Section B: Differential Calculus Basics

Unit 7: Foundations of Calculus

Section B: Differential Calculus Basics

Welcome

Welcome to Section B: Differential Calculus Basics! In this section, you’ll explore the concept of the derivative, which is a fundamental tool in calculus. The derivative helps us understand how functions change, providing insights into rates of change, motion, and optimization.

Imagine

Imagine you’re an engineer analyzing the speed of a roller coaster at different points along its track. The concept of the derivative allows you to calculate how the speed changes over time, helping you design a safe and thrilling ride.

Context

You’ve previously studied algebra, functions, and limits. Now, we’ll build on these concepts by introducing derivatives, which help you understand how functions change and how to analyze and optimize these changes in various contexts.

Overview

This section covers the derivative and its interpretation, techniques of differentiation, derivatives of trigonometric functions, using the chain rule, and applications of derivatives to motion and optimization. You’ll learn to apply these concepts to analyze and solve problems involving rates of change and optimization.

Objectives

  • Understand the concept of the derivative and its significance in calculus.
  • Learn and apply various techniques of differentiation, including the power rule, product rule, quotient rule, and chain rule.
  • Differentiate trigonometric functions, understanding how these functions change and how to analyze their behavior.
  • Use the chain rule to differentiate composite functions, mastering this essential tool in differential calculus.
  • Explore real-world applications of derivatives, using them to analyze motion, optimize functions, and solve various problems.

Preparatory Guidance

Definitions and Pronunciations
  • Derivative: The rate at which a function is changing at any given point, representing the slope of the tangent line to the function’s graph.
  • Chain Rule: A formula for computing the derivative of the composition of two or more functions, expressed as \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x).
Verbal Reading of Equations
  • For the derivative notation f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, say “f prime of x equals the limit as h approaches zero of f of x plus h minus f of x over h.”
  • For the chain rule, say “if y equals f of g of x, then the derivative of y with respect to x is f prime of g of x times g prime of x.”
Problem-Solving Strategies
  • Understand and calculate derivatives, analyzing how functions change and applying these concepts to various scenarios.
  • Apply techniques of differentiation, including the power rule, product rule, quotient rule, and chain rule, to solve problems involving rates of change.
  • Differentiate trigonometric functions, understanding their behavior and how they change in various contexts.
  • Use the chain rule to differentiate composite functions, mastering this essential tool in differential calculus.
  • Explore real-world applications of derivatives, using them to analyze motion, optimize functions, and solve various problems.

Considerations

How can understanding derivatives help you analyze and optimize functions in calculus? Why is the concept of the derivative important for understanding rates of change and motion? In what ways can you apply derivatives to solve real-world problems and optimize various processes?