PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
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1 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Water seeps out of a conical filter at the constant rate of 5 cc / sec. When the height of water level in the cone is 5 cm, find the rate at which the height decreases. The filter is 0 cm high and the radius of its base is 0 cm. Ans: 5 cm / sec ( ) A ship travels southwards at the speed of km / hr and another ship 8 km to its south travels eastwards at 8 km / hr. ( a ) Find the rate at which the distance between the two ships increases after hour. ( b ) Find the rate at which their distance increases after two hours. ( c ) Eplain the difference in the signs of the two rates. [ Ans: ( a ) - 8. km / hr, ( b ) 8 km / hr, ( c ) distance between the ships initially decreases till the ships are closest to each other. This happens at time, t =.8 hr when the rate of change of distance between them is zero. ] ( ) The period T of simple pendulum of length l is given by the formula T = the length is increased by %, what is the approimate change in the period? [ Ans: % increase ] ( ) Find the approimate value of sec - ( - 0 ). Ans: - 00 l. If g ( 5 ) A formula for the amount of electric current passing through the tangent galvanometer is i = k tan θ, where θ is the variable and k is a constant. Prove that the relative error in i is minimum when θ =. ( 6 ) Find the radian measure of the angle between the tangents to y = a and = ay at their point of intersection other than the origin. Ans : - tan ( 7 ) Prove that the portion of any tangent to the curve y = a which lies between the co-ordinate aes is of constant length. ( a > 0 ).
2 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( 8 ) If λ λ, then prove that the curves y a λ b λ = and other orthogonally. y a λ b λ ( 9 ) Verify Rolle s Theorem for f ( ) = sin - sin, [ 0, ]. = intersect each ( 0 ) For f ( ) = - 6 a b, it is given that f ( ) = f ( ) = 0. Find a and b and (, ) such that f ( ) = 0. Ans : a =, b = - 6, = ± ( ) Apply mean value theorem to f ( ) = log ( ) over the interval [ 0, ] and prove that 0 < - <, ( > 0 ). log ( ) ( ) Apply mean value theorem to f ( ) = e over the interval [ 0, ] and prove that e - 0 < log <, ( > 0 ). ( ) Prove that for > 0, < log ( ) <. ( ) Length of each of the three sides of a trapezium is 5a. What should be the length of its fourth side if its area is maimum possible? [ Ans: 0a ] ( 5 ) A 8 metre long wire is to be cut into pieces. One piece is to be bent to form a square and another piece is to be bent to form a circle. If the total area of the square and the circle has to be minimized, where should the wire be cut? 8 Ans: m for square and m for circle
3 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( 6 ) A window is in the shape of a semicircle over a rectangle. If the total perimeter of the window is to be kept constant and the maimum amount of light has to pass through the window, then prove that the length of the rectangle should be twice its height. ( 7 ) The illumination due to an electric bulb at any point varies directly as the candlepower of the bulb and inversely as the square of the distance of the point from the bulb. Two electric bulbs of candlepowers C and 8C are placed 6 metres apart. At what point between them, the total illumination is minimum? [ Ans: m from the bulb having candlepower C ] ( 8 ) The perimeter of an isosceles triangle is constant and it is 00 cm. Its base increases at the rate of 0. cm / sec. Find the rate at which the altitude on the base increases, when the length of the base is 0 cm. Ans : - 0 cm / sec ( 9 ) A boy flies a kite at a height of 50 m. The kite moves away from the boy at a horizontal velocity of 6 m / sec. Find the rate at which the string is released when the kite is 0 m away from the boy. Ans : 7 m / sec ( 0 ) If a 5 % error is committed in the measurement of the radius of a sphere, what percentage of error will be committed in the calculation of its volume? [ Ans: 5 % ] ( ) If the error in measuring the radius of the base of a cone is δr and if its height is constant, what is the error committed in calculating the total surface area of the cone? ( r h r radius of base = Ans: r δr, r h h = height of cone ( ) In the calculation of the area of a triangle using the formula = bc sin A, A was taken as / 6. Actually, an % error crept into this measure of A. If b, c are constants, what is the percentage error in calculation of the area? Ans: % 6
4 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Prove that y = and 6y = 7 intersect orthogonally. ( ) Find the equation of the tangent to 9 y = 6 which is perpendicular to the line - y = 0. [ Ans: y = ± 6 ] ( 5 ) Find the measure of the angle between y = a and y = a. [ Ans: Angle is zero as both touch each other at ( a, a ) and ( - a, - a ) ] ( 6 ) Prove that 9y = 5 and - y = 5 intersect orthogonally. ( 7 ) Determine whether Rolle s theorem is applicable in the following cases and if so, determine c such that f ( c ) = 0. ( i ) f ( ) =, - < = 7 -, < 6 [ Ans: f is not differentiable for =, f ( ) 0 for any ( -, 6 ) ] ( ii ) f ( ) =, < = 5 -,, [, 5 ] [ Ans: f is not differentiable for =, f ( ) 0 for any (, 5 ) ] ( iii ) f ( ) = l l, [ -, ] [ Ans: f is not differentiable for = 0, f ( ) 0 for any ( -, ) ] ( 8 ) Apply mean value theorem to f ( ) = a b c d on [ 0, ], a, b > 0. Ans : - b a a ab b ( 9 ) Find where is increasing and where it is decreasing. Ans : f increases in l 7 7 <, R, f increases in l >, R
5 PROBLEMS - APPLICATIONS OF DERIVATIVES Page 5 ( 0 ) Prove that e >, R - { 0 }. ( ) Prove that tan is increasing over 0, ( ) If > 0, prove that log ( ) > -.. Check for global or local etreme values: ( to 7 ) ( ) f ( ) = , [ -, ]. Ans : global mam. f ( - ) = 57, global min. f ( ) = -, local min. f ( 0 ) = 5 and f ( ) = -, local mam. f ( ) f ( ) = - -, [ -, ]. Ans : f ( - ) = 9 6 = 87 6 is a local and global maimum, f ' ( ) = 0, but ( -, ), f ( ) = - is a global minimum. ( 5 ) f ( ) = 50-0, [ 0, ]. Ans : global maimum local minimum is f ( 0 ) = 0. For = 5 5 f ( ) = , ( 6 ) f ( ) = sin cos, R. Ans: Ma. f k, = 5 Min. f k = -
6 PROBLEMS - APPLICATIONS OF DERIVATIVES Page 6 ( 7 ) f ( ) = sin, [ 0, ]. Ans: local ma. f, local min. f, = = - global min. f ( 0 ) = 0, global ma. f ( ) =. ( 8 ) If the length of the hypotenuse of a right triangle is given, prove that its area is maimum when it is an isosceles right triangle. ( 9 ) If the volume of a right circular cone with given oblique height is maimum, prove that the radian measure of its semi vertical angle is tan. ( 0 ) Find approimate value of log 999. [ Ans: ] 0 ( ) Prove that the length of the tangent is constant for the curve t = a cos t log tan, y = a sin t. ( ) Prove that for the curve y = a, the mid-point of the segment of a tangent intercepted between the two aes is precisely the point of contact ( i.e., the point of tangency ). ( ) Using mean value theorem, prove that if > 0, then ( ) Prove that if > 0, then log ( ) is a decreasing function. - < tan <. ( 5 ) By cutting equal squares from the four corners of a 6 0 tin sheet, a bo is to be constructed. What should be the length of each square if the volume of the bo is to be maimum? [ Ans: ]
7 PROBLEMS - APPLICATIONS OF DERIVATIVES Page 7 ( 6 ) Prove that is minimum when = e. ( 7 ) The cost of manufacturing TV sets per day is 5 5 and the sale price of one TV set then is How many TV sets should be manufactured to maimize profit? Prove that then the cost of making one TV set is minimum. ( 8 ) Find the minimum distance of (, ) from y = 8. [ Ans: ] ( 9 ) Which line passing through (, ) makes a triangle of minimum area with the two aes in the first quadrant? [ Ans: y = ] ( 50 ) The equation of motion of a particle moving on a line is s = t at bt c, where displacement s is in m, time t is in sec., and a, b, c, are constants. t = s = 7 and velocity, v = 7 m/s acceleration, a = m / s. Find a, b, c. [ Ans: a =, b = -, c = 5 ] ( 5 ) Find the rate of increase of the volume, V, of a sphere with respect to its surface dv area, S. Ans: = S / ds ( 5 ) Find the area of an equilateral triangle with respect to its perimeter, p. Ans: p / 6 ( 5 ) A m tall man walks towards a source of light situated at the top of a pole 8 m high, at the rate of.5 m / s. How fast is the length of his shadow changing when he is 5 m away from the pole? [ Ans: - m / s ]
8 PROBLEMS - APPLICATIONS OF DERIVATIVES Page 8 ( 5 ) Two sides of a triangle are 5 and 0. How fast is the third side increasing when the angle between the given sides is 60 and is increasing at the rate of per sec.? Ans: 9 unit / s ( 55 ) A metal ball of radius 90 cm. is coated with a uniformly thick layer of ice, which is melting at the rate of 8 cc / min. Find the rate at which the thickness of the ice is decreasing when the ice is 0 cm thick. [ Ans: cm / min ] ( 56 ) Obtain the equations of the tangent and the normal to the curve = a cos θ and y = b sin θ at θ-point. y a by Ans: cos θ sin θ, a b = - = - a b cos θ sin θ ( 57 ) Obtain the equations of the tangent and the normal to the curve n n y a b = at a point where = a. y Ans:, a by a b = - = - a b ( 58 ) Obtain the equations of the tangent and the normal to the curve = e t, y = e t at t = 0. [ Ans: - y = 0, y - 5 = 0 ] ( 59 ) Find the lengths of the sub-tangent, sub-normal, tangent and normal at θ = curve = a ( θ sin θ ) and y = a ( - cos θ ). to the a a Ans: l a l,, l a l,
9 PROBLEMS - APPLICATIONS OF DERIVATIVES Page 9 ( 60 ) For f ( ) = ( - a ) m ( - b ) n, [ a, b ] and m, n N, determine whether Rolles theorem is applicable and if so, determine c ( a, b ) such that f ( c ) = 0. Ans: Rolles' theorem is applicable and c = mb m na n ( 6 ) Apply the mean value theorem to f ( ) = cos, 0, tan <. ( 6 ) Show that 0 < < < ( 6 ) For > 0, prove that - ( 6 ) Using f ( ) = - sin < - sin. < tan - <., > 0, which is greater, e or and prove that e? [ Ans: e ] ( 65 ) Apply Mean value theorem to f ( ) = log ( ) over the interval [ a, b ], where b - a b b a 0 < a < b and prove that log - < < b a. a ( 66 ) Using f ( ) = tan sin -, 0 < < tan, prove that <. sin ( 67 ) Prove that the rectangle inscribed in a given semi-circle has maimum area, if its length is double its breadth. ( 68 ) Fuel cost of running a train is proportional to square of its speed in km / hr. and is Rs. 00 / hr at 0 km / hr. What is its most economical speed, if the fied charges are Rs. 00 per hour? [ Ans: 0 km / hr ]
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