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 ]