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Overview of Exam Preparation

• Last week of classes focuses on the hardest material.

• Midterms include chapters four and eight.

• Students will have spring break to study the material before exams.

Midterm Grading Policies

• No specific midterm curve but overall curve for the class is applied.

• Grades based on individual performance, but overall averages affect grading scale.

• Guaranteed grades provided in syllabus; e.g. 90% = A-, 80% = B-.

• The final exam heavily influences overall grades.

Introduction to Optimization in Calculus

Purpose of the Class

• Study calculus as a method for modeling real-world applications.

• Understand that models have inaccuracies but can provide useful insights.

Definition of Optimization

• Finding maximum or minimum values of a function over a specified domain.

• Importance across various fields, particularly business and economics.

Example of Optimization

Function Example: f(x) = x²

• Domain: All real numbers (−∞ to +∞).

• Maximum value: None (the function is not bounded above).

  • As x increases beyond 1, f(x) > x, indicating no maximum.

• Minimum value: Achieved at x = 0 (f(0) = 0).

  • x² is always greater than or equal to 0.

Graphical Interpretation

• Function is bounded below by zero (the minimum value).

• Range: [0, ∞).

Calculus Approach to Optimization

Utilizing Derivatives

• Derivative of f(x) = x²: f'(x) = 2x.

• Identify regions of increase/decrease based on the sign of f'.

  • f' > 0 for x > 0 (function increasing).

  • f' < 0 for x < 0 (function decreasing).

Sign Chart Analysis

• Local minimum occurs at x = 0 due to sign change of derivative.

• Visualizing sign chart helps in understanding local minima and maxima.

Interval Notation for Domains

• Closed intervals: [a, b] (including endpoints).

• Open intervals: (a, b) (excluding endpoints).

• Critical points are evaluated at endpoints or where the derivative is zero or undefined.

Local Vs. Global Maxima/Minima

Local Extrema

• Defined within a small neighborhood of a point.

• Local max occurs when the function is higher than surrounding points.

Global Extrema

• Extreme values over the entire function's domain.

• Confusion may arise about endpoints being local extrema.

First Derivative Test

Analyzing Critical Points

• Identify critical points (derivative = 0 or undefined).

• Evaluate signs of the derivative around critical points.

  • Positive -> increasing; Negative -> decreasing.

Local Maximum and Minimum Conditions

• Local min: f' changes from negative to positive at critical point.

• Local max: f' changes from positive to negative at critical point.

Non-Examples of Local Extrema

Stationary Points

• Function can be stationary (derivative = 0) and still not have local max/min (e.g., f(x) = x³).

Behavior at Critical Points

• Need to evaluate surrounding behavior to classify extremum.

Conclusion

• Understanding critical points and first derivative tests is crucial for optimization in functions.

• The next session will tackle specific examples for finding maximum/minimum values.