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Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
Concept: Simple Problems on Applications of Derivatives
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
Concept: Simple Problems on Applications of Derivatives
Find the angle of intersection of the curves \[y^2 = 4ax \text { and } x^2 = 4by\] .
Concept: Tangents and Normals
The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of the sum of their volumes.
Concept: Graph of Maxima and Minima
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Concept: Tangents and Normals
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
Concept: Simple Problems on Applications of Derivatives
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Concept: Graph of Maxima and Minima
Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x2 + 5x + 3.
Concept: Approximations
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Concept: Increasing and Decreasing Functions
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Concept: Maxima and Minima
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Concept: Increasing and Decreasing Functions
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Concept: Maxima and Minima
Find the equation of a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis
Concept: Tangents and Normals
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Concept: Simple Problems on Applications of Derivatives
Find the equation of the tangent line to the curve `"y" = sqrt(5"x" -3) -5`, which is parallel to the line `4"x" - 2"y" + 5 = 0`.
Concept: Tangents and Normals
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.
Concept: Maximum and Minimum Values of a Function in a Closed Interval
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Concept: Increasing and Decreasing Functions
The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.
Concept: Tangents and Normals
The maximum value of `(1/x)^x` is ______.
Concept: Maxima and Minima
A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
Concept: Rate of Change of Bodies or Quantities
