Page 1Math 217Exam 3Name:ID:Section:This exam has 16 questions:• 15 multiple choice worth 5 points each.• 1 hand graded worth 25 points.Important:• No graphing calculators!• For the multiple choice questions, mark your answer on the answer card.• Show all your work for the written problems. You will be graded on the ease of reading yoursolution.• You are allowed a 3 × 5 note card for the exam.Function Transform Function Transformf0(t) sF (s) − f(0) 11sf00(t) s2F (s) − sf(0) − f0(0) t1s2Rt0f(τ) dτF (s)stnn!sn+1eatf(t) F (s − a) taΓ(a + 1)sa+1u(t − a)f(t − a) e−asF (s)1√πt1√s(f ∗ g)(t) =Zt0f(τ)g(t − τ ) dτ F (s)G(s) cos ktss2+ k2tf(t) −F0(s) sin ktks2+ k21tf(t)R∞sF (σ) dσ cosh ktss2− k2f(t), period p11 − e−psZp0e−stf(t) dt sinh ktks2− k2u(t − a)e−ass12k3(sin kt − kt cos k t)1(s2+ k2)2eat1s − at2ksin kts(s2+ k2)2tneatn!(s − a)n+112k(sin kt + kt cos kt)s2(s2+ k2)2sin(A + B) = sin A cos B + cos A sin Bcos(A + B) = cos A cos B − sin A sin B2 cos A cos B = cos(A − B) + cos(A + B)2 sin A sin B = cos(A − B) − cos(A + B)2 sin A cos B = sin(A − B) + sin(A + B)Page 2Math 217Exam 31. Use partial fractions to decompose the function3s2− 4s + 2(s − 1)(s2− 4s + 4)(a)−1s − 1+s + 2(s − 2)2(b)1s − 1+1s − 2+2s + 1(s − 2)2(c)1s − 1+s + 3s2− 4s + 4(d)1s − 1+2s − 2+6(s − 2)2(e)−1s − 1+6s − 2(f)1s − 1+1s2− 4s + 4(g)1s − 1+6(s − 2)2(h)1s − 1+1s − 2+1(s − 2)2(i)−1s − 1+2s − 2+2(s − 2)2(j) None of the abovePage 3Math 217Exam 32. Let f(t) = 3 cos t. The graph of f(t) is the top graph and the function g(t) is on the bottom.-3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 8 9f(t)=3cos(t)-3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 8 9g(t)Which of the following functions is g(t)?(a) u(t)f(t)(b) u(t + 2)f(t + 2)(c) u(t − 2)f(t + 2)(d) u(t + 2)f(t − 2)(e) u(t − 2)f(t − 2)(f) u(t)f (3t)(g) u(t + 2)f(3t + 2)(h) u(t − 2)f(3t + 2)(i) u(t + 2)f(3t − 2)(j) u(t − 2)f(3t − 2)(k) None of the abovePage 4Math 217Exam 33. Let f(t) = sin 3t and g(t) = 1. Find the convolution product (f ∗ g)(t).(a) t(b) −t(c)13cos 3t(d) −13cos 3t(e)13(1 − cos 3t)(f) −13(1 − cos 3t)(g) cos 3t(h) −cos 3t(i) 1 − cos 3t(j) −1 + cos 3t(k) None of the above4. Letf(t) =Zt0g(t − τ )h(τ ) dτLet F (s) = L{f(t)}, G(s) = L{g(t)} and H(s) = L{h(t)}.Suppose you know G(2) = 8 and H(2) = 3.Find F (2), choose the closest answer.(a) 2(b) 4(c) 4(d) 6(e) 10(f) 16(g) 20(h) 24(i) 40(j) 48(k) 96Page 5Math 217Exam 35. Let f(t) = L−1{F (s)} whereF (s) =1s − 4+2(s − 1)2Find f(1), choose the closest answer.(a) 5(b) 57(c) 49(d) 55(e) 57(f) 60(g) 64(h) 93(i) 101(j) 143(k) F (s) is not the Laplace transform of a function of exponential order6. Let f(t) = L−1{F (s)} whereF (s) =1s2− 2s + 5Find f(1), choose the closest answer.(a) 0.0(b) 0.2(c) 0.8(d) 1.2(e) 1.9(f) 2.3(g) 2.5(h) 2.7(i) 3.2(j) 3.3(k) F (s) is not the Laplace transform of a function of exponential orderPage 6Math 217Exam 37. Let f(t) = L−1{F (s)} wherelns − 1s + 3Find f(1), choose the closest answer.(a) −15.2(b) −7.6(c) −3.8(d) −2.7(e) −1.9(f) 0.0(g) 1.9(h) 2.7(i) 3.8(j) 7.6(k) F (s) is not the Laplace transform of a function of exponential order8. Let f(t) = L−1{F (s)} whereF (s) =s − 1s + 1Find f(1), choose the closest answer.(a) −20(b) −15(c) −10(d) −5(e) 0(f) 5(g) 10(h) 15(i) 20(j) 25(k) F (s) is not the Laplace transform of a function of exponential orderPage 7Math 217Exam 39. Let f(t) = L−1{F (s)} whereF (s) =s2+ 1s(s − 1)2Find f(2), choose the closest answer.(a) 0.0(b) 27.5(c) 28.0(d) 28.5(e) 29.0(f) 29.5(g) 30.0(h) 30.5(i) 31.0(j) 31.5(k) F (s) is not the Laplace transform of a function of exponential order10. LetF (s) = L(2t − 3)2Find F (0.5).(a) 0(b) 2(c) 6(d) 18(e) 22(f) 26(g) 30(h) 34(i) 38(j) 42(k) The given function is not of exponential order and has no Laplace transformPage 8Math 217Exam 311. LetF (s) = L{t sin 2t}Find F (1.9).(a) 0.00(b) 0.06(c) 0.07(d) 0.08(e) 0.09(f) 0.10(g) 0.11(h) 0.12(i) 0.13(j) 0.14(k) The given function is not of exponential order and has no Laplace transform12. Let F (s) = L{f(t)} wheref(t) =(sin t if t < π0 if t ≥ πFind F (0.4).(a) 0.0(b) 0.5(c) 0.6(d) 0.7(e) 0.8(f) 0.9(g) 1.0(h) 1.1(i) 1.2(j) 1.5(k) The given function is not of exponential order and has no Laplace transformPage 9Math 217Exam 313. Use the Laplace transform to turn the initial value problem into an algebraic equation in X(s).x00+ 8x0+ 15x = etx(0) = 1, x0(0) = 2(a) (s2+ 8s)X(s) =1s − 1− 15(b) X(s) = (s2+ 8s + 15)(1s − 1+ 2s + 9)(c) (X(s) + 3)(X(s) + 5) =1s − 1(d) s2+ 8s + 15X(s) =1s − 1(e) (s2+ 8s + 15)X(s) = s + 10(f) (s2+ 8s + 15)X(s) = 2s + 9(g) (s2+ 8s + 15)X(s) = 2s + 9 +1s − 1(h) (s2+ 8s + 15)X(s) =1s − 1+ s + 10(i) (s2+ 8s + 15)X(s) =1s − 1(j) (s2+ 8s + 15)X(s) =1s + 1(k) None of the abovePage 10Math 217Exam 314. Use the Laplace transform to turn the initial value problem into an algebraic equation in X(s).Solve for X(s).2x00− 5x0− 12x = cos t x(0) = x0(0) = 0Find the Laplace transform of the solution.(a) X(s) =ss2+ 1(b) X(s) =s(s2+ 1)(s − 4)(2s + 3)(c) X(s) =1(s2+ 1)(s − 4)(2s + 3)(d) X(s) =s − 1(s − 4)(2s + 3)(e) X(s) =1s2+ 1(f) X(s) =(s − 4)(2s + 3)s2+ 1(g) X(s) =1(s − 4)(2s + 3)(h) X(s) =s(s − 4)(2s + 3)(i) X(s) =s(s − 4)(2s + 3)s2+ 1(j) None of the abovePage 11Math 217Exam 315. Use the Laplace transform to turn the initial value problem into a differential equation in X(s).x00− (2t + 1)x0− 2tx = 0 x(0) = 1, x0(0) = 2(a) 2sX0+ (s2− s)X = 0(b) s(s3− s − 4)X = s5+ 2s4− s − 2(c) 2sX0+ (s2− s)X = 3 − s(d) 2sX0+ (s2+ s)X = 0(e) X00− (2s2+1s)X0−2sX = 0(f) 4X0− (s2+ s)X = 3s − 1(g) (2s + 2)X0+ (s2− s + 2)X = s + 1(h) 4X0− (s2− s)X = 3 − s(i) 2sX0+ (s2+ s)X = 3 − s(j) 4X0− (s2− s)X = 0(k) None of the aboveWRITTEN PROBLEM—SHOW YOUR WORKMath 217Exam 3Name:ID:Section:Note: You will be graded on the readability of your work. Use the last (blank) page, if necessary.16. For this problem you are to solve the differential equationx00+ x = f (t), x(0) = 0, x0(0) = 2wheref(t) =(cos t if 0 ≤ t ≤ π0 if t > π(a) Write f(t) using unit s tep functions.(b) Find F (s), the Laplace transform of f(t).(c) Transform the differential equation into an algebraic equation in X(s).(d) Find X(s) (solve your algebraic equation for X(s)).(e) Solve the differential equation; find x(t).(f) Write your solution as a piecewise
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