## 辅导案例-MAS241

• August 11, 2020

❉❛t❛ Pr♦✈✐❞❡❞✿ ❋♦r♠✉❧❛ ❙❤❡❡t MAS241 SCHOOL OF MATHEMATICS AND STATISTICS August Supplementary Exam Period 2019-2020 Mathematics II (Electrical) 1 hour (nominal) ❚❤✐s ✐s ❛♥ ♦♣❡♥ ❜♦♦❦ ❡①❛♠✳ ❆♥s✇❡r ❛❧❧ q✉❡st✐♦♥s✳ ❚❤❡ s✉❜♠✐ss✐♦♥ ❞❡❛❞❧✐♥❡ ✐s ✶✵ ❛♠ ✭❇❙❚✮✱ t✇❡♥t②✲❢♦✉r ❤♦✉rs ❛❢t❡r ✐t ✐s r❡❧❡❛s❡❞✳ ▲❛t❡ s✉❜✲ ♠✐ss✐♦♥ ✇✐❧❧ ♥♦t ❜❡ ❝♦♥s✐❞❡r❡❞ ✇✐t❤♦✉t ❡①t❡♥✉❛t✐♥❣ ❝✐r❝✉♠st❛♥❝❡s✳ ■t ✐s ❡①♣❡❝t❡❞ t❤❛t ②♦✉ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ❝♦♠♣❧❡t❡ t❤✐s ❡①❛♠ ✐♥ ❛♣♣r♦①✐♠❛t❡❧② ♦♥❡ ❤♦✉r ❛♥❞ ✐t ✐s r❡❝♦♠♠❡♥❞❡❞ t❤❛t ②♦✉ s✉❜♠✐t t❤❡ ✇♦r❦ ✇✐t❤✐♥ ❢♦✉r ❤♦✉rs✳ ❨♦✉ ✇✐❧❧ ♥♦t ❜❡ ♣❡♥❛❧✐s❡❞ ❢♦r t❛❦✐♥❣ ❧♦♥❣❡r✱ ❤♦✇❡✈❡r✳ ❯♥❧❡ss ✐t ✐s ❡①♣❧✐❝✐t❧② st❛t❡❞ ♦t❤❡r✇✐s❡✱ ✐t ✐s ✐♥t❡♥❞❡❞ t❤❛t ❝❛❧❝✉❧❛t✐♦♥s ❛r❡ ♣❡r❢♦r♠❡❞ ❜② ❤❛♥❞ ✭♣♦ss✐❜❧② ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✮✳ ❚♦ ❣❛✐♥ ❢✉❧❧ ♠❛r❦s✱ ②♦✉ ✇✐❧❧ ♥❡❡❞ t♦ s❤♦✇ ②♦✉r ✇♦r❦✲ ✐♥❣✳ ❨♦✉ ✇✐❧❧ ♥♦t ❣❡t ❢✉❧❧ ♠❛r❦s ✐❢ ②♦✉ s✐♠♣❧② ✇r✐t❡ ❞♦✇♥ ♦✉t♣✉t ❢r♦♠ ❛ ❝♦♠♣✉t❡r ♣❛❝❦❛❣❡✳ ❇② ✉♣❧♦❛❞✐♥❣ ②♦✉r s♦❧✉t✐♦♥s ②♦✉ ❞❡❝❧❛r❡ t❤❛t ②♦✉r s✉❜♠✐ss✐♦♥ ❝♦♥s✐sts ❡♥t✐r❡❧② ♦❢ ②♦✉r ♦✇♥ ✇♦r❦✱ t❤❛t ❛♥② ✉s❡ ♦❢ s♦✉r❝❡s ♦r t♦♦❧s ♦t❤❡r t❤❛♥ ♠❛t❡r✐❛❧ ♣r♦✈✐❞❡❞ ❢♦r t❤✐s ♠♦❞✉❧❡ ✐s ❝✐t❡❞ ❛♥❞ ❛❝❦♥♦✇❧❡❞❣❡❞ ❛♥❞ t❤❛t ♥♦ ✉♥❢❛✐r ♠❡❛♥s ❤❛✈❡ ❜❡❡♥ ✉s❡❞✳ ❚♦t❛❧ ♠❛r❦s ✹✵✳ Please leave this exam paper on your desk Do not remove it from the hall ❘❡❣✐str❛t✐♦♥ ♥✉♠❜❡r ❢r♦♠ ❯✲❈❛r❞ ✭✾ ❞✐❣✐ts✮ t♦ ❜❡ ❝♦♠♣❧❡t❡❞ ❜② st✉❞❡♥t MAS241 1 Turn Over MAS241 Blank MAS241 2 Continued MAS241 ✶ ✭✐✮ ❚❤❡ ❝❤❛r❣❡ q(t) ♦❢ ❛ ❝✉rr❡♥t ✐♥ ❛ ❝✐r❝✉✐t ✐s ❞❡s❝r✐❜❡❞ ❜② q′′(t) + 2q′(t) + 5q(t) = δ(t) s✉❜❥❡❝t t♦ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s q(0) = q′(0) = 0✱ ✇❤❡r❡ δ(t) st❛♥❞s ❢♦r t❤❡ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝❤❛r❣❡ q(t) ❛t t✐♠❡ t > 0✳ ✭✶✵ ♠❛r❦s✮ ✭✐✐✮ ▲❡t f(t)←→ 2e−jω 1 + ω2 ❜❡ ❛ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♣❛✐r✳ ❋✐♥❞ f(t)✳ ✭✺ ♠❛r❦s✮ ✭✐✐✐✮ ▲❡t f(t) = e−|t| ❛♥❞ g(t) = r❡❝t2(t)✳ ❋✐♥❞ F{f ∗ g(t)}✳ ✭✺ ♠❛r❦s✮ ✷ ✭✐✮ ▲❡t D ⊂ R2 ❜❡ t❤❡ r❡❣✐♦♥ ❜♦✉♥❞❡❞ ❜② t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s (0, 0)✱ (1, 0)✱ (1, 1)✳ ❋✐♥❞ ∫∫ D ex 2 dA. ✭✶✵ ♠❛r❦s✮ ✭✐✐✮ ▲❡t f(x, y) = √ x2 + y2 ❛♥❞ D := {(x, y) : x2 + y2 ≤ 1}✳ ❋✐♥❞ ∫∫ D f(x, y)dA. ✭✶✵ ♠❛r❦s✮ End of Question Paper MAS241 3 MAS241 FORMULA SHEET Laplace transform: The Laplace transform of a function f(t) is given by: L{f(t)}(s) := ∫ ∞ 0 e−stf(t)dt. Properties of the Laplace transform: L{f(t)} = F (s) in the following table. L{af(t) + bg(t)} = aL{f(t)}+ bL{g(t)} linearity L{f ′(t)} = sF (s)− f(0) differentiation w.r.t. t L{f ′′(t)} = s2F (s)− sf(0)− f ′(0) second differentiation w.r.t. t L{e−ktf(t)} = F (k + s) frequency shift L{f(t− a)H(t− a)} = e−asF (s) (for a > 0) time shift L{f(at)} = 1 a F ( s a ) (for a > 0) scaling L{f ∗ g(t)} = L{f(t)}L{g(t)} (for f(t), g(t) causal) convolution Table of standard Laplace transforms: f(t) L{f(t)}(s) Region of validity tn (for n ≥ 0) n! sn+1 Re(s) > 0 sin(kt) k s2+k2 Re(s) > 0 cos(kt) s s2+k2 Re(s) > 0 H(t− T ) (for T ≥ 0) e −sT s Re(s) > 0 δ(t− T ) (for T ≥ 0) e−sT s ∈ C Fourier transform: The Fourier transform and inverse Fourier transforms are given by: F{f(t)}(ω) = F (ω) := ∫ ∞ −∞ f(t)e−jωtdt, f(t) = F−1{F (ω)} = 1 2pi ∫ ∞ −∞ F (ω)ejωtdω. Properties of the Fourier transform: F{f(t)} = F (ω) in the following table: F{ejθtf(t)} = F (ω − θ) frequency shift F{f(t− T )} = e−jωTF (ω) time shift F{f (n)(t)} = (jω)nF (ω) differentiation F{F (t)} = 2pif(−ω) symmetry F{f(at)} = 1 |a| F (ω a ) scaling F{f ∗ g(t)} = F{f(t)}F{g(t)} convolution Table of standard Fourier transforms: f(t) F{f(t)}(ω) e−a|t| (for a > 0) 2a a2+ω2 rectT (t) sinc( Tω 2 ) 1 2piδ(ω) Fourier series: The Fourier series of a periodic function f(t) with fundamental period T is given by S[f ] = a0 2 + ∞∑ n=1 ( an cos(ωnt) + bn sin(ωnt) ) where ωn = 2pin T , an = 2 T ∫ T/2 −T/2 f(t) cos(ωnt)dt, bn = 2 T ∫ T/2 −T/2 f(t) sin(ωnt)dt. Coordinate systems: Cylindrical polar coordinates (x, y, z) = (r cos(θ), r sin(θ), z) (r, θ, z) = (√ x2 + y2, arctan( y x ), z ) dV = rdrdθdz. Spherical polar coordinates (x, y, z) = (ρ sin(φ) cos(θ), ρ sin(φ) sin(θ), ρ cos(φ)) (ρ, θ, φ) = (√ x2 + y2 + z2, arctan( y x ), arccos( z ρ ) ) dV = ρ2 sin(φ)dρdφdθ.

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