Introducing Calculus: Can Change Occur at An Instant?
Abigail Young
6 min read
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Study Guide Overview
This study guide covers limits, the foundation of calculus. It explains how secant lines represent average rate of change and how limits help determine instantaneous rate of change. The guide emphasizes evaluating limits graphically, numerically, and algebraically, and connects limits to the concept of derivatives. Practice questions on limits and their applications are included.
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#Introduction: Embracing Change
Most of the math you've seen so far has dealt with constant rates of change (aka, good ol' slope). But the real world? It's all about change that's not constant. That's where calculus comes in, and it all starts with understanding limits.
Limits are the foundation of calculus, allowing us to analyze how functions behave at specific points and as they approach certain values. They are the key to understanding derivatives and integrals.
#1. Finding the Rate of Change: Secant Lines
#1.1 The Slope Formula
Remember the classic slope formula? It's the foundation for understanding rates of change:
This formula gives us the slope of a secant line, which represents the average rate of change between two points on a curve. Think of it as a straight line cutting through the function at two distinct points.
#1.2 Visualizing Secant Lines
Look at the image below. Notice how each secant line has a different slope? That's because we're using different intervals to measure the rate of change.
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