The rate of change of the temperature of a bowl of soup (SSS) cooling on a table is directly proportional to the difference between the temperature of the soup and the temperature of the room (18∘C18^{\circ}\text{C}18∘C).
Write down a differential equation for this relationship.
Show that S=18+AektS = 18 + Ae^{kt}S=18+Aekt where A A\,A and k k\,k are constants.
Given that the initial temperature of the soup is 98∘C98^{\circ}\text{C}98∘C, write down the value of AAA.
After 5 minutes, the temperature of the soup is 58∘C58^{\circ}\text{C}58∘C. Show that k=−15ln2\displaystyle k = -\frac{1}{5} \ln 2k=−51ln2.
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.