Class 11 Physics: Basic Mathematical Tools for Physics

Introduction

• Purpose: Understanding basic mathematical tools needed for Class 11 Physics.
• Importance: Essential for solving physics problems, particularly concepts like instantaneous velocity and acceleration.
• Applications: Used extensively in physics (and limitedly in chemistry).

Differentiation

• Concept: Breaking a big function into smaller units.
• Explanation: Similar to breaking a puzzle into smaller pieces.
• Mathematical Expression:
  • Represented as dy/dx where y and x are variables related by a function y = f(x).
  • x is the independent variable and y the dependent variable.

Basic Rules and Formulas

1. Power Rule:
   • d(u^n) / dx = n * u^(n-1) * du/dx
   • E.g.: d(x^3) / dx = 3 * x^2
2. Differentiation of a Constant:
   • Always zero.
   • E.g.: d(5) / dx = 0
3. Exponential Functions:
   • d(e^x) / dx = e^x
   • d(e^y) / dx = e^y * dy/dx
4. Logarithmic Functions:
   • d(ln(x)) / dx = 1/x
   • d(ln(y)) / dx = (1/y) * dy/dx
5. Trigonometric Functions:
   • d(sin(x)) / dx = cos(x)
   • d(cos(x)) / dx = -sin(x)
   • d(tan(x)) / dx = sec^2(x)
   • d(sec(x)) / dx = sec(x) * tan(x)
   • d(cot(x)) / dx = -csc^2(x)
   • d(csc(x)) / dx = -csc(x) * cot(x)

Application in Physics

• Instantaneous Velocity: v = dx/dt
• Instantaneous Acceleration: a = dv/dt = d²x/dt²
  • Use differentiation to find slope and rate of change concepts.
• Example Problems: Calculating velocity or acceleration using provided functions and differentiation rules.

Integration

• Concept: Opposite of differentiation; joining small units into a large one.
• Explanation: Finding area under curves, summing up small rectangles to get the total area.
• Mathematical Expression:
  • ∫y * dx
  • If given limits: ∫ from a to b (y * dx)

Basic Rules and Formulas

1. Power Rule:
   • ∫x^n * dx = (x^(n+1)/(n+1)) + C (where n ≠ -1)
   • E.g.: ∫x^2 * dx = (x^3/3) + C
2. Integration of 1/x:
   • ∫(1/x) * dx = ln|x| + C
3. Integration of dx:
   • ∫dx = x + C
4. Definite Integration:
   • With limits: Plug in upper and lower limits.
   • E.g.: ∫ from 2 to 3 (x^2) dx = [x^3/3] from 2 to 3
5. Exponential Functions:
   • ∫(e^x) * dx = e^x + C
6. Constants:
   • Pull constants outside before integrating.

Application in Physics

• Finding Area Under Curves: Vital in calculating displacement, area under vt graphs, etc.

Example Problem

• Given a position function, finding instantaneous velocities at specific times using differentiation.
• Example: x = a + bt², find v = dx/dt = 2bt
• Specific calculations for given values.

Conclusion

• Understanding differentiation and integration is crucial for solving physics problems.
• Upcoming topics: Motion in Plane and Vectors.