Mathematics and Statistics Flashcards

The Laplace transform of f(t) = t·e^(2t) is:

Answer: 1/(s-2)²

Using the Laplace transform formula L{t·e^(at)} = 1/(s-a)², with a=2, the result is 1/(s-2)².

Which numerical method uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n) to find roots?

Answer: Newton-Raphson method

The Newton-Raphson method iteratively refines root estimates using the function and its derivative.

A dataset has mean μ = 50 and standard deviation σ = 5. What percentage of data falls within one standard deviation of the mean, assuming a normal distribution?

Answer: 68%

The empirical rule states that approximately 68% of data in a normal distribution falls within ±1σ of the mean.

The determinant of the matrix [[3,1],[2,4]] equals:

Answer: 10

det = (3)(4) - (1)(2) = 12 - 2 = 10.

Which of the following is the correct expression for the Taylor series expansion of e^x about x=0?

Answer: Σ x^n/n! for n=0 to ∞

The Maclaurin series for e^x is 1 + x + x²/2! + x³/3! + … = Σ x^n/n!.

Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what is P(A ∩ B)?

Answer: 0.20

For independent events, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.20.

What is the general solution to the differential equation dy/dx = 3y?

Answer: y = Ce^(3x)

Separating variables and integrating gives ln|y| = 3x + C₁, so y = Ce^(3x).