Calculus studies change and accumulation. It has two main parts:
1. Differential Calculus
How things change
- Focus: Rates of change, slopes, derivatives.
- Key idea: Finds how fast something is changing at any point.
- Example: If you drive a car, differential calculus tells you your instantaneous speed at any moment (the derivative of your position with respect to time).Derivative = slope of the tangent line to a curve
- Example: If f(x) = x^2, then f’(x) = 2x, which tells how fast f(x) changes at each x
2. Integral Calculus
How things accumulate
- Focus: Areas under curves, totals, integrals.
- Key idea: Finds the total amount accumulated over time or space.
- Example: If you know your speed at each moment, integral calculus tells you the total distance you’ve traveled.Integral =
- Example: \int x^2 \, dx = \frac{1}{3}x^3 + C
Fundamental Theorem of Calculus
This theorem connects the two parts:
- Differentiation and integration are opposites.
- If you integrate a function and then differentiate the result, you get the original function back.
Algebra uses symbols (like x, y) to represent numbers and relationships. It helps us solve problems where some values are unknown.
