Numerical Differentiation and Integration

These flashcards cover key concepts from numerical differentiation and integration, including definitions, rules, formulas, and error estimates.

What is numerical differentiation?

It is the process of approximating the derivative of a function using discrete data points.

What approaches can be used for numerical differentiation?

1. Taylor Series Expansion 2. Passing a polynomial through the given data and differentiating the interpolating polynomial.

What is the first derivative of a function defined by?

f �(c) = lim h→0 f (c + h) − f (c) / h, for all c in (a, b).

What is the Forward Formula in numerical differentiation?

f �(x0) ≈ (f(x0 + h) - f(x0)) / h, using Taylor series expansion.

What does Absolute Error represent in numerical differentiation?

Error = |true value − approximate value|.

What is the formula for the backward difference?

f �(x0) = (f(x0) − f(x0 − h)) / h.

What is the central difference formula?

f �(x0) = (f(x0 + h) - f(x0 − h)) / (2h).

What is the second derivative approximation using central differences?

f ��(x0) ≈ (f(x0 + h) - 2f(x0) + f(x0 − h)) / h².

What are Newton’s divided differences?

A method used for calculating the derivative of a function using discrete data.

When is the Lagrange interpolating polynomial used?

When data points are not equispaced.

What does the Newton-Cotes formula involve?

Polynomial interpolation to approximate the integral of a function.

What are the types of Newton-Cotes formulae?

1. Trapezoidal Rule 2. Simpson's Rule 3. Simpson's Three-Eighths Rule.

What is the Trapezoidal Rule formula?

∫[a,b] f(x)dx ≈ h/2 [f(a) + f(b)] - (h³/12)f ″(ξ) for some ξ in (a, b).

What is the error term of the Trapezoidal Rule?

Error = -(b-a)³ / 12 f ″(ξ) where ξ is some point in (a, b).

What is Simpson's 1/3 Rule?

Approximation defined as ∫[a,b] f(x)dx ≈ (h/3)[f(a) + 4f(m) + f(b)] where m = (a+b)/2.

What is the error for Simpson's Rule?

Error = -(b-a)⁵ / 90 f (4)(ξ), where ξ is in (a, b).

What is the purpose of composite rules in numerical integration?

To reduce the error by dividing the interval [a, b] into equal subintervals.

What does the Composite Simpson’s Rule formula look like?

∫[a,b] f(x)dx = (b-a)/3n/2 + 4f(x1) + f(x2)) - (b-a)⁵ / 180f (4)(µ).

What does the Composite Trapezoidal Rule state?

∫[a,b] f(x)dx = (b-a)/2[f(a) + 2Σf(xj) + f(b)] - (b-a)² / 12f ″(µ).

What are common application scenarios for numerical integration?

Approximation of integrals that are difficult to solve analytically.

What is Simpson's Three-Eighths Rule?

Approximation for n = 3 given as ∫[a,b] f(x)dx = (3h/8)[f(x0) + 3f(x1) + 3f(x2) + f(x3)].

How is the error in the Composite Simpson's Rule expressed?

E(f) = - (b-a)/180 * h⁴ * f (4)(µ).

What is a defining characteristic of the Fundamental Theorem of Calculus?

It relates antiderivatives to definite integrals: ∫[a,b] f(x)dx = F(b) - F(a).

What is an antiderivative?

A function F such that F' = f, where f is integrable.

What indicates that a function is described with discrete data?

The function values are defined only at specific points, typically represented in a table.

What is the basic error correction for numerical methods?

Error can be reduced by choosing a smaller step size or increasing the number of intervals.