Find the first order partial derivatives
See Answer →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].
See Answer →Verify means values theorem for the function f(x) = 4x³-4x in the interval [a,b], where a=0, and b=3.
See Answer →Explain why Lagrange's mean value theorem is not applicable to the following functions in the respective intervals: f(x) = 13x +11, x∈ [1,3].
See Answer →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].
See Answer →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.
See Answer →Find the height of a right cylinder with greatest lateral surface area that may be inscribed in a given sphere of radius R.
See Answer →An open storage bin with square base and vertical sides is to be constructed from a given amount of material. Determine its dimensions if its volume is to be maximum neglecting the thickness of material and waste in constructing it.
See Answer →Show that the conical tent of given capacity will require the least amount of canvas if its height is √2 times its base radius.
See Answer →Using vectors, find the area of the triangle ABC with vertices A(1,2,3), B(2, -1, 4) and C(4,5,-1).
See Answer →If A,B,C,D are the points with position vectors , respectively. Find the projection of
along
.
If , show that
x
=
x
=
x
.
Find the angle between the vectors and
.
Find a vector of magnitude 6, which is perpendicular to both the vectors and
Show that area of a parallelogram whose diagonals are given by and
is
Also find the area of the parallelogram whose diagonals are
and
, and
find vector
such that
and