The output signal RRR, measured in millivolts, of a high-precision sensor is modeled by the function
R(t)=12tln(4t)t>0 R(t) = 12\sqrt{t} \ln(4t) \quad t > 0 R(t)=12tln(4t)t>0where ttt is the time in milliseconds after the sensor is triggered. The signal is zero at time TTT.
State the value of TTT.
The signal reaches its minimum value at the point MMM.
Use calculus to find the exact coordinates of MMM.
Hence find the range of the function S(t)S(t)S(t) where
S(t)=−5R(t) S(t) = -5R(t) S(t)=−5R(t)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.