List of top Mathematics Questions on Applications of Derivatives asked in CBSE CLASS XII

Find the equation of the normal at the point (am2, am3 ) for the curve ay2 = x3.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \(2m\) and volume is \(8m^3\).If building of tank costs Rs 70per sq meters for the base and Rs 45per square metre for sides.What is the cost of least expensive tank?

 

Let I be any interval disjoint from (−1, 1). Prove that the function f given by \(f(x)=x+\frac{1}{x}\) is strictly increasing on I.

For the curve \(y = 4x^3 − 2x^5\) , find all the points at which the tangents passes through the origin.

Find the least value of a such that the function f given \(f(x)=x^2+ax+1\) is strictly increasing on \((1, 2)\).

Prove that the function f given by \(f(x) = x^2 − x + 1\) is neither strictly increasing nor strictly decreasing,on \((−1, 1)\).

Prove that the logarithmic function is strictly increasing on \((0, ∞)\).

Prove that \(y=\frac{ 4sinθ}{(2+cosθ)}-θ \)is an increasing function of \(θ\) in \([0,\frac π2]\).

Find the values of x for which \(y=[x(x-2)]^2\) is an increasing function.

Show that \(y = log(1+x) - \frac {2x}{2+x}, \ x>-1\),is an increasing function of x throughout its domain.

The total revenue in Rupees received from the sale of \(x\) units of a product is given by \(R(x)=3x^2+36x+5\).The marginal revenue, when \(x=15\) is
The rate of change of the area of a circle with respect to its radius \(r\) at \(r = 6 cm\) is
A ladder \(5m\) long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of \(2 cm/s\). How fast is its height on the wall decreasing when the foot of the ladder is \(4 m\) away from the wall?
Find the intervals in which the function \(f\) given by \(f(x) = 2x^2 − 3x\) is (a) strictly increasing (b) strictly decreasing
Show that the function given by \(f(x) = sin x\) is (a) strictly increasing in \((0,\frac{π}{2})\) (b) strictly decreasing in \((\frac{π}{2},π)\) (c) neither increasing nor decreasing in \((0, π)\)
Show that the function given by \(f(x) = e^{2x}\) is strictly increasing on \(R\).
A balloon, which always remains spherical, has a variable diameter \(\frac{3}{2}(2x+1)\) Find the rate of change of its volume with respect to \(x\)
The radius of an air bubble is increasing at the rate of \(\frac{1}{2} cm/s\). At what rate is the volume of the bubble increasing when the radius is \(1 cm\)?
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm
The length \(x\) of a rectangle is decreasing at the rate of \(5 cm/minute\) and the width \(y\) is increasing at the rate of \(4 cm/minute\). When \(x = 8 cm\) and \(y = 6 cm\), find the rates of change of \((a)\) the perimeter, and \((b)\) the area of the rectangle.
The radius of a circle is increasing at the rate of \(0.7 cm/s.\) What is the rate of increase of its circumference?
A stone is dropped into a quiet lake and waves move in circles at the speed of \(5 cm/s\). At the instant when the radius of the circular wave is \(8 cm\), how fast is the enclosed area increasing?
The radius of a circle is increasing uniformly at the rate of \(3 cm/s\). Find the rate at which the area of the circle is increasing when the radius is \(10 cm\).
The volume of a cube is increasing at the rate of \(8 cm^3 /s\). How fast is the surface area increasing when the length of an edge is \(12 cm\)?
Find the rate of change of the area of a circle with respect to its radius r when (a) r=3 cm (b) r=4 cm