Introduction to Calculus
Calculus is the mathematics of change and accumulation. Two big ideas: the derivative (rate of change) and the integral (accumulated area).
Limits
A limit describes what a function approaches as x gets close to a value — even if the function is undefined there.
lim(x→2) of (x² − 4)/(x − 2) = lim(x→2)(x+2) = 4
Derivatives
The derivative measures instantaneous rate of change — the slope of the tangent line at any point.
If f(x) = x², then f'(x) = 2x (power rule)
At x = 3: slope = 2(3) = 6
Power rule: d/dx(xⁿ) = nxⁿ⁻¹
Common Derivative Rules
| Function | Derivative |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| eˣ | eˣ |
Integrals
The integral accumulates values — geometrically, the area under a curve. It is the inverse of differentiation (Fundamental Theorem of Calculus).
∫x² dx = x³/3 + C (reverse power rule: add 1, divide by new power)
FAQ
What is C in an integral? The constant of integration — any constant disappears when you differentiate, so we add C back.
What is calculus used for? Physics (motion), engineering (optimization), economics (marginal cost), biology (population growth).
Quick Quiz
Test what you just learned. Choose the best answer for each question.