Basic Calculus: A Quick-Start Guide
1. The Core Idea: What is Calculus?
Traditional math (Algebra and Geometry) is great for things that stay still. Calculus is
the mathematics of change. It is generally divided into two main branches:
1. Differential Calculus: Cutting something into tiny pieces to find the rate of
change (slopes).
2. Integral Calculus: Joining tiny pieces together to find the total amount (area).
2. Limits: The Foundation
Before you can find a slope at a single point, you need a Limit. A limit asks: "As $x$ gets
infinitely close to a value, what does $y$ look like?"
The Notation
$$\lim_{x \to a} f(x) = L$$
This means as $x$ approaches $a$, the function $f(x)$ approaches $L$.
3. Derivatives (The Slope)
The derivative tells you the instantaneous rate of change. If you are driving, your
speedometer shows your derivative (velocity) at that exact moment.
The Power Rule
This is the most essential shortcut in calculus. If you have a function $f(x) = x^n$, the
derivative is:
$$f'(x) = nx^{n-1}$$
Example:
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Function: $f(x) = x^3$
Derivative: $f'(x) = 3x^2$
4. Integrals (The Area)
Integration is the opposite of differentiation. It calculates the area under a curve.
The Indefinite Integral
When you integrate, you are finding the "Anti-derivative." Because constants disappear
during differentiation, we add a $+ C$.
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
Example:
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Problem: $\int 3x^2 dx$
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Answer: $x^3 + C$
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5. The Fundamental Theorem of Calculus
This is the "bridge" that connects derivatives and integrals. It states that the area under
a curve from point $a$ to $b$ can be found using the anti-derivative:
$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$
Summary Cheat Sheet
Operation
Derivative
Integral
Purpose
Find Slope / Speed
Find Area / Total
Key Rule (Power Rule)
$nx^{n-1}$
$\frac{x^{n+1}}{n+1}$
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Would you like me to expand on any of these sections with more practice problems or
specific rules like the Product Rule or Chain Rule?
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