MA527_HW10_2023_Fall_Mossing_GADED

MA52700 distance Homework 10 Joseph Mossing TOTAL POINTS 41 / 45 QUESTION 1 1 498 8 4 / 5 + 5 pts Correct ✓ + 4 pts small mistake/Wrong error + 3 pts mistake + 2 pts several mistakes + 1 pts something is right QUESTION 2 2 498 11 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right QUESTION 3 3 503 7 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right QUESTION 4 4 503 13 5 / 5 ✓ + 5 pts Full credit: $$ y=e^{mt}, m =-4+(-)\sqrt{- \lambda} $$. Case 1: $$\lambda = \mu^2 .$$ $$\lambda= n^2, y=e^{-4t}sinnt$$ Case 2: $$\lambda = 0, y= 0$$ Case 3: $$\lambda = -\mu^2 .$$ also $$y=0$$ + 2 pts does not obtain $$ y=e^{mt}, m =-4+(- )\sqrt{-\lambda} $$. + 4.5 pts minor error + 3.5 pts obtain $$ y=e^{mt}, m =-4+(-)\sqrt{- \lambda} $$ but does not find the nontrivial solutions QUESTION 5 5 509 1 5 / 5 ✓ + 5 pts Correct: $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ $$f(x) = 8(P_1(x)-P_3(x)+P_5(x))$$ + 2 pts does not have $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ correct + 4 pts single minor error + 3 pts have $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ correct and have at least 2 errors.

(including the case of not writing clearly that the ans is a finite sum of n<=5) QUESTION 6 6 509 5 5 / 5 ✓ + 5 pts Full credit. Reason: 1. Pn(x) is odd if n is odd, and is even if n is even. 2. If f(x)Pn(x) is odd the integration will be zero. 3. oddxodd=even, evenxeven=even, evenxodd=odd. Example: the previous two problems - 2 pts no/incorrect proof - 2 pts no/incorrect example (an example of a odd function with its F-L series and a even function with its F-L series) - 1 pts minor error QUESTION 7 7 517 1 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts No answer or nothing is correct QUESTION 8 8 517 11 4 / 5 + 5 pts Correct ✓ + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts No answer or nothing is correct f(x) not complete QUESTION 9 9 517 18 3 / 5 + 5 pts Correct + 4 pts small mistake ✓ + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts No answer or nothing is correct Wrong last step. Also, it should be dw instead of dx. Page 2

1 498 8 4 / 5 + 5 pts Correct ✓ + 4 pts small mistake/Wrong error + 3 pts mistake + 2 pts several mistakes + 1 pts something is right Page 5

2 498 11 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right Page 7

4 503 13 5 / 5 ✓ + 5 pts Full credit: $$ y=e^{mt}, m =-4+(-)\sqrt{-\lambda} $$. Case 1: $$\lambda = \mu^2 .$$ $$\lambda= n^2, y=e^{-4t}sinnt$$ Case 2: $$\lambda = 0, y= 0$$ Case 3: $$\lambda = -\mu^2 .$$ also $$y=0$$ + 2 pts does not obtain $$ y=e^{mt}, m =-4+(-)\sqrt{-\lambda} $$. + 4.5 pts minor error + 3.5 pts obtain $$ y=e^{mt}, m =-4+(-)\sqrt{-\lambda} $$ but does not find the nontrivial solutions Page 11

5 509 1 5 / 5 ✓ + 5 pts Correct: $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ $$f(x) = 8(P_1(x)-P_3(x)+P_5(x))$$ + 2 pts does not have $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ correct + 4 pts single minor error + 3 pts have $$C_m = \frac{2m+1}{2}\int_{-1}^1 f(x)P_m(x)dx $$ correct and have at least 2 errors. (including the case of not writing clearly that the ans is a finite sum of n<=5) Page 14

6 509 5 5 / 5 ✓ + 5 pts Full credit. Reason: 1. Pn(x) is odd if n is odd, and is even if n is even. 2. If f(x)Pn(x) is odd the integration will be zero. 3. oddxodd=even, evenxeven=even, evenxodd=odd. Example: the previous two problems - 2 pts no/incorrect proof - 2 pts no/incorrect example (an example of a odd function with its F-L series and a even function with its F-L series) - 1 pts minor error Page 16

7 517 1 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts No answer or nothing is correct Page 19