In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
The cross-section of a high-tensile structural support beam is modeled by the curve defined by the equation
3y2−4xy+2x2+6x=20 3y^2 - 4xy + 2x^2 + 6x = 20 3y2−4xy+2x2+6x=20where xxx and yyy are coordinates in decimetres. The boundary of the cross-section intersects the positive xxx-axis at the point RRR.
State the coordinates of RRR.
The curve has two stationary points, PPP and QQQ, where the tangent to the curve is horizontal.
Show that, at points PPP and QQQ,
ax2+bx+c=0 ax^2 + bx + c = 0 ax2+bx+c=0where aaa, bbb and ccc are integers to be found.
Find the xxx-coordinate of point QQQ, given that xQ>0x_Q > 0xQ>0, giving your answer to 3 decimal places.
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.