A robotic welding head follows a path C C\,C in the xyxyxy-plane described by the parametric equations
x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤π x = 2\sin \theta + 5\cos \theta, \quad y = 4\cos^2 \theta + 2\sin \theta, \quad 0 \le \theta \le \pi x=2sinθ+5cosθ,y=4cos2θ+2sinθ,0≤θ≤πShow that dydx=1\displaystyle \frac{dy}{dx} = 1dxdy=1 where θ=0\theta = 0θ=0.
The point P P\,P lies on C C\,C where θ=0\theta = 0θ=0.
Find the equation of the tangent to the path C C\,C at the point PPP, giving your answer in the form y=mx+cy = mx + cy=mx+c.
The tangent to the path at P P\,P intersects the curve C C\,C again at the point QQQ.
Show that the value of θ \theta\,θ at point Q Q\,Q satisfies the equation
4cos2θ−5cosθ+1=0 4\cos^2 \theta - 5\cos \theta + 1 = 0 4cos2θ−5cosθ+1=0Hence find the exact value of the yyy-coordinate of QQQ.
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.