The relationship between the pressure ppp and temperature ttt of a specific gas model is described by the equation
p3t+2pt3=12 p^3t + 2pt^3 = 12 p3t+2pt3=12Prove that the curve representing this relationship does not intersect the coordinate axes in the ppp-ttt plane.
Show that
dtdp=−3p2t+2t3p3+6pt2 \frac{dt}{dp} = -\frac{3p^2t + 2t^3}{p^3 + 6pt^2} dpdt=−p3+6pt23p2t+2t3Prove that the model has no stationary points.
In the case when p>0p > 0p>0, find the equation of the tangent line to the curve at the point where t=1t = 1t=1. Give your answer in the form at+bp=cat + bp = cat+bp=c, where a,b,ca, b, ca,b,c are integers.
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.