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Homework 7
- Section 6.3 Fourier Series: Scalar Product Spaces
- Read Section 6.3. Do Problem 6.37. WeBWorK reading question due April 21 @ 9:50 am.
- A Geogebra activity on Fourier approximation
- In class (group work): 6.38. Final due date: May 5
- Homework: 6.39, 6.40, 6.41 (use cosines), 6.43, 6.45, 6.46, 6.50, 6.51, 6.53
- Hand In: 6.41*, 6.43*, 6.50
- 6.41*. Complete Problem 6.41 on Page 351, but with the function \(g(x)\) defined to be \[g(x) = \left\{\matrix{2\vert x\vert&\mbox{for}\ -0.5\leq x\leq 0.5\cr 1&\mbox{otherwise}}\right.\] for \(-1 \leq x\leq 1\). (This is NOT the wave shown in Figure 6.12a.) (Hints: (a) Make use of the even nature of the functions you are integrating to halve the amount of calculation (but don't forget to incorporate the values of the integrals you don't need to work out) and (b) There may be a need to consider various cases of \(n\) beyond \(n\) odd or \(n\) even; thus you may find yourself considering when \(n\) has the forms \(4m, 4m+2\) (two cases when \(n\) is even), \(4m+1, 4m+3\) (two cases when \(n\) is odd; these two cases may later combine simply to when \(n\) is odd).)
- 6.43*. Complete Problem 6.43 on Page 352, but where the underlying space is \(\mathcal{F}((0,2])\), the space of piecewise continuous functions on the interval \((0,2]\). Thus the inner product is \[\langle f(x),g(x)\rangle = \int_0^2 f(x)g(x)\ dx.\] Use the same initial set \(\mathcal{W}\).
- 6.50. Complete Problem 6.50, as stated in the text.
- Read Section 6.4. Do the True/False questions (pp. 363-364). Do Problems 6.37. WeBWorK reading question due April 26 @ 9:50 am.
- In class (group work): 6.60(a), 6.73(b). Final due date: May 8
- Homework: 6.60(c), 6.61, 6.62, 6.64, 6.67, 6.73(a)(c), 6.75, 6.76
- Hand In: 6.61(b)*, 6.73(c)*
- 6.61(b)*. Complete Problem 6.61(b) on Page 365, but with the matrix \[A=\frac{1}{2}\left[\matrix{1&1&\sqrt{2}&a\cr -1&1&0&b\cr -1&-1&\sqrt{2}&c\cr 1&-1&0&d\cr}\right].\]
- 6.73(c)*. Complete Problem 6.73(c) on Page 365, but with the hyperplane \(3x-2y+2z-w=0\).
- Read Section 6.5. Do Problems 6.79. WeBWorK reading question due May 1 @ 9:50 am.
- In class (group work): 6.82, 6.85. Final due date: May 8
- Homework: 6.83, 6.85, 6.86, 6.89, 6.91, 6.92, 6.93
- Hand In: 6.86*, 6.93, LS1 (below)
- 6.86*. Complete Problem 6.86 on Page 378, but with \(B_1=[1,3,-1,1,-1]^t\ \mbox{and}\ B_2 = [1,0,2,1,2]^t.\)
- 6.93. Complete Problem 6.93 on Pages 379-380, as stated in the text.
- LS1. Find the least squares best fit line \(y = a + bt\) to the four points \((t,y) = (0,1),(1,6),(3,14),(4,18)\). Also compute the predicted values vector \(B_0\) of \(y\) for each of the given \(t\) values, and then compute the error vector \(B_1 = B - B_0\).
- Read Section 6.6. Do the True/False questions (pp. 391-392). Do Problems 6.95, 6.96. WeBWorK reading question due May 3 @ 9:50 am.
- In class (group work): 6.97(a), 6.98(b), 6.102. Final due date: May 10
- Homework: 6.97(c)(e) (identify the surface types), 6.98(c)(d), 6.100, 6.104, 6.105, 6.112
- Hand In: 6.97(e)* (identify the surface type), 6.100*
- 6.97(e)*. Complete Problem 6.97(e) on Page 392, but with the quadratic variety \(3x^2+2y^2+2z^2+4yz=3\) (identify the surface type).
- 6.100*. Complete Problem 6.100 on Page 393, but giving an equation for an ellipse centered at the origin with a length 18 major cord parallel to the vector \([5,12]^t\) and a length 4 minor axis. Pay attention to the definition (given in the problem) of the major cord of an ellipse.
- Read Section 6.7. Do the True/False questions (pp. 403-404). Do Problems 6.116(a), 6.117(a). WeBWorK reading question due May 5 @ 9:50 am.
- In class (group work): 6.116(c), 6.117(e). Final due date: May 10
- Homework: 6.116(b)(d), 6.117(c)(b)(d), 6.119(a)(b), 6.120, 6.123, 6.124
- Hand In: none