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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.