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COMEDK202510 May 2025Evening ShiftMathematicsApplication of DerivativesActual

x=a( + ) and y=a(1- ) represents the equation of a curve. If changes at a constant rate k then the rate of change of the slope of the tangent to the curve at = 3 is

Options

  1. A2 k 3
  2. B2 k 3
  3. C2 k
  4. Dk 3

Correct answer

B. 2 k 3

Step-by-step solution

Given x = a( + ) and y = a(1 - ) . The slope of the tangent is m = dy dx = dy/d dx/d . Calculating derivatives with respect to : dy d = a dx d = a(1 + ) Thus, m = a a(1 + ) = 2 ( /2) ( /2) 2 ^2( /2) = ( /2) . We need the rate of change of the slope with respect to time t , which is dm dt = dm d d dt . Given d dt = k , we have dm d = d d ( /2) = 1 2 ^2( /2) . At = 3 , /2 = 6 . dm d = 1 2 ^2( /6) = 1 2 ( 2 3 )^2 = 1 2 4 3 = 2 3 . Therefore, dm dt = 2 3 k = 2k 3 . Answer: 2 k 3

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