(a) Find all the second order partial derivatives of the following function.
f(x, y) = x² - 8xy + y²
For the given function ( f(x, y) = x^2 - 8xy + y^2 ), we first find the first-order partial derivatives. The partial derivative with respect to (x) is ( _______ ________ ____ _____ _________ _________ _____ __________ __________ ___ ______.
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∫((1+ln x)³)/x dx
Solve the following system of equations using Cramer's rule.
x + y - z = 6
3x - 2y + z = -5
x + 3y - 2z = 14
(dy/dx)² = (7x)/(4y²)
(a) Define Error and their types in quantitative chemical analysis.
(a) Evaluate the following limit.
(b) If the law of motion of a particle is given as: s = -t³ + 3t² + 25, then (2+1)
i) find its velocity and acceleration.
ii) find the distance covered by the particle in time t = 5 units
(a) Which of the following sets are finite, and which are infinite?
(i) The set of points on the circumference of a circle.
(ii) ]0, 1[
(iii) [-1, 1]
(iv) {1, 2, ..., 100}
(b) If A = {1, 2, 3}, B = {2, 3, 4, 5}, and C = {1}, determine A ∪ B ∪ C and also verify
A ∪ B ∪ C = (A ∪ B) ∪ C = A ∪ (B ∪ C)
Solve the following differential equation:
(1/y²) (dy/dt) = 1 - e⁻³ᵗ
(c) Show that the equation
(y - 2x³)dx = x(1 - xy)dy
becomes exact on multiplication by x⁻² and solve it.