Chapter 12: Limits and Derivatives

Summary

• Introduction to Calculus as the study of change in functions.
• Intuitive idea of derivatives and limits.
• Derivatives of standard functions are explored.
• Algebra of limits and derivatives is discussed.

Key Formulas and Definitions

• Derivative Definition:
  • For a function f, the derivative f'(x) is defined as:
    $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Limit Definition:
  • The limit of a function f(x) as x approaches a is:
    $\lim_{x \to a} f(x) = L$
• Standard Limits:
  • $\lim_{x \to 0} \frac{\sin x}{x} = 1$
  • $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$

Learning Objectives

• Understand the concept of limits and derivatives.
• Apply the definition of derivatives to find the slope of functions.
• Evaluate limits using algebraic manipulation.
• Differentiate standard functions using rules of differentiation.

Common Mistakes and Exam Tips

• Mistake: Confusing the limit of a function at a point with the function's value at that point.
  • Tip: Always check if the function is defined at the limit point.
• Mistake: Forgetting to apply the quotient rule correctly when differentiating.
  • Tip: Write down the quotient rule formula before applying it.

Important Diagrams

• Graph of a Function: Illustrates the relationship between a function and its tangent line, showing how the derivative represents the slope at a point.
• Limit Evaluation Table: Displays values of functions approaching a limit to visually confirm limit behavior.

Miscellaneous Exercises

• Find the derivative of various functions from first principles.
• Evaluate limits for given functions as x approaches specific values.

Chapter 12: Limits and Derivatives

12.1 Introduction

• Introduction to Calculus, a branch of mathematics dealing with the study of change in the value of a function as the points in the domain change.
• Topics covered:
  • Intuitive idea of derivative
  • Naive definition of limit
  • Algebra of limits
  • Definition of derivative
  • Algebra of derivatives
  • Derivatives of certain standard functions

12.2 Intuitive Idea of Derivatives

• Example: A body dropped from a tall cliff covers a distance of 4.9t² metres in t seconds.
  • Distance function: S = 4.9t²
  • Objective: Find the velocity at t = 2 seconds.
  • Average velocity formula:
    • Average velocity between t₁ and t₂ = (Distance travelled between t₁ and t₂) / (t₂ - t₁)

12.3 Limits

• Understanding the limiting process is crucial.
• Example: For the function f(x) = x², as x approaches 0, f(x) approaches 0.
  • Notation: lim (x→0) f(x) = 0
• Example: For g(x) = |x|, lim (x→0) g(x) = 0.

Important Examples

• Example 14: Derivative of f(x) = 1 + x + x² + ... + x⁵⁰ at x = 1 is 1275.
• Example 15: Derivative of f(x) = (x + 1)/(x) using the quotient rule.
• Example 16: Derivative of sin x using first principles.
• Example 17: Derivative of tan x using first principles.

Standard Limits

• Theorem: For functions f and g:
  • lim (x→a) [f(x) + g(x)] = lim (x→a) f(x) + lim (x→a) g(x)
  • lim (x→a) [f(x) * g(x)] = lim (x→a) f(x) * lim (x→a) g(x)

Exercises

1. Find the derivative of the following functions from first principles:
   • (i) -x
   • (ii) (-x)⁻
   • (iii) sin(x + 1)
   • (iv) cos(5)
2. Evaluate the following limits:
   • (i) lim (x→3) (x + 3)
   • (ii) lim (x→π) [(x - 22)/7]
   • (iii) lim (x→4) (4x + 3)/(x - 2)
   • (iv) lim (x→0) [(x + 1)⁵ - 1]/x

Conclusion

• Calculus is essential in various fields such as Physics, Chemistry, Economics, and Biological Sciences.

Common Mistakes and Exam Tips

Common Pitfalls

• Misunderstanding Limits: Students often confuse the limit of a function at a point with the value of the function at that point. Remember,

   If both the limit and the function value exist and are equal, then the function is continuous at that point.

• Incorrect Application of Derivative Rules: When applying the product or quotient rule, ensure that you correctly identify the functions involved and their derivatives.

   For example, for the product rule, if u = f(x) and v = g(x), then the derivative is given by (uv)' = u'v + uv'.
• Ignoring Undefined Points: When evaluating limits, be cautious of points where the function is undefined. For instance, if you encounter a 0/0 form, factor and simplify before applying the limit.

Exam Tips

• Practice Derivatives from First Principles: Be comfortable with finding derivatives using the definition, as it reinforces understanding of the concept.
• Use Tables for Limits: When evaluating limits, especially for functions that approach a point, create a table of values to visualize the behavior of the function.

• Memorize Standard Limits: Familiarize yourself with common limits, such as

   lim (sin x)/x = 1 as x approaches 0.
• Check for Continuity: Always verify if a function is continuous at a point before applying limit properties.
• Review Algebra of Limits: Remember the properties of limits, such as the limit of a sum being the sum of the limits, and practice applying them in various scenarios.