About Course

Move beyond calculating derivatives to wielding them as versatile problem-solving tools. In this mini-course, you’ll discover how derivatives unlock the shape and behavior of any curve, guarantee the existence of critical points through cornerstone theorems, drive optimal solutions in design and resource management, and link multiple changing quantities in real time. With clear, intuitive explanations and real-world scenarios, you’ll gain the confidence to apply derivatives in science, engineering, economics, and beyond.

What Will You Learn?

  • Translate derivative signs and curvature into precise curve sketches
  • Apply the Mean Value Theorem and its precursor to guarantee critical points
  • Set up and solve optimization problems in diverse contexts
  • Connect and compute multiple changing rates in related-rates scenarios
  • Interpret and articulate results clearly for scientific and engineering applications

Course Content

1. Curve Sketching & Shape Analysis
Learn to read a curve’s story directly from its rate of change and the way that rate itself evolves. You’ll identify where a function climbs or descends, detect bending and inflection, and turn derivative information into accurate hand-drawn sketches without trial and error.

  • 1.1 Identifying Increasing & Decreasing Intervals
  • 1.2 Recognizing Concavity & Inflection Points
  • 1.3 Constructing a Complete Sketch from Derivative Clues

2. The Mean Value Theorem & Its Foundations
Understand why, under gentle conditions, some instant always matches the average change over an interval. You’ll explore the guarantee behind this principle, see how its simpler predecessor ensures a stationary point when endpoints coincide, and apply both ideas to practical situations like motion and error bounding.

3. Optimization
Transform derivative techniques into decision-making tools. You’ll learn to frame real situations—minimizing cost, maximizing area, or balancing efficiency—set up the right relationships, and identify the best possible outcome by locating and testing critical points.

4. Related-Rates Scenarios
Tackle problems where multiple quantities change together over time—like filling tanks, moving shadows, or expanding heat zones. You’ll learn to link variables through their relationships, differentiate with respect to time, and extract each quantity’s individual rate to interpret dynamic systems.

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