Mathematical Methods · Unit 3 · Further applications of differentiation · The second derivative and applications of differentiation
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A particle moves along a straight line. Its displacement from a fixed origin at time \( t \) seconds is given by \( s(t) = 2t^3 - 9t^2 + 12t + 5 \), where \( s \) is measured in metres. Determine the rate of change of the velocity at \( t = 2 \) seconds.
If the first derivative of a function is f'(x) = 3x² - 6x, what does the second derivative f''(x) represent?
Consider the function \( f(x) = x^4 - 8x^3 + 18x^2 \). (a) Determine the derivative \( f'(x) \). (1 mark) (b) Determine the second derivative \( f''(x) \). (1 mark) (c) Find all values of \( x \) for which the rate of change of \( f'(x) \) is zero. (1 mark) (d) Hence, determine the coordinates of all points on the graph of \( y = f(x) \) where the rate of change of the gradient function is zero. (2 marks)