MATH 125 Final: MATH 125 UWashington Quiz125FinalA13

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turquoisegnu128

15 Feb 2019

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MATH 125

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$MATH 125 Full Course Notes$

MATH 125 Full Course Notes

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Mathematical economics final, december 7, 2012: let a = (cid:18) 3 1. Answer: the characteristic equation is 2 6 + 8 = 0, yielding eigenvalues (a) = The corresponding eigenvectors are v2 = (1, 1)t and v4 = (1, 1)t. let p = 2 (cid:18) 2 + 2 2 + 2. 0: consider the di erential equation x + 4x = 0. 2(cid:19): find the solution to the above equation with initial conditions x(0) = x0 and. Answer: we nd the characteristic equation by substituting e t for x. Using the initial conditions we nd = x0 and = x1/2. A rather harder approach is to let y = x. That yields the system d dt (cid:18) x y(cid:19) = (cid:18) 0. The characteristic equation is again 2 + 4 = 0 with eigenvalues 2i. Then nding the eigenvectors v1 and v2, let p = [v1, v2], and the solution is y(cid:19) = p 1(cid:18) e2it (cid:18) x.

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[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7<-3 < 7 which gives -4 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al <, then f(c) >

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