Given that y=e−2t(sin2t+cos2t)y = \mathrm{e}^{-2t}(\sin 2t + \cos 2t)y=e−2t(sin2t+cos2t), find dydt\frac{\mathrm{d}y}{\mathrm{d}t}dtdy. Simplify your answer.
Hence, find
∫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 aaa is a rational number.
The displacement sss (in μm\mu\text{m}μm) of a micro-mechanical resonator at time ttt (in seconds) is modeled by s(t)=e−2tsin2ts(t) = \mathrm{e}^{-2t} \sin 2ts(t)=e−2tsin2t for t≥0t \ge 0t≥0. The areas of the finite regions bounded by the curve and the ttt-axis are denoted by A1,A2,…,An,…A_1, A_2, \dots, A_n, \dotsA1,A2,…,An,… where A1A_1A1 is the area of the region from t=0t=0t=0 to the first positive root.
(i) Find the exact value of the area A1A_1A1.
(ii) Show that An+1An=e−π\frac{A_{n+1}}{A_n} = \mathrm{e}^{-\pi}AnAn+1=e−π.
(iii) Show that the exact value of the total area enclosed between the curve and the ttt-axis for t≥0t \ge 0t≥0 is
1+e−π4(1−e−π) \frac{1 + \mathrm{e}^{-\pi}}{4(1 - \mathrm{e}^{-\pi})} 4(1−e−π)1+e−πor equivalently eπ+14(eπ−1)\frac{\mathrm{e}^{\pi} + 1}{4(\mathrm{e}^{\pi} - 1)}4(eπ−1)eπ+1
825 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.