(a) r = 3 cm (b) r = 4 cm

The volume of a cube is increasing at the rate of 8 cm³/s. How fast is the surface area increasing when the length of an edge is 12 cm?

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

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 at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

A ladder 5 m 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?

The radius of an air bubble is increasing at the rate of  cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Show that the function given by f(x) = sin x is

(a) strictly increasing in  (b) strictly decreasing in 

(c) neither increasing nor decreasing in (0, π)

Find the intervals in which the function f given by f(x) = 2x² − 3x is

(a) strictly increasing (b) strictly decreasing

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is

(a) strictly increasing (b) strictly decreasing

Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x² + 2x − 5 (b) 10 − 6x − 2x²

(c) −2x³ − 9x² − 12x + 1 (d) 6 − 9x − x²

(e) (x + 1)³ (x − 3)³

Show that, is an increasing function of x throughout its domain

Prove that the function f given by f(x) = log sin x is strictly increasing on and strictly decreasing on

Prove that the function f given by f(x) = log cos x is strictly decreasing on  and strictly increasing on

Find the equation of all lines having slope −1 that are tangents to the curve 

Find points on the curve  at which the tangents are

(i) parallel to x-axis (ii) parallel to y-axis

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x⁴ − 6x³ + 13x² − 10x + 5 at (0, 5)

(ii) y = x⁴ − 6x³ + 13x² − 10x + 5 at (1, 3)

(iii) y = x³ at (1, 1)

(iv) y = x² at (0, 0)

(v) x = cos t, y = sin t at 

Prove that the curves x = y² and xy = k cut at right angles if 8k² = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = (2x − 1)² + 3        (ii) f(x) = 9x² + 12x + 2

(iii) f(x) = −(x − 1)² + 10    (iv) g(x) = x³ + 1

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = |x + 2| − 1 (ii) g(x) = − |x + 1| + 3

(iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3|

(v) h(x) = x + 1, x (−1, 1)

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i). f(x) = x²           

(v). f(x) = x³ − 6x² + 9x + 15

(vi). 

(vii). 

(viii). 

Prove that the following functions do not have maxima or minima:

(i) f(x) = e^x (ii) g(x) = logx

(iii) h(x) = x³ + x² + x + 1

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i)  (ii) 

(iii) 

(iv) 

Find the maximum profit that a company can make, if the profit function is given by

p(x) = 41 − 72x − 18x²

Find both the maximum value and the minimum value of

3x⁴ − 8x³ + 12x² − 48x + 25 on the interval [0, 3]

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

What is the maximum value of the function sin x + cos x?

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum

A square piece of tin of side 18 cm is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Find the intervals in which the function f given by

is (i) increasing (ii) decreasing

Find the intervals in which the function f given byis

(i) increasing (ii) decreasing

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Find the points at which the function f given byhas

(i) local maxima (ii) local minima

(ii) point of inflexion

Find the absolute maximum and minimum values of the function f given by