Please select a subject first
Advertisements
Advertisements
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
Concept: Logarithmic Differentiation
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x
Concept: Derivatives of Composite Functions - Chain Rule
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
Concept: Maxima and Minima
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
Concept: Maxima and Minima
The surface area of a spherical balloon is increasing at the rate of 2cm2/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?
Concept: Derivatives as a Rate Measure
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
Concept: Maxima and Minima
A car is moving in such a way that the distance it covers, is given by the equation s = 4t2 + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?
Concept: Derivatives as a Rate Measure
A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall
Concept: Derivatives as a Rate Measure
Find the values of x, for which the function f(x) = x3 + 12x2 + 36ЁЭСе + 6 is monotonically decreasing
Concept: Increasing and Decreasing Functions
A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which
(i) his shadow is lengthening
(ii) the tip of the shadow is moving
Concept: Derivatives as a Rate Measure
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Concept: Methods of Integration: Integration by Parts
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Concept: Methods of Integration: Integration by Parts
Evaluate the following:
`int x tan^-1 x . dx`
Concept: Methods of Integration: Integration by Parts
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
Concept: Methods of Integration: Integration Using Partial Fractions
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
Concept: Methods of Integration: Integration Using Partial Fractions
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
Concept: Methods of Integration: Integration Using Partial Fractions
Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x) "d"x`
Concept: Methods of Evaluation and Properties of Definite Integral
Evaluate:
`int_0^(pi/2) cos^3x dx`
Concept: Methods of Evaluation and Properties of Definite Integral
Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`
Concept: Methods of Evaluation and Properties of Definite Integral
Find the area of the region lying between the parabolas y2 = 4ax and x2 = 4ay.
Concept: Area Between Two Curves
