Calculus with Numbers

Sometimes, we can't find the derivative or integral of a function using calculus rules. That's where numerical differentiation and integration come in handy!

Numerical Differentiation

We estimate the slope of a function at a point using nearby values. The simplest formula is:

\[ f'(x) \approx \frac{f(x+h) - f(x)}{h} \]

where \( h \) is a small number.

Numerical Integration

We estimate the area under a curve by breaking it into small pieces.

• Trapezoidal Rule: Approximates the area using trapezoids.
• Simpson's Rule: Uses parabolic arcs for better accuracy.

Why Use These Methods?

• Functions may be too complex for standard calculus.
• Data might only be available as a table, not as a formula.

Everyday Uses

From predicting how much fuel a car uses, to finding the area under a curve on a graph, these methods are everywhere!