Mathematics & Engineering Fundamentals Flashcards

Which interpolation method fits a polynomial of degree n-1 exactly through n data points?

Answer: Lagrange interpolation

Lagrange interpolation constructs a unique polynomial of degree at most n-1 that passes exactly through all n given data points.

If A and B are mutually exclusive events, then P(A ∪ B) equals:

Answer: P(A) + P(B)

Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0 and the addition rule simplifies to P(A) + P(B).

The determinant of a 2×2 matrix [[a, b],[c, d]] is:

Answer: ad - bc

The 2×2 determinant is computed as the product of the main diagonal minus the product of the anti-diagonal: ad - bc.

The solution to the first-order linear ODE dy/dt + 2y = 0 with y(0) = 5 is:

Answer: y = 5e^(-2t)

Separating variables gives dy/y = -2dt, integrating yields ln|y| = -2t + C, and applying y(0) = 5 gives y = 5e^(-2t).

The trapezoidal rule approximates a definite integral by treating the integrand as:

Answer: A straight line between adjacent points

The trapezoidal rule connects adjacent function values with straight lines, computing the area of resulting trapezoids.

For a complex number z = a + bi, the modulus |z| is:

Answer: √(a² + b²)

The modulus is the distance from the origin in the complex plane, given by √(a² + b²) by the Pythagorean theorem.

Which property distinguishes an exact differential equation M(x,y)dx + N(x,y)dy = 0?

Answer: ∂M/∂y = ∂N/∂x

An equation is exact when the cross-partial derivatives are equal: ∂M/∂y = ∂N/∂x, guaranteeing existence of a potential function F where dF = Mdx + Ndy.