An industrial laser cutter follows a trajectory C C\,C defined by the parametric equations
x=2p2,y=23p3+4p2−14p+k x = 2p^2, \quad y = \frac{2}{3}p^3 + 4p^2 - 14p + k x=2p2,y=32p3+4p2−14p+kwhere k k\,k is a constant and p≠0p \neq 0p=0.
Find dydx\displaystyle \frac{dy}{dx}dxdy in terms of ppp.
The line l l\,l is the normal to the curve C C\,C at the point A A\,A where p=1p = 1p=1.
Given that the tangent to C C\,C at the point B B\,B is parallel to lll,
show that the parameter p p\,p at point B B\,B is a solution of the equation
p2+2p−7=0 p^2 + 2p - 7 = 0 p2+2p−7=0Hence find the value of p p\,p at BBB, justifying your choice given that the xxx-coordinate of B B\,B is greater than 10.
Given that the yyy-intercept of l l\,l is 23\displaystyle \frac{2}{3}32,
determine the value of kkk.
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.