MA527_HW09_2023_Fall_Mossing_GADED

MA52700 distance Homework 9 Joseph Mossing TOTAL POINTS 29.5 / 30 QUESTION 1 1 490 11 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers QUESTION 2 2 490 20 5 / 5 ✓ + 5 pts Correct + 2.5 pts did not set x = 1 (or x = 0) in the solution of problem 11 + 3 pts major error QUESTION 3 3 490 24 4.5 / 5 + 5 pts Correct + 3.5 pts (a) incorrect, (b) correct (a):$$f(x)=\frac{1}{2}-\frac{2}{\pi}(cos\frac{\pi x}{4}-\frac{1}{3}cos\frac{3\pi x}{4}+...)$$ $$=\frac{1}{2}-\frac{2}{\pi}\sum_{n=0}^\infty \frac{1}{2n+1}(-1)^ncos\frac{(2n+1)\pi x}{4}$$ $$=\frac{1}{2}-\frac{2}{\pi}\sum_{n=1}^\infty \frac{1}{n}sin(\frac{n\pi}{2})cos\frac{n\pi x}{4}$$ + 3.5 pts (a) correct, (b) incorrect. (b):$$f(x)=\frac{2}{\pi}(sin\frac{\pi x}{4}- sin\frac{2\pi x}{4}+\frac{1}{3}sin\frac{3\pi x}{4}+...)$$ $$=-\frac{2}{\pi}\sum_{n=1}^\infty \frac{1}{n}((- 1)^n-cos(\frac{n\pi}{2}))sin\frac{n\pi x}{4}$$ + 2.5 pts both (a) (b) incorrect ✓ + 4.5 pts minor error x 1 + 4 pts minor error X 2 + 3.5 pts major error QUESTION 4 4 490 29 5 / 5 ✓ + 5 pts Correct. (a): $$\frac{2}{\pi}-\sum_{n=1}^\infty \frac{4}{\pi}\frac{1}{(2n-1)(2n+1)}cos2nx$$ $$=\frac{2}{\pi}-\sum_{n=2}^\infty \frac{2}{\pi}\frac{1+(-1)^{n}}{(n-1)(n+1)}cosnx$$ $$=\frac{2}{\pi}-\frac{4}{\pi}(\frac{1}{1\times 3}cos2x+\frac{1}{3\times 5}cos4x+...)$$ (b)$$sinx$$

+ 4.5 pts minor error x 1 + 4 pts minor error x 2 + 3 pts minor error >=3 or major error + 0 pts no answers QUESTION 5 5 494 2 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers QUESTION 6 6 494 6 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers Page 2

1 490 11 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers Page 4

3 490 24 4.5 / 5 + 5 pts Correct + 3.5 pts (a) incorrect, (b) correct (a):$$f(x)=\frac{1}{2}-\frac{2}{\pi}(cos\frac{\pi x}{4}-\frac{1}{3}cos\frac{3\pi x}{4}+...)$$ $$=\frac{1}{2}-\frac{2}{\pi}\sum_{n=0}^\infty \frac{1}{2n+1}(-1)^ncos\frac{(2n+1)\pi x}{4}$$ $$=\frac{1}{2}-\frac{2}{\pi}\sum_{n=1}^\infty \frac{1}{n}sin(\frac{n\pi}{2})cos\frac{n\pi x}{4}$$ + 3.5 pts (a) correct, (b) incorrect. (b):$$f(x)=\frac{2}{\pi}(sin\frac{\pi x}{4}-sin\frac{2\pi x}{4}+\frac{1}{3}sin\frac{3\pi x}{4}+...)$$ $$=-\frac{2}{\pi}\sum_{n=1}^\infty \frac{1}{n}((-1)^n-cos(\frac{n\pi}{2}))sin\frac{n\pi x}{4}$$ + 2.5 pts both (a) (b) incorrect ✓ + 4.5 pts minor error x 1 + 4 pts minor error X 2 + 3.5 pts major error Page 10

4 490 29 5 / 5 ✓ + 5 pts Correct. (a): $$\frac{2}{\pi}-\sum_{n=1}^\infty \frac{4}{\pi}\frac{1}{(2n-1)(2n+1)}cos2nx$$ $$=\frac{2}{\pi}-\sum_{n=2}^\infty \frac{2}{\pi}\frac{1+(-1)^{n}}{(n-1)(n+1)}cosnx$$ $$=\frac{2}{\pi}-\frac{4}{\pi}(\frac{1}{1\times 3}cos2x+\frac{1}{3\times 5}cos4x+...)$$ (b)$$sinx$$ + 4.5 pts minor error x 1 + 4 pts minor error x 2 + 3 pts minor error >=3 or major error + 0 pts no answers Page 14

5 494 2 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers Page 17

6 494 6 5 / 5 ✓ + 5 pts Correct + 4 pts small mistake + 3 pts mistake + 2 pts several mistakes + 1 pts something is right + 0 pts no answers Page 19