The vertical displacement, y y\,y millimetres, of a vibrating plate in a laboratory experiment is modelled by the function y=f(t)y = f(t)y=f(t), where
f(t)=(t−4)(2t+1)2 f(t) = (t - 4)(2t + 1)^2 f(t)=(t−4)(2t+1)2for t≥−1t \ge -1t≥−1, where t t\,t is the time in seconds.
The graph of y=f(t)y = f(t)y=f(t) touches the ttt-axis at the point P P\,P and crosses the ttt-axis at the point QQQ.
State the coordinates of the point PPP.
Find f′(t)f'(t)f′(t).
Hence show that the equation of the tangent to the curve at the point where t=2.5t = 2.5t=2.5 can be expressed in the form y=ky = ky=k, where k k\,k is a constant to be found.
The displacement is modified to y=f(t+b)y = f(t + b)y=f(t+b), where b b\,b is a constant. This new curve passes through the origin (0,0)(0, 0)(0,0).
State the possible values of bbb.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.