MATH 151 Lecture 9: Section 3.4 - Complex Numbers(1)

3t to compute the integral R(z + y dA where R is the parallelogram with (1 point) In this problem we use the change of variables z = 4s + t y = 8 vertices (x, y) (0,0), (8,2), (11,-7), and (3,-9) a,y) s,t) First find the magnitude of the Jacobian Then, with a = c= , and d s+ t+

7. Let A--2 1 3 1 1 -1 (a) Find the eigenvalues of A. (6 points) (b) For each eigenvalue, find a basis for the corresponding eigenspace. (6 points) (c) Factor A into a produet XDx-1, where D is a diagonal matrix. Use the factorization to compute A10 and eA. (10 points-Bonus)

2/3 (15x)/(15x) 4/5x - 6 (15x) rightarrow 2 (5x)/(3) 4 - 90x rightarrow 10x/12 - 90x we can see that all three terms of this expression are divisible by 2, but we know that we can't cancel terms, we can only cancel factors, so we factor out a 2 from the numerator and denominator. 2 5x/2 (6 - 45x) rightarrow 2 5x/2 (6 - 45x) rightarrow 5x/6 - 45x b) 3/x + 8/7/x^2 - 2/x There are 4 total terms in this expression (2 in the numerator and 2 in the denominator). Of these terms, 3 are mini-fractions, with denominators of x, x^2, and x. The LCM of these denominators is x^2. So we multiply all four terms by x^2 and simplify. Don't forget to multiply the 8 by x^2 also!! x^2 3/x + 8 x^2/x^2 7/x^2 - 2/x x^2 rightarrow^Cancel x^2 3/x + 8 x^2/x^2 7/x^2 - 2/x x^2 rightarrow 3x + 8x^2/7 - 2x c) 15/1/x - 3 - 2/x + 1