SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.2 - Applications of Derivatives [Latest edition]

The slope of the tangent to the curve x = 2 sin³θ, y = 3 cos³θ at θ = `pi/4` is ______.

The slope of the normal to the curve y = x² + 2e^x + 2 at (0, 4) is ______.

If the line y = 4x – 5 touches the curve y² = ax³ + b at the point (2, 3) then a + b is

If the tangent at (1, 1) on y² = x(2 − x)² meets the curve again at P, then P is

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. The time when the acceleration is 14 ft/sec², is 

Let f(x) = x³ − 6x² + 9𝑥 + 18, then f(x) is strictly decreasing in ______

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______

The edge of a cube is decreasing at the rate of 0.6 cm/sec then the rate at which its volume is decreasing when the edge of the cube is 2 cm, is

A particle moves along the curve y = 4x² + 2, then the point on the curve at which y – coordinate is changing 8 times as fast as the x – coordinate is

The function f(x) = x log x is minimum at x = ______.

Find the slope of tangent to the curve y = 2x³ – x² + 2 at `(1/2, 2)`

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. Find the velocity when 𝑡 = 2 sec

Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function

Show that f(x) = x – cos x is increasing for all x.

Show that the function f(x) = x³ + 10x + 7 for x ∈ R is strictly increasing

Find the slope of normal to the curve 3x² − y² = 8 at the point (2, 2)

Find the slope of tangent to the curve x = sin θ and y = cos 2θ at θ = `pi/6`

Find the equation of normal to the curve y = 2x³ – x² + 2 at `(1/2, 2)` 

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?

A man of height 2 metres walks at a uniform speed of 6 km/hr away from a lamp post of 6 metres high. Find the rate at which the length of the shadow is increasing.

Water is being poured at the rate of 36 m³/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising

Test whether the function f(x) = x³ + 6x² + 12x − 5 is increasing or decreasing for all x ∈ R

Test whether the following function f(x) = 2 – 3x + 3x² – x³, x ∈ R is increasing or decreasing

Find the values of x for which the function f(x) = 2x³ – 6x² + 6x + 24 is strictly increasing

Find the point on the curve y = `sqrt(x - 3)` where the tangent is perpendicular to the line 6x + 3y – 5 = 0.

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?

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?

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

Find the values of x for which the function f(x) = x³ – 6x² – 36x + 7 is strictly increasing

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.

Divide the number 30 into two parts such that their product is maximum.

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.

Find points on the curve given by y = x³ − 6x² + x + 3, where the tangents are parallel to the line y = x + 5.

The volume of the spherical ball is increasing at the rate of 4π cc/sec. Find the rate at which the radius and the surface area are changing when the volume is 288 π cc

The volume of a sphere increases at the rate of 20 cm³/sec. Find the rate of change of its surface area, when its radius is 5 cm

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

Find the values of x for which f(x) = 2x³ – 15x² – 144x – 7 is

(a) Strictly increasing
(b) strictly decreasing

Find the local maximum and local minimum value of  f(x) = x³ − 3x² − 24x + 5

A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.