13%20-%20Applications%20of%20Derivatives

13%20-%20Applications%20of%20Derivatives - PROBLEMS 13 -...

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PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 ( 1 ) 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 15 cm, find the rate at which the height decreases. The filter is 20 cm high and the radius of its base is 10 cm. 4 Ans: cm sec 45  /   π ( 2 ) A ship travels southwards at the speed of 24 km / hr and another ship 48 km to its south travels eastwards at 18 km / hr. ( a ) Find the rate at which the distance between the two ships increases after 1 hour. ( b ) Find the rate at which their distance increases after two hours. ( c ) Explain the difference in the signs of the two rates. [ Ans: ( a ) - 8.4 km / hr, ( b ) 18 km / hr, ( c ) distance between the ships initially decreases till the ships are closest to each other. This happens at time, t = 1.28 hr when the rate of change of distance between them is zero. ] ( 3 ) The period T of simple pendulum of length l is given by the formula T = 2 π g l . If the length is increased by 2 % , what is the approximate change in the period ? [ Ans: 1 % increase ] ( 4 ) Find the approximate value of sec - 1 ( - 2 01 ). 21 Ans: 3 200 3 π - ( 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 = 4 π . ( 6 ) Find the radian measure of the angle between the tangents to y 2 = 4ax and x 2 = 4ay at their point of intersection other than the origin. 4 3 tan : Ans 1 - ( 7 ) Prove that the portion of any tangent to the curve 3 2 3 2 3 2 a y x = + which lies between the co-ordinate axes is of constant length. ( a > 0 ).
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PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 2 ( 8 ) If λ 1 λ 2 , then prove that the curves 22 11 xy 1 a b + = and 1 + = intersect each other orthogonally. ( 9 ) Verify Rolle’s Theorem for f ( x ) = sin x - sin 2x, x [ 0, π ]. ( 10 ) For f ( x ) = x 3 - 6x 2 + ax + b, it is given that f ( 1 ) = f ( 3 ) = 0. Find a and b and x ( 1, 3 ) such that f ’ ( x ) = 0. ± = = = 3 1 2 x 6, b 11, a : Ans - ( 11 ) Apply mean value theorem to f ( x ) = log ( 1 + x ) over the interval [ 0, x ] and prove that 1 x 1 ) x 1 ( log 1 0 < + < - , ( x > 0 ). ( 12 ) Apply mean value theorem to f ( x ) = e x over the interval [ 0, x ] and prove that x x 1 e log 0 x < < - , ( x > 0 ). ( 13 ) Prove that for x > 0, x ) x 1 ( log x 1 x < + < + . ( 14 ) 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 maximum possible ? [ Ans: 10a ] ( 15 ) A 28 metre long wire is to be cut into pieces. One piece bent form 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 ?
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This note was uploaded on 11/28/2011 for the course PHYSICS 300 taught by Professor Smith during the Spring '06 term at ITT Tech Flint.

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13%20-%20Applications%20of%20Derivatives - PROBLEMS 13 -...

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