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Sine Function The function f(x) = sin(x) is a continuous function over its entire domain, with its domain consisting of all the real numbers. The range of this function is [-1, 1] as can be seen in the graph below: So, if the limit of sine function is calculated at any given real number it’s always defined and lies between [-1, 1]. f(x) = sin(x) limx→a f(x) = limx→a sin(x)                  = sin(a), where a is a real number Cosine Function The function f(x) = cos(x) is a continuous function over its entire domain, with its domain consisting of all the real numbers. The range of this function is [-1, 1] as can be seen in the graph below: So, if the limit of the cosine function is calculated at any given real number it’s always defined and lies between [-1, 1]. f(x) = cos(x) lim x→a  f(x) = lim x→a  cos(x)                    = cos(a), where a is a real number Ta...

13.3.1 Algebra of limits This section demonstrates the output of sum, difference, product and quotient of limits. Let p and q be two functions such that their limits limx→a p(x) and limx→a q(x) exist. Limit of the sum of two functions is the sum of the limits of the functions. limx→a [p(x) + q(x)] = limx→a p(x) + limx→a q(x). Limit of the difference of two functions is the difference of the limits of the functions. limx→a [p(x) − q(x)] = limx→a p(x) − limx→a q(x). Limit of product of two functions is the product of the limits of the functions. limx→a [p(x) × q(x)] = [limx→a p(x)] × [limx→a q(x)]. Limit of quotient of two functions is the quotient of the limits of the functions. limx→a [p(x) ÷ q(x)] = [limx→a p(x)] ÷ [limx→a q(x)]. Limit of product of a function p(x) with a constant, q(x) = α is α times the limit of p(x). limx→a [α.p(x))] = α. limx→a p(x). 13.3.2 Limits of polynomials and rational functions The limits of polynomials and rational functions are elaborated along with solve...

Trigonometric Functions of Sum and Product of two ANGLEs

In our daily life, the word work has different meanings. Mental work, hardwork, etc.But physics word 'work' is used whenever there is a displacement for a particle or for a body under action of a force. Scalar product ( dot product) Scalar product of two vectors Ā and B can be defined as Ā.B.  It is read as  Ā dot B and hence also called dot product of Ā and B. It is scalar quantity, hence called scalar product. The dot product is commuting. i.e,  A. B=B. A If A and B are magnitudes of vectors and θ the angle between A  and B   then   A. B= ABcosθ case I when θ=0⁰ A.B= ABcos0= AB i.e when the two vectors are in the same direction the dot product will be maximum case II θ=90⁰ A.B = ABCOS0 when two vectors are at right angles  the dot product will be zero case III θ=180⁰ A.B = ABCOS180=-AB When the two vectors are in opposite directions the dot product will be negative and minimum SCALAR PRODUCT OF ORTHOGONAL UNIT VECTORS Let i, j and k be the unit ...

If an object of mass m has velocity v, its kinetic energy K is K=1/2(mv.v) = 1/2(mv²) KE is scalar quantity. Kinetic energy of an object is a measure of the work an object can do by virtue of its motion Workdone by variable energy 

Kinetic Energy Workdone by variable force A body moving from xᵢ to xf under variable force. Graph shows variation of force with position. Consider Δx=AB, The force in this interval is nearly constant  ie; ΔW=F(x)Δx F(x)Δx  Area of rectangle when we add successive rectangular areas, we get total work as W = ∑ˣₓᵢ F(x)Δx When we take Δx tends to zero the summation can be replaced integration 

HOOKS LAW The restoring force developed in the spring is proportional to the displacement x and it os opposite to the displacement ie, Fα -x F=-kx Where k is a constant called spring constant. Potential energy stored in spring C onsider a massless spring fixed to a rigid support at one end and body attached to the other end. The body moves on frictionless surface. If a body is displaced by a distance dx, The workdone for this displacement dw=Fdx The total workdone to move the body from x=0 to x. This workdone is stored a potential energy in a spring. P. E.= 1/2 (kx²) Spring force is conservative force If the spring is displaced from initial position xi to x f and again to xi; Total workdone = W=0 This zero workdone means, spring force is conservative Energy of oscillating spring at any point .  If the block of mass ( attached to massless spring)is extended to xₘ and released, it will oscillate in between +xₘ and -xₘ. The total mechanical energy a...

input 3.2  ANGLES Trigonometry functions are measured in terms of radian for a circle drawn in XY plane. Radian is nothing but the measure of an angle just like a degree. The difference between the degree and radian is; Degree :If rotation from the initial side to terminal side is (1/360)th of revolution, then the angle is said to measure 1 degree. 1 degree=60minutes 1 minute=60 second Radian:  If an angle is subtended at the center by an arc of length ‘l’ and then the angle is measured as radian. Suppose  Θ  is the angle formed at the center, then, θ = Length of the arc/radius of the circle. θ = l/r Relation between Degree and Radian: 2π radian=360° Or π radian=180° Where π = 22/7 TABLE FOR  DEGREE AND RADIAN RELATIONS Consider a unit circle with centre at the origin(0,0).Let P(x,y) be any point on the circle.Let angle AOP be θ radians.Then we define cos θ = x and sin θ = y. So we define cos θ as x coordinate of a...