Application Of Derivatives - Calculus - Mathematics Previous Year Questions

The point on the curve \(y^2=x\), the tangent at which makes an angle 45\(^\circ\) with X-axis is

The length of the subtangent to the curve \({x^2}{y^2} = {a^4}\) at \(( - a,a)\) is

The range in which \(y = - {x^2} + 6x - 3\) is increasing, is

OA and OB are two roads enclosing an angle of 120\(^\circ\). X and Y start from O at the same time. X travels along OA with a speed of 4 km/h and Y travels along OB with a speed of 3 km/h. The rate at which the shortest distance between X and Y is increasing after 1 hour is

Find the maximum value of \(f(x) = {1 \over {4{x^2} + 2x + 1}}\).

The approximate value of \(f(5.001)\), where \(f(x)=x^3-7x^2+15\) is

If the tangent to the curve \(xy + ax + by = 0\) at (1, 1) is inclined at an angle \({\tan ^{ - 1}}2\) with X-axis, then

Let \(f(x) = a - {(x - 3)^{8/9}}\), then maxima of \(f(x)\) is

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is

$$f(x)=2 x-\tan ^{-1} x-\log (x+\sqrt{x^2+1}) \text { is monotonically increasing, when }$$

The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minimum at

The altitude of a cone is \(20 \mathrm{~cm}\) and its semi vertical angle is \(30^{\circ}\). If the semi vertical angle is increasing at the rate of \(2^0\) per second, then the radius of the base is increasing at the rate of

The dimensions of the largest rectangle of side \(x\) and \(y\) that can be inscribed in the right angled triangle of sides \(\mathrm{a}\) and \(\mathrm{b}\) is

$$\text { The function } y=\tan x-x \text { is }$$

$$\text { If } f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x} \text { is decreasing for all } x \text {, then }$$

If \(f(x)=\log x+b x^2+a x, x \neq 0\) has extreme values (or turning points) at \(x=-1\) and \(x=2\) then the values of \(\mathrm{a}\) and \(\mathrm{b}\) are

If \((x-a)^2+(y-b)^2=c^2\), where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are some constants, \(c>0\) then \(\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}\) is independent of

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

What is the nature of the function \(f(x)=x^3-3 x^2+4 x\) on real numbers?

The turning point of the function \(y=\frac{a x-b}{(x-1)(x-4)}\) at the point \(P(2,-1)\) is