Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied.
i. f(x) = sin2 x on the interval [0, π].
ii. f (x) = x² + 4 on [-2, 2].
iii. f(x) = sinx + cos x on
iv. f(x) = x³- 2x on [0, 1].
Use the method of Laplace transforms to find the solution of the initial value problem y" + 9y = 6 cos 3t, y(0) = 2, y'(0) = 0
Reduce the equation
xy" + y' + xy = 0, x > 0
into Bessel's equation and hence write its general solution in terms of Bessel's functions.
, and
find vector
such that
and
If A is a square matrix, such that A² =A, then find (I + A)³ - 7A.
Given a point on the axis of the parabola y² = 2px at a distance a from the vertex, find the abscissa of the point of the curve closest to it.
Verify means values theorem for the function f(x) = 4x³-4x in the interval [a,b], where a=0, and b=3.
Find 'c' of Cauchy's mean value theorem for the function f(x) = 2.ln(x) and g(x) = x²-1 in the interval [2,3].