Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007

Introduction

This tutorial provides a step-by-step guide to the fundamental concepts introduced in the first lecture of MIT's Single Variable Calculus course. It focuses on derivatives, slope, velocity, and the rate of change, essential topics for anyone looking to understand calculus and its applications.

Step 1: Understanding Derivatives

• Definition: A derivative represents the instantaneous rate of change of a function. It can be thought of as the slope of the tangent line to the function at a given point.
• Notation: The derivative of a function ( f(x) ) is often denoted as ( f'(x) ) or ( \frac{df}{dx} ).
• Geometric Interpretation:
  • If you have a curve on a graph, the derivative at a point tells you how steep the curve is at that point.
  • A positive derivative indicates the function is increasing, while a negative derivative means it is decreasing.

Step 2: Calculating the Slope

• Definition of Slope: The slope between two points on a line can be calculated using the formula: [ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} ]
• Example:
  • Given points ( (x_1, y_1) ) and ( (x_2, y_2) ):
    • If ( (1, 2) ) and ( (3, 4) ), the slope is: [ \text{slope} = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1 ]

Step 3: Understanding Velocity

• Concept: Velocity is the derivative of position with respect to time. It indicates how position changes as time progresses.
• Formula: If ( s(t) ) represents position as a function of time, the velocity is given by: [ v(t) = \frac{ds}{dt} ]
• Practical Example: If a car travels along a straight road, knowing its position at different times allows you to calculate its velocity.

Step 4: Rate of Change

• Definition: The rate of change of a quantity refers to how much that quantity changes with respect to another variable.
• Application: Rate of change can apply to various contexts, such as population growth, speed of a vehicle, or changes in revenue over time.
• Example: If a population grows from 100 to 150 over 5 years, the average rate of change is: [ \text{Rate of Change} = \frac{150 - 100}{5} = 10 \text{ individuals per year} ]

Conclusion

In this tutorial, we covered the foundational concepts of derivatives, slope, velocity, and rate of change. These concepts are crucial for understanding single variable calculus and have broad applications in science, economics, and engineering. As a next step, consider practicing these concepts through problems and examples to solidify your understanding.