The vertical profile of a sculpted roller coaster rail is modeled by the function h(x)=(x−4)(2x+5)2h(x) = (x - 4)(2x + 5)^2h(x)=(x−4)(2x+5)2 for x≥−3x \ge -3x≥−3, where hhh is the height in decimetres and xxx is the horizontal distance from a sensor.
The rail touches the baseline at point PPP and crosses the baseline at point QQQ.
State the coordinates of the point PPP.
Determine h′(x)h'(x)h′(x).
Hence show that the equation of the tangent to the rail at the point where x=116x = \frac{11}{6}x=611 can be expressed in the form y=ky = ky=k, where kkk is a constant to be found.
A modification shifts the track horizontally so the equation becomes y=h(x+b)y = h(x + b)y=h(x+b), where bbb is a constant. The modified track now passes through the sensor's origin O(0,0)O(0,0)O(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.