MATH 202 Exam 1, Spring 2008

MATHEMATICS 202 SPRING SEMESTER 2006-2007 MAKEUP-QUIZ I Time: 80 MINUTES. Date: March 26, 2007. Name:—————————————————————————- ID Number:———————————————————————– Section Number:—————————————————————— Course Instructors: Professors A. Lyzzaik and H. Yamani Question Grade 1 /14 2 /14 3 /14 4 /14 5 /14 6 /14 7 /18 TOTAL /100 Answer The Following Seven Questions On The Page Al- located For Each Question (You May Use The Back Of The Pages If Needed). 1 2 1. Solve the differential equation by finding an appropriate integrat- ing factor (y2 + xy3) dx + (5y2 − xy + y3 sin y) dy = 0. (14 points) 3 2. Solve the initial-value problem (14 points) dy dx = y(xy3 − 1), y(0) = 1. 4 3. Solve the differential equation (14 points) y(ln x − ln y)dx = (x ln x − x ln y − y)dy. 5 4. Solve the initial-value problem (14 points) dy dx = cos(x + y), y(0) = π/4. 6 5. The functions y(x) = x4/16, −∞ < x < ∞, and y(x) = � 0, x < 0 x4/16, x ≥ 0 have the same domain but are clearly different. Show that both func- tion are solutions to the initial-value problem dy/dx = xy1/2, y(2) = 1. Does this contradict the existence and uniqueness theorem? Justify your answer. (14 points) 7 6. Use Stoke’s theorem to evaluate the flux of the vector field F(x, y, z) = x2yi + 2y3zj + 3zk across the surface with parametric equations r(r, θ) = (r cos θ)i + (r sin θ)j + rk. 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π. (14 points) 8 7. Find the area of the surface cut from the hemisphere x2+y2+z2 = 4, z ≥ 0, by the cylinder x2 + y2 = 2x. (16 points)