IIT JEE Main / Advanced Physics 2013 – Differentiation & Integration in graphs.

Differentiation and Integration plays a key role in analysing and intrepreting the theoretical as well as the applicational principles of Physics. Consequently , understanding the representation of the same in a graphical format is an inevitable task for any student of Physics , more so for an advanced exam like IIT JEE Main / Advanced Physics 2013. Presented in this post are some of the basic aspects of the same .

1. Slope / Gradient of a Straight line

  • Case where the straight line is parallel to the x – axis . Here , θ = 0° . Slope is given by  m =  tanθ  = o

  • Case where the straight line makes an angle θ such that 0° <  θ < 90°  . As  θ lies in the first quadrant , the slope m = tanθ  is + ve.

  • Case where the straight line makes an angle θ such that , π/2 < θ < π . Here , the slope of m = tan θ is negative.

  • Case where the straight line is parallel to the y – axis  i.e,  θ = 90°  . Here the slope , m = tan 90° = ∞  ( undefined )

2. Graphical representation of  dy / dx

Consider a curve y = f (x) where P ( x , y ) and  Q ( x + Δx , y + Δy ) are any two points on the curve . P is joined to Q .

Considering Δ PQR ,   Δy/Δx  = tan ψ . Because Δx → 0 , point Q reaches point P . In the limiting position as Q →  P , the line PQ becomes tangent PT to the curve at P .  Therefore,

limΔx→0[ Δy/Δx ]   =  limψ→θtan ψ      ,  hence   dy/dx = tan θ

Therefore , dy/dx  at any point ( x , y ) is equal to the slope or gradient of the tangent to the curve at point P.

3. The Second Derivative

d²y/dx²  =  d/dx[ dy/dt ]  = d/dx ( tan θ )  =  d/dx ( slope ) . Note that d²y/dx² is the rate of change of the slope  y = f ( x ) curving with respect to x .

  • When the slope of the curve is increasing , then ,  d²y /dx²  >  0
  • When the slope of the curve is constant , then , d²y /dx²  = 0
  • When the slope of the curve is decreasing , then , d²y /dx²  <  o

Case where θ  > 90°  , θ2 > θ1  and the slope of tan θ is negative and it’s increasing.

Case where dy/dx  > 0  and   d²y/dx² < 0 . Here the slope of the curve is + ve and decreasing .

Case where  dy/dx  <  0 and   d²y/dx²  <  0  . Here the slope of the curve is negative and it’s decreasing .

4 . Definitive  Integration

Consider an elementary strip at a distance x and thickness dx from a curve  y = f ( x ) .

Here ,   ∫x=bx=a represents the summation of all elements  between the lines x = a and x = b .

Hence ,   ∫x=bx=a y.dx gives the total area under the curve  y = f ( x ) created by the x- axis between the lines x = a and x = b . [ Definitive Integration deals with the area enclosed by the curves ]

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