Calculate derivatives of power and polynomial functions. Mathematical Methods 2025 v1.3

Given $f(x) = 5x^4 - 3x^3 + 2x - 7$, calculate $f'(x)$ and hence find the value of $f'(2)$.

A particle moves along a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is given by \[ s(t) = 2t^4 - 9t^3 + 12t^2 + 5 \] where \(s\) is measured in metres. (a) Calculate the velocity function \(v(t)\) for the particle. (1 mark) (b) Calculate the acceleration function \(a(t)\) for the particle. (1 mark) (c) Determine the velocity of the particle when \(t = 2\) seconds. Give your answer in m/s. (1 mark) (d) Determine the acceleration of the particle when \(t = 2\) seconds. Give your answer in m/s². (1 mark)

A rectangular storage tank has a square base with side length \(x\) metres and height \(h\) metres. The volume of the tank is fixed at \(512\) cubic metres. (a) Show that the surface area \(S\) of the tank (including the base and all four walls, but not the top) can be expressed as \(S(x) = x^2 + \frac{2{,}048}{x}\). (b) Calculate \(\frac{dS}{dx}\). (c) Use your result from part (b) to determine the value of \(x\) for which the surface area is stationary.