MATH135 Final: MATH135 Final Exam 2006 Fall

1. Let : 1 +2i, w 2-i, and 4+3i. Compute (a) : + 3w;(b) -2w+ Use the quadratic formula to solve these equations; express the answers as (d) zz-z = 1; (e) z-22. 3. Sketch the locus of those points w with (a)|w 3:(b)lw-21- 1:(e)|w +22 4. Find Re(1/2) and Im(1/z) if z = x + iy, 0. Show that Re(iz)--Im z and 5. Give the polar representation for (a)-1+ i; (b) I + iV3; (c)-|; (d) (2-i) 6. Give the complex number whose polar coordinates (r, 0) are (a) (V3, x/4); 7, Let a, b, and c be real numbers with a ψ 0 and b2 4ac. Show that the two roots 8. Suppose that A is a real number and B is a complex number. Show that (g) Re[zl1 +] Im(iz)- Re z. (b) (1 y 2, π); (c) (4,-π/2); (d) (2,-24); (e) (1,4ç §(f) (v/2, 9-4). of ax + bx c0 are complex conjugates of each other. and 9. Show that |z 1 if and only if 1/z- 10. Let z and w be complex numbers with zw 0. Show that either z or w is zero. 11. Show that lz + w12-12-w12 = 4 Re(zi) for any complex numbers z, w. 12. Let z1, z2 z, be complex numbers. Establish the following formulas by mathematical induction: (b) Re(zit zit , , , + z.) = Re(z.) + Reiz) + + Re(%) 13. Determine which of the following sets of three points constitute the vertices of 14. Show that cos θ-cos ψ and sine-sin ψ if and only if θ-ψ is an integer 15. Show that the triangle with vertices at 0, z, and w is equilateral if and only if a right triangle: (a) 3+5i, 2+2i,5 +i; (b)2+ i, 35i,4 + i; (c) 6+4i,7+Si, 8 + 4i. multiple of 2 10 Chapter 1 The Complex Plane 16. Let zo be a nonzero complex number. Show that the locus of points tzo. -ao

Step 3 Multiply both sides of the differential equation by t and get [ty] = dt Step 4 Integrating both sides and solving for y, we obtain C1C- 2 Int C To satisfy the initial condition, we must have 21n(1)+C-C [In(1)-이. Hence, the solution of the initial-value problem is 2Int 1 Figure 3 Solution of the initial-value problem fy + ty 2, y(1)= 1, t > 0 t > 0 Among all the solution curves that are shown in Fig. 3, th problem is the curve that goes through the point (1, 1) e solution of the > Now Try Exercise In the following section, we present several interesting applications of fre linear differential equations. Check Your Understanding 10.3 1. Using an integrating factor, solve yty 1e 2. Find an integrating factor for the differential equi EXERCISES 10.3 in Exercises 1-6, find an integrating factor for each equation Taket0. In Exercises 21-28, solve the initial-value problem. 22. ty, + y = In t, y(e) = 0, t > 0 Int-1 23. y+ 24, y, 2(10-y), y(0) = 1 10-9e-2t 10+10f In Exercises 7-20, solve the given equation using an integrating factor. Taket>0. 26. ty'-y=-1, y(1) = 1, t>01 27, y, + 2y cos(2t)-2 cos(2t), y(5) = 0 y 28, ty' + y-sin t, y()-0, t > 0-1 cost Technology Exercises 29. Consider the initial-value -1- - 9. 10· y' = 2(20-y) 20 + C'e-2t 12. e +1 11、 y' = .5(35-y) 13. y, + 10+1-0y=府14.3-h-e2t tr" + Cee 15. (1 + tly, + y--I 17. 6y'+ty t 19,y, + y = 2-et y'=-th+10, (a) Is the solution increasing or decreasing whie y(0)=50. 16, y, = e-"(y + 1) _1+ 18. ety' + y = 1 1 + Ce", C'e-e-t 20 , + y = 1 [Hint: Compute y'(0),1 i + Ce-i(t+1)3/2 (b) Find the solution and plot it for OStS t k indicates answers that are in the back of the book. 19. y = 2-ne, + Ce-r 29. (a) y'(0)--40, y(t) is decreasing at t = 0