Jac and Jill have built a ramp for their toy car. They will release the car at the top of the ramp and the car will jump off the end of the ramp. The cross-section of the ramp is modelled by the function \(f\), where \(f(x)= \begin{cases}\displaystyle \ 40 & 0 \leq x<5 \\ \dfrac{1}{800}\left(x^3-75 x^2+675 x+30\ 375\right) & 5 \leq x \leq 55\end{cases}\) \(f(x)\) is both smooth and continuous at \(x=5\). The graph of \(y=f(x)\) is shown below, where \(x\) is the horizontal distance from the start of the ramp and \(y\) is the height of the ramp. All lengths are in centimetres. --- 2 WORK AREA LINES (style=lined) --- --- 1 WORK AREA LINES (style=lined) --- --- 1 WORK AREA LINES (style=lined) --- --- 1 WORK AREA LINES (style=lined) --- Jac and Jill decide to use two trapezoidal supports, each of width \(10 cm\). The first support has its left edge placed at \(x=5\) and the second support has its left edge placed at \(x=15\). Their cross-sections are shown in the graph below. --- 5 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
Calculus, MET1 ADV 2004 1b
Differentiate `x^4 + 5x^(−1)` with respect to `x`. (2 marks)