Calaus — Functions and Graphs, Limits and Continuity + 5 more topics

Functions are central to math. A function assigns each input value (from the domain) to exactly one output value (in the range). This "exactly one output" rule is crucial. If an input gives more than one output, it's not a function.

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Differentiation is all about finding the derivative of a function. The derivative tells you the instantaneous rate of change of a function. Geometrically, this means finding the slope of the tangent line to the function's graph at a given point. We use $f'(x)$ or $\frac{dy}{dx}$ to denote the derivative of a function $y = f(x)$.

When we talk about a limit, we're asking what value a function $f(x)$ gets closer and closer to as $x$ gets closer and closer to some number, say $a$. It doesn't matter what the function actually is at $a$, just what it's approaching.

We write this as $\lim_{x \to a} f(x) = L$. This means as $x$ approaches $a$, $f(x)$ approaches $L$.

Integration is fundamentally about two things:
1. Indefinite Integration (Antiderivatives): Finding a function whose derivative is the given function.
2. Definite Integration: Calculating the numerical value of the accumulated change of a function over a specific interval, often representing area.