NCERT Solutions for class 12th Mathematics Chapter 6 Application of Derivatives

Exercise 6.1

Question 1. Find the rate of change of the area of a circle with respect to its radius r when: (a) r = 3 cm (b) r= 4cm

Question 2. 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?

Question 3. 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.

Question 4. 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?

Question 5. 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?

Question 6. The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Question 7. 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 = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Question 8. 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.

Question 9. A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

Question 10. 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 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4m away from the wall?

Question 11. A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Question 12. The radius of an air bubble is increasing at the rate of 1/2cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Question 13. A balloon, which always remains spherical, has a variable diameter3/2(2x + 1). Find the rate of change of its volume with respect to x.

Question 14. S and 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?

Question 15. The total cost C (x) in Rupees associated with the production of x units of an item is given by C (x) = 0.007×3 – 0.003×2 + 15x + 4000. Find the marginal cost when 17 units are produced.

Question 16. The total revenue in Rupees received from the sale of x units of a product is given by R (x) = 13×2 + 26x + 15. Find the marginal revenue when x = 7.

Choose the correct answer in the Questions 17 and 18

Exercise 6.2

Question 1. Show that the function given by f (x) = 3x + 17 is strictly increasing on R.

Question 2. Show that the function given by f (x) = e2x is strictly increasing on R.

Question 3. Show that the function given by f (x) = sin x is

Question 4. Find the intervals in which the function f given by f (x) = 2×2 – 3x is (a) strictly increasing (b) strictly decreasing.

Question 5. Find the intervals in which the function f given by f (x) = 2×3 – 3×2 – 36x + 7 is (a) strictly increasing (b) strictly decreasing.

Question 6. Find the intervals in which the following functions are strictly increasing or decreasing:

Question 7. Show that y = log (1 + x) = 2x/(2 + x) , x > – 1, is an increasing function of x throughout its domain.

Question 8. Find the values of x for which y = [x (x – 2)]2 is an increasing function.

Question 9.

Question 10. Prove that the logarithmic function is strictly increasing on (0, ¥).

Sol. Let f (x) = log x, defined for x > 0 \ f ¢ (x) =1x> 0, when x > 0Hence, f (x) is an increasing function for x > 0 i.e. f (x) is increasing function whenever is defined.

Question 11. Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (–1, 1).

Question 12. Which of the following functions are strictly decreasing on ?

(a) cos x (b) cos 2x (c) cos 3x (d) tan x

Sol. (a, b)

Question 13. On which of the following intervals is the function f given by f (x) = x100 + sin x – 1 strictly decreasing?

Question 14. Find the least value of a such that the function f given by f (x) = x2 + ax+ 1 is strictly increasing on [1, 2].

Sol. We have f(x) = x2 + ax + 1, f ‘(x) = 2x + a. In interval (1, 2) i.e., 1 < x <2 Þ 2 < 2x < 4 Þ (2 + a) < (2x + a) < (4 + a)

Since, f (x) is strictly increasing function, then (2 + a) > 0

[For this f ¢ (x) > 0 and (2x + a) > (2 + a)]\ (2 + a) > 0 Þ a > –2.Thus the least value of a is –2.

Question 15. Let I be any interval disjoint from (–1, 1). Prove that the function f given by f (x) = x +1/x is strictly increasing on I.

Question 16.

Question 17.

Question 18. Prove that the function given by f (x) = x3 – 3×2 + 3x – 100 is increasing in R.

Question 19. The interval in which y = x2 e–x is increasing in

Exercise 6.3

Question 1. Find the slope of the tangent to the curve y = 3×4 – 4x at x = 4.

Question 2. Find the slope of the tangent to the curve y = (x – 1)/(x – 2 ), x not= 2 at x = 10.

Question 3. Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x-coordinate is 2.

Question 4. Find the slope of the tangent to the curve y = x3 – 3x + 2 at the point whose x-coordinate is 3.

Question 5. Find the slope of the normal to the curve

Question 6. Find the slope of the normal to the curve x = 1 – a sin q, y = b cos2 q at q =pi/ 2.

Question 7. Find points at which the tangent to the curve y = x3 – 3×2 – 9x + 7 is parallel to the x-axis.

Question 8. Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Question 9. Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11

Question 10. Find the equation of all lines having slope – 1 that are tangents to the curve y =1/(x-1), x not= 1.

Question 11. Find the equation of all lines having slope 2 which are tangents to the curve y =1/(x-3) , x not= 3.

Question 12. Find the equations of all lines having slope 0 which are tangent to the curve y = 1/(x2-2x+3).

Question 13.

Question 14. Find the equations of the tangent and normal to the given curves at the indicated points:

Question 15. Find the equation of the tangent line to the curve y = x2 – 2x + 7 which is (a) parallel to the line 2x – y + 9 = 0 (b) perpendicular to the line 5y – 15x = 13.

Question 16. Show that the tangents to the curve y = 7×3 + 11 at the points where x = 2 and x = – 2 are parallel

Question 17. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Question 18. For the curve, y = 4×3 – 2×5, find all the points at which the tangent passes through the origin.

Question 19. Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis

Question 20. Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.

Question 21. Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0

Question 22. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).

Question 23. Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1

Question 24.

Question 25.

Choose the correct answer in Questions 26 and 27

Exercise 6.4

Question 1. Using differentials, find the approximate value of each of the following up to 3 places of decimal.

Question 2. Find the approximate value of f (2.01), where f (x) = 4×2 + 5x + 2

Question 3. Find the approximate value of f (5.001), where f(x) = x3 – 7×2 + 15

Question 4. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%

Question 5. Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Question 6. If the radius of a sphere is measured as 7m with an error of 0.02 m, then find the approximate error in calculating its volume.

Question 7. If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area

Question 8. If f (x) = 3×2 + 15x + 5, then the approximate value of f (3.02) is(a) 47.66 (b) 57.66 (c) 67.66 (d) 77.66

Question 9. The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

(a) 0.06 x3 m3 (b) 0.6 x3 m3 (c) 0.09 x3 m3 (d) 0.9 x3 m3

Exercise 6.5

Question 1. Find the maximum and minimum values, if any, of the following functions

Question 2. Find the maximum and minimum values, if any, of the following functions

Question 3. 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 :

Note: If f'(x) = 0 at only one point and no close interval for x is given, then this point is either local as well as absolute minima or local as well as absolute maxima

Note: If f ‘(x) = 0 at only two points and no closed interval of x is given, then one of them is the point of local as well as absolute minima and the other is the point of local as well as absolute maxima

Question 4. Prove that the following functions do not have maxima or minima:

Question 5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

Question 6. Find the maximum profit that a company can make, if the profit functionis given by p(x) = 41 – 24x – 18x2

Question 7. Find both the maximum value and the minimum value of 3x4 – 8x3 + 12x2 – 48x + 25 on the interval [0, 3]

Question 8. At what points in the interval [0, 2p], does the function sin 2x attain its maximum value?

Question 9. What is the maximum value of the function sin x + cos x?

Question 10. Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1]

Question 11. It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a

Question 12. Find the maximum and minimum values of x + sin 2x on [0, 2pi]

Question 13. Find two numbers whose sum is 24 and whose product is as large as possible.

Question 14. Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Question 15. Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum

Question 16. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum

Question 17. A square piece of t in of side 18 cm is to be 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.

Question 18. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box withouttop, 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 maximum?

Question 19. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area

Question 20. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Question 21. 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 ?

Question 22. 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 ?

Question 23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

Question 24. Show that the right circular cone of least curved surface and given volume has an altitude equal to root 2 times the radius of the base

Question 25. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan–1 root 2.

Question 26. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin–1 (1/3).

Choose the correct answer in the Questions 27 and 29

Miscellaneous Exercise

Question 1. Using differentials, find the approximate value of each of the following:

Question 2. Show that the function given by f (x) =log x /x has maximum at x = e.

Question 3. 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?

Question 4. Find the equation of the normal to curve y2 = 4x at the point (1, 2)

Question 5. Show that the normal at any point q to the curve x = a cos q + a q sin q, y = a sin q – aq cos q is at a constant distance from the origin.

Question 6. Find the intervals in which the function f given by

Note: cos x is +ve in first and fourth quadrant and –ve in second and third quadrant

Question 7. Find the intervals in which the function f given by f (x) = x3 + 13x, x ¹ 0 is

Question 8. Find the maximum area of an isosceles triangle inscribed in the ellipse

Question 9. A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is 8 m3. If building of tank costs ` 70 per sq metres for the base and ` 45 per square metre for sides. What is the cost of least expensive tank?

Question 10. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Question 11. A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.

Question 12. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Question 13. Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has

(i) local maxima (ii) local minima (iii) point of inflexion

Question 14. Find the absolute maximum and minimum values of the function f given

Question 15. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3

Question 16. Let f be a function defined on [a, b] such that f ‘(x) > 0, for all x E (a, b).Then prove that f is an increasing function on (a, b).

Question 17. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/root3. Also find the maximum volume

Question 18. Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle a is one-third that of the cone and the greatest volume of cylinder is 4/27 pih3 tan2 a.

Choose the correct answer in questions from 19 to 24

Question 19. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

(a) 1 m3/h (b) 0.1 m3/h (c) 1.1 m3/h (d) 0.5 m3/h

Sol. (a) Let h be the height of the wheat filled in the cylindrical tank at any instant

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