Numerical Differentiation & Integration

Approximate derivatives using difference quotients and integrals using trapezoidal and Simpson's rules.

Numerical Differentiation

Approximate the derivative f'(x) using finite difference formulas. The exact derivative is the limit as h → 0.

Function

f(x)

Point x = 0.7854

Step size h = 0.1000

Exact f'(x): 0.707107

Forward:

0.670603 ε = 0.03650381

Backward:

0.741255 ε = 0.03414796

Central:

0.705929 ε = 0.00117792

━━ Function

f(x)

- - - Exact tangent (green)

━━ Forward difference secant

━━ Backward difference secant

━━ Central difference secant

Practice Problems

Use central difference with h = 0.01 to approximate f'(x). Enter your answer to 3 decimal places.

Problem 1: $f(x)$ = cos(x), x = 0.524

$f'(0.524) \approx \frac{f(0.534) - f(0.514)}{0.02}$

Problem 2: $f(x)$ = x^2, x = 2.000

$f'(2.000) \approx \frac{f(2.010) - f(1.990)}{0.02}$

Problem 3: $f(x)$ = x^3, x = 1.500

$f'(1.500) \approx \frac{f(1.510) - f(1.490)}{0.02}$

Implementation

"hljs-comment">// Forward difference
f'(x) ≈ (f(x + h) - f(x)) / h

"hljs-comment">// Backward difference
f'(x) ≈ (f(x) - f(x - h)) / h

"hljs-comment">// Central difference (most accurate)
f'(x) ≈ (f(x + h) - f(x - h)) / (2h)