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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.