Welcome to calculus! Calculus is the branch of mathematics that deals with change. While algebra works with fixed values and straight lines, calculus helps us understand curves, changing speeds, and dynamic processes. Think of it as the math of motion and accumulation.
Differential calculus is all about rates of change. You already know how to find the slope of a straight line using rise over run. Differential calculus extends this idea to curved lines. It finds the instantaneous rate of change at any point on a curve by finding the slope of the tangent line at that point.
Integral calculus is about accumulation and finding areas under curves. You know how to find the area of rectangles and triangles. Integral calculus extends this to find the area under any curve by conceptually breaking it into infinitely many tiny rectangles and adding up their areas. This process of accumulation has many applications.
The Fundamental Theorem of Calculus is one of the most important discoveries in mathematics. It shows that differentiation and integration are inverse operations, just like addition and subtraction or multiplication and division. This deep connection means that these two seemingly different processes actually undo each other, making calculus problems much more manageable.
To summarize what we've learned: Calculus is the powerful mathematical tool for understanding change and accumulation. Differential calculus gives us rates of change, integral calculus gives us accumulation, and the Fundamental Theorem shows they're connected as inverse operations. This makes calculus essential for many fields including physics, engineering, and economics.