Lecture Notes: Single Variable Calculus - Lecture 3

Introduction

• Confirmation of audio and video clarity.
• Continuation of the 7 p.m. lecture series on Single Variable Calculus.

Previous Class Review

• Completed several lectures on the concepts of limits and derivatives.
• Today's focus: Applications of differentiation.

Key Topics Discussed

Applications of Differentiation

• Introduction to concepts like Mean Value Theorem and Maxima/Minima.

Standard Limits

• Review of standard limits and their applications in derivatives.

Important Limits

1. Limit of polynomials:
   • [ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} ]
   • [ \lim_{x \to a} \frac{x^m - a^m}{x^n - a^n} = \frac{m}{n} a^{m-n} ]
2. Trigonometric Limits:
   • [ \lim_{x \to 0} \frac{\sin(ax)}{x} = a ]
   • [ \lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2} ]
3. Exponential Limits:
   • [ \lim_{x \to 0} \frac{e^{ax} - 1}{x} = a ]
4. Function of Functions:
   • If ( f(x) ) approaches 1 and ( g(x) ) approaches infinity, then [ \lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} g(x)(f(x)-1)} ]
5. Special Cases:
   • Limit of ( x ) approaching zero for certain functions leading to specific behaviors.

Homework Assignment

1. Calculate the limit:
   • [ \lim_{x \to 0} \frac{e^{2x} - 1}{\sin(4x)} ]
2. Determine if the function ( f(x) = |\sin(2\pi x)| ) is continuous and differentiable.

Mean Value Theorem (MVT)

• Definition: If ( f(x) ) is continuous on ( [a, b] ) and differentiable on ( (a, b) ), then there exists at least one ( c ) in ( (a, b) ) such that:
  [ f'(c) = \frac{f(b) - f(a)}{b - a} ]
• Geometric interpretation of MVT:
  • The slope of the tangent at any point ( c ) (i.e., ( f'(c) )) equals the slope of the line connecting points ( a ) and ( b ).
• Physical example:
  • Speed limits enforced by average speed checks between two points on a road.

Conclusion

• Reminders for the next session:
  • Doubt revision scheduled for tomorrow at 7:00 p.m.
• Homework submission and practice expected.
• Recap of mean value theorem's importance in calculus.