An electronic sensor measures a damped oscillation signal given by the function V(t)=e−2tsin(2t)V(t) = \mathrm{e}^{-2t} \sin(2t)V(t)=e−2tsin(2t) for t≥0t \ge 0t≥0, where t t\,t is time in milliseconds.
Given that y=e−2t(sin2t+cos2t)y = \mathrm{e}^{-2t}(\sin 2t + \cos 2t)y=e−2t(sin2t+cos2t), find dydt\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}dtdy. Simplify your answer.
Hence, show that
∫e−2tsin2t dt=ae−2t(sin2t+cos2t)+C \int \mathrm{e}^{-2t} \sin 2t \, \mathrm{d}t = a \mathrm{e}^{-2t}(\sin 2t + \cos 2t) + C ∫e−2tsin2tdt=ae−2t(sin2t+cos2t)+Cwhere a a\,a is a rational number to be determined.
The areas of the finite regions bounded by the signal curve and the ttt-axis are denoted by A1,A2,…,An,… A_1, A_2, \dots, A_n, \dots\,A1,A2,…,An,… where A1 A_1\,A1 is the area of the first pulse (the region between t=0t=0t=0 and the first root of V(t)=0V(t) = 0V(t)=0 for t>0t > 0t>0).
Find the exact value of the area A1A_1A1.
Show that the ratio of successive areas An+1An\displaystyle \frac{A_{n+1}}{A_n}AnAn+1 is constant and find its value in terms of e\mathrm{e}e.
Show that the exact value of the total area enclosed between the signal curve and the ttt-axis for all t≥0 t \ge 0\,t≥0 is
1+e−π4(1−e−π) or equivalently eπ+14(eπ−1) \frac{1 + \mathrm{e}^{-\pi}}{4(1 - \mathrm{e}^{-\pi})} \text{ or equivalently } \frac{\mathrm{e}^{\pi} + 1}{4(\mathrm{e}^{\pi} - 1)} 4(1−e−π)1+e−π or equivalently 4(eπ−1)eπ+1825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.