution of Vectors Class 11 Notes | EduRev

When D =10m, the area is changing at rate (π/2) = 5π m2/m. This means that a small change ΔD m in the diameter would result in a change of about 5p ΔD m2 in the area of the circle.

Physical Example : 

Ex.54 Boyle's Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant : PV = C. Find the rate of change of volume with respect to pressure. 

Sol. Resolution of Vectors Class 11 Notes | EduRev

Ex.55 (a) Find the average rate of change of the area of a circle with respect to its radius r as r changed from 

(i) 2 to 3                      (ii) 2 to 2.5                     (iii) 2 to 2.1 

(b) Find the instantaneous rate of change when r = 2. 

(c) Show that there rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically when this is true by drawing a circle whose radius is increased by an amount Δr. How can you approximate the resulting change in area ΔA if Δr is small ? 

Sol. (a) (i) 5π (ii) 4.5 π (iii) 4.1 π

(b) 4π

(c) ΔA ≈ 2 πrΔr

21. MAXIMA & MINIMA

Suppose a quantity y depends on another quantity x in a manner shown in figure. It becomes maximum at x1 and minimum at x2. At these points the tangent to the curve is parallel to the x-axis and hence its slope is tanθ = 0. Thus, at a maximum or a minima slope

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Maxima 

Just before the maximum the slope is positive, at the maximum it is zero and just after the maximum it is negative. Thus, Resolution of Vectors Class 11 Notes | EduRev decrease at a maximum and hence the rate of change of Resolution of Vectors Class 11 Notes | EduRev is negative at a maximum i.e., Resolution of Vectors Class 11 Notes | EduRev at maximum. The quantity Resolution of Vectors Class 11 Notes | EduRev is the rate of change of the slope. It is written as Resolution of Vectors Class 11 Notes | EduRev. Conditions for maxima are : (a) Resolution of Vectors Class 11 Notes | EduRev (b) Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Minima 

Similarly, at a minimum the slope changes from negative to positive, Hence with the increases of x. The slope is increasing that means the rate of change of slope with respect to x is positive.

Hence Resolution of Vectors Class 11 Notes | EduRev

Conditions for minima are : 

(a) Resolution of Vectors Class 11 Notes | EduRev                  (b) Resolution of Vectors Class 11 Notes | EduRev

Quite often it is known from the physical situation whether the quantity is a maximum or a minimum. The test on Resolution of Vectors Class 11 Notes | EduRev may then be omitted.

Ex.56 Find maximum or minimum values of the functions : 

(A) y = 25x2 + 5 -10x          (B) y = 9 -(x -3)2 

Sol. (A) For maximum and minimum value, we can put Resolution of Vectors Class 11 Notes | EduRev

or Resolution of Vectors Class 11 Notes | EduRev                     Resolution of Vectors Class 11 Notes | EduRev x = Resolution of Vectors Class 11 Notes | EduRev

Further, Resolution of Vectors Class 11 Notes | EduRev

or Resolution of Vectors Class 11 Notes | EduRev has positive value at x = Resolution of Vectors Class 11 Notes | EduRev. Therefore, y has minimum value at x = Resolution of Vectors Class 11 Notes | EduRev. Therefore, y has minimum value at x = Resolution of Vectors Class 11 Notes | EduRev. Substituting x =Resolution of Vectors Class 11 Notes | EduRev in given equation, we get

ymin = Resolution of Vectors Class 11 Notes | EduRev

(B) y = 9 -(x -3)2 = 9 -x2 +-9 6x

or y = 6x -x2

Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev

For minimum or maximum value of y we will substitute Resolution of Vectors Class 11 Notes | EduRev

or 6 -2x = 0

x = 3

To check whether value of y is maximum or minimum at x = 3 we will have to check whether Resolution of Vectors Class 11 Notes | EduRev is positive or negative.

Resolution of Vectors Class 11 Notes | EduRev

or Resolution of Vectors Class 11 Notes | EduRev is negative at x = 3. Hence, value of y is maximum. This maximum value of y is,

ymax = 9 -(3 -3)2 = 9

22. INTEGRATION 

Definitions : 

A function F(x) is a antiderivative of a function f(x) if

F'(x) = f(x)

for all x in the domain of f. The set of all antiderivatives of f is the indefinite integral of f with respect to x, denoted by

 Resolution of Vectors Class 11 Notes | EduRev

The symbol Resolution of Vectors Class 11 Notes | EduRev is an integral sign. The function f is the integrand of the integral and x is the variable of integration.

For example f(x) = x3 then f'(x) = 3x2

So the integral of 3x2 is x3

Similarly if f(x) = x3 + 4

there for the general integral of 3x2 is x3 + c where c is a constant

One antiderivative F of a function f, the other antiderivatives of f differ from F by a constant. We indicate this in integral notation in the following way :

Resolution of Vectors Class 11 Notes | EduRev                 .....(i)

The constant C is the constant of integration or arbitrary constant, Equation (1) is read, "The indefinite integral of f with respect to x is F(x) + C." When we find F(x) + C, we say that we have integrated f and evaluated the integral.

Ex.57 Evaluate Resolution of Vectors Class 11 Notes | EduRev

Sol.Resolution of Vectors Class 11 Notes | EduRev

The formula x2 + C generates all the antiderivatives of the function 2x. The function x2+ 1, x2 -π, and

x2+Resolution of Vectors Class 11 Notes | EduRev are all antiderivatives of the function 2x, as you can check by differentiation.

Many of the indefinite integrals needed in scientific work are found by reversing derivative formulas.

Integral Formulas

Indefinite Integral    Reversed derivated formula

1. Resolution of Vectors Class 11 Notes | EduRev,n ¹ -1, n rational    

Resolution of Vectors Class 11 Notes | EduRev (special case)

Resolution of Vectors Class 11 Notes | EduRev = xn

Resolution of Vectors Class 11 Notes | EduRev

2. Resolution of Vectors Class 11 Notes | EduRev  Resolution of Vectors Class 11 Notes | EduRev
3. Resolution of Vectors Class 11 Notes | EduRev    Resolution of Vectors Class 11 Notes | EduRev
4. Resolution of Vectors Class 11 Notes | EduRev  Resolution of Vectors Class 11 Notes | EduRev
5. Resolution of Vectors Class 11 Notes | EduRev Resolution of Vectors Class 11 Notes | EduRev(-cot x) = cosec2 x
6. Resolution of Vectors Class 11 Notes | EduRevResolution of Vectors Class 11 Notes | EduRev = sec x tan x
7. Resolution of Vectors Class 11 Notes | EduRev = -cosec x +CResolution of Vectors Class 11 Notes | EduRev(-cosec x) = cosec x cot x

 Ex.58 Examples based on above formulas :

(a) Resolution of Vectors Class 11 Notes | EduRev

(b) Resolution of Vectors Class 11 Notes | EduRev                                                                    Formula 1 with n = 5

(c) Resolution of Vectors Class 11 Notes | EduRev                           Formula 1 with n = Resolution of Vectors Class 11 Notes | EduRev

(d) Resolution of Vectors Class 11 Notes | EduRev                                                      Formula 2 with k = 2

(e) Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev     Formula 3 with k = Resolution of Vectors Class 11 Notes | EduRev

Ex.59 Right :Resolution of Vectors Class 11 Notes | EduRev = x sin x + cos x C

Reason : The derivative of the right-hand side is the integrand :

Check : Resolution of Vectors Class 11 Notes | EduRev = x cos x + sin x -sin x + 0 = x cos x.

Wrong :Resolution of Vectors Class 11 Notes | EduRev = x sin x +C

Reason : The derivative of the right-hand side is not the integrand :

Check :Resolution of Vectors Class 11 Notes | EduRev = x cos x + sin x + 0 Resolution of Vectors Class 11 Notes | EduRev x cos x

 

Rule No. 1 Constant Multiple Rule

  • A function is an anti derivative of a constant multiple k of a function f if and only if it is k times an antiderivative of f.

                 Resolution of Vectors Class 11 Notes | EduRev
Ex.60 Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev 

 

Rule No.2   Sum And Difference Rule

  • A function is an anti derivative of a sum or difference f ± g if and only if it is the sum or difference of an anti derivative of f an anti derivative of g.

                 Resolution of Vectors Class 11 Notes | EduRev               
Ex.61 Term-by-term integration 

Evaluate : Resolution of Vectors Class 11 Notes | EduRev 

Sol. If we recognize that (x3/3) -x2 +5x is an anti derivative of x2 -2x +5, we can evaluate the integral as

Resolution of Vectors Class 11 Notes | EduRev

If we do not recognize the anti derivative right away, we can generate it term by term with the sum and difference Rule :

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

This formula is more complicated than it needs to be. If we combine C1, C2 and C3 into a single constant

C = C1 + C2 + C3, the formula simplifies to

Resolution of Vectors Class 11 Notes | EduRev

and still gives all the anti derivatives there are. For this reason we recommend that you go right to the final form even if you elect to integrate term by term. Write
Resolution of Vectors Class 11 Notes | EduRev

Find the simplest anti derivative you can for each part add the constant at the end.

Ex.62 We can sometimes use trigonometric identities to transform integrals we do not know how to evaluate into integrals. The integral formulas for sin2 x and cos2 x arise frequently in applications.

(a) Resolution of Vectors Class 11 Notes | EduRev   = Resolution of Vectors Class 11 Notes | EduRev Resolution of Vectors Class 11 Notes | EduRev
  = Resolution of Vectors Class 11 Notes | EduRev
 Resolution of Vectors Class 11 Notes | EduRev

(b) Resolution of Vectors Class 11 Notes | EduRev    = Resolution of Vectors Class 11 Notes | EduRev  Resolution of Vectors Class 11 Notes | EduRev
Resolution of Vectors Class 11 Notes | EduRev As in part (a), but with a sign change

23. Some Indefinite integrals (An arbitrary constant should be added to each of these integrals. 

(a) Resolution of Vectors Class 11 Notes | EduRev (provided n ¹ --1) C    
(b) Resolution of Vectors Class 11 Notes | EduRev

(c) Resolution of Vectors Class 11 Notes | EduRev 
(d) Resolution of Vectors Class 11 Notes | EduRev
(e) Resolution of Vectors Class 11 Notes | EduRev   
(f)Resolution of Vectors Class 11 Notes | EduRev 

Ex.63 (a) Resolution of Vectors Class 11 Notes | EduRev 
(b)Resolution of Vectors Class 11 Notes | EduRev

(c) Resolution of Vectors Class 11 Notes | EduRev 
(d)Resolution of Vectors Class 11 Notes | EduRev

(e) Resolution of Vectors Class 11 Notes | EduRev 
(f) Resolution of Vectors Class 11 Notes | EduRev

(g) Resolution of Vectors Class 11 Notes | EduRev     
(h) Resolution of Vectors Class 11 Notes | EduRev

24. DEFINITE INTEGRATION OR INTEGRATION WITH LIMITS 

Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

Ex.64 Resolution of Vectors Class 11 Notes | EduRev  = 3 [4 -(-1)] = (3) (5) = 15

Resolution of Vectors Class 11 Notes | EduRev    = Resolution of Vectors Class 11 Notes | EduRev + cos (0) = -0 + 1 = 1

Ex.65 (1) Resolution of Vectors Class 11 Notes | EduRev

(2) Resolution of Vectors Class 11 Notes | EduRev

(3) Resolution of Vectors Class 11 Notes | EduRev

25. APPLICATION OF DEFINITE INTEGRAL 

Calculation Of Area Of A Curve. 

Resolution of Vectors Class 11 Notes | EduRev

From graph shown in figure if we divide whole area in infinitely small strips of dx width.

We take a strip at x position of dx width.

Small area of this strip dA = f(x) dx

So, the total area between the curve and x-axis = sum of area of all strips = Resolution of Vectors Class 11 Notes | EduRev

Let f(x) > 0 be continuous on [a,b]. The area of the region between the graph of f and the x-axis is

Resolution of Vectors Class 11 Notes | EduRev

Ex.66 Using an area to evaluate a definite integral 

Evaluate Resolution of Vectors Class 11 Notes | EduRev 0 < a < b. 

Sol. We sketch the region under the curve y = x, a £ x £ b (figure) and see that it is a trapezoid with height (b -a) and bases a and b.

Resolution of Vectors Class 11 Notes | EduRev

The value of the integral is the area of this trapezoid :

Thus =

Resolution of Vectors Class 11 Notes | EduRev

and so on.

Notice that x2/2 is an antiderivative of x, further evidence of a connection between antiderivatives and summation.

(i) To find impulse

Resolution of Vectors Class 11 Notes | EduRev so implies = Resolution of Vectors Class 11 Notes | EduRev

Ex.67 If F = kt then find impulse at t = 3 sec.

so impulse will be area under f - t curve

Resolution of Vectors Class 11 Notes | EduRev                               

Resolution of Vectors Class 11 Notes | EduRev = Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

2. To calculate work done by force :

 Resolution of Vectors Class 11 Notes | EduRev

Resolution of Vectors Class 11 Notes | EduRev

So area under f - x curve will give the value of work done.

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