Study of Maximum and Minimum Values in Calculus

Introduction

• We have reached an advanced level in the calculus course.
• Importance of calculus: crucial in solving various types of mathematical problems.
• Emphasis on practice and homework.
• Discussion on the importance of real-life applications.

Today's Topic

• Maximum and Minimum Values of Functions
• Finding Absolute Extremes
• Summary of Absolute Extremes on Continuous Functions
• Real-Life Applications
• Problems Involving Finite Closed Intervals

Absolute Maximum and Minimum

• Absolute Maximum: The highest value of the function
• Absolute Minimum: The lowest value of the function
• Definition: A point of the function where the function's value is greater/lesser than all other values.

Extreme Value Theorem

• If a function is continuous and defined on a closed interval, it will certainly have a maximum and minimum value.

Finding Maximum and Minimum Values

• Determining Critical Points
• Evaluating End Points of the Interval
• Determining the Largest and Smallest Values

Applied Problems

• Example 1: Finding the maximum area of a rectangle with a perimeter of 100 feet.
  • Formula: Area = x * (50 - x)
  • Use of derivatives and critical points
  • Maximum area: 625 square feet (Dimensions: 25x25 feet)
• Example 2: Making a box from cardboard
  • Volume maximization problem
  • Formula: Volume = (16 - 2x) * (30 - 2x) * x
  • Determining critical points
  • Maximum volume: 19600/27 (at critical points)

Conclusion

• A deep understanding of calculus and its applications is essential
• Homework and practice are necessary
• Detailed study in the next lecture

Homework

• Complete assignments and discuss any issues.

Thank you! See you in the next lecture. Allah Hafiz.