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COMEDK2024Morning ShiftMathematicsApplication of DerivativesActual

For a given curve y=2 x-x^2 , when x increases at the rate of 3 units/sec, then how does the slope of the curve change?

Options

  1. AIncreasing at 3 units/sec
  2. BDecreasing at 6 units/sec
  3. CDecreasing at 3 units/sec
  4. DIncreasing at 6 units/sec

Correct answer

B. Decreasing at 6 units/sec

Step-by-step solution

The equation of the curve is given by y = 2x - x^2 . The slope of the curve at any point x is given by m = dy dx = 2 - 2x . To find the rate of change of the slope with respect to time t , we differentiate m with respect to t using the chain rule: dm dt = d dt (2 - 2x) = -2 dx dt . Given that the rate of increase of x is dx dt = 3 units/sec, we substitute this value into the expression: dm dt = -2 3 = -6 units/sec. The negative sign indicates that the slope is decreasing at a rate of 6 units/sec. Answer: Decreasing

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