1. Evaluate the limit: lim(x->0) [(sin(2x))/(x)].
2. Solve the differential equation: dy/dx = y^2 + e^x.
3. Determine whether the series converges or diverges: Σ(n=1 to ∞) (1/n^(2/3)).
4. Find the value of x that minimizes the function f(x) = x^3 - 4x^2 + 5x + 6.
5. Calculate the curl of the vector field F(x, y, z) = (z^2, xz, y^3).
6. In linear algebra, find the eigenvalues and eigenvectors of the matrix A = [[4, -1], [2, 3]].
7. Determine the area bounded by the curves y = x^2 and y = 2x - x^2.
8. Find the Taylor series expansion of f(x) = ln(1 + 2x) centered at x = 0, up to the 5th-degree term.
9. Solve the system of equations: 2x + y - z = 4, x - 3y + 2z = -1, 3x + 4y - 6z = 7.
10. Prove by induction: For all positive integers n, 1^2 + 2^2 + 3^2 + ... + n^2 = (n(n+1)(2n+1))/6.
Pure Furies ~ Whereabouts of Hearts♪
Absurd Responses=Report/Termination.
2. Solve the differential equation: dy/dx = y^2 + e^x.
3. Determine whether the series converges or diverges: Σ(n=1 to ∞) (1/n^(2/3)).
4. Find the value of x that minimizes the function f(x) = x^3 - 4x^2 + 5x + 6.
5. Calculate the curl of the vector field F(x, y, z) = (z^2, xz, y^3).
6. In linear algebra, find the eigenvalues and eigenvectors of the matrix A = [[4, -1], [2, 3]].
7. Determine the area bounded by the curves y = x^2 and y = 2x - x^2.
8. Find the Taylor series expansion of f(x) = ln(1 + 2x) centered at x = 0, up to the 5th-degree term.
9. Solve the system of equations: 2x + y - z = 4, x - 3y + 2z = -1, 3x + 4y - 6z = 7.
10. Prove by induction: For all positive integers n, 1^2 + 2^2 + 3^2 + ... + n^2 = (n(n+1)(2n+1))/6.
Pure Furies ~ Whereabouts of Hearts♪
Absurd Responses=Report/Termination.
Answers & Comments
Answer:
To evaluate the limit: lim(x->0) [(sin(2x))/(x)], you can use L'Hôpital's Rule, which gives a result of 2.
The solution to the differential equation dy/dx = y^2 + e^x is y(x) = -1/(Ce^(-x) - 1), where C is the constant of integration.
The series Σ(n=1 to ∞) (1/n^(2/3)) converges because the exponent (2/3) is greater than 1, making it a p-series with p > 1.
To find the value of x that minimizes the function f(x) = x^3 - 4x^2 + 5x + 6, take the derivative f'(x) and set it equal to 0. Solving f'(x) = 0 gives you the critical points, and you can find the minimum from there.
The curl of the vector field F(x, y, z) = (z^2, xz, y^3) is ∇ × F = (2z - y, -x, -x).
The eigenvalues of the matrix A = [[4, -1], [2, 3]] are λ1 = 5 and λ2 = 2. The corresponding eigenvectors are v1 = (1, 2) and v2 = (-1, 1).
To determine the area bounded by the curves y = x^2 and y = 2x - x^2, you need to find the points of intersection and then integrate the absolute difference of the functions over that interval.
The Taylor series expansion of f(x) = ln(1 + 2x) centered at x = 0 is ln(1 + 2x) = 2x - (4/2)x^2 + (8/3)x^3 - (16/4)x^4 +
To solve the system of equations: 2x + y - z = 4, x - 3y + 2z = -1, 3x + 4y - 6z = 7, you can use various methods like substitution or elimination to find the values of x, y, and z that satisfy all three equations.
To prove by induction that for all positive integers n, 1^2 + 2^2 + 3^2 + ... + n^2 = (n(n+1)(2n+1))/6, you can use mathematical induction, showing that it holds for n = 1 and then assuming it holds for n = k and proving it for n = k + 1
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