MATH 102 Final, Fall 2007

Math 102 Final Exam Dr. H. Yamani Fall 2007 Name: Do each of the following problems. Show all your work. No work shown. No credit. ****GOOD LUCK**** Problem 1. (Answer on pages 1,2,3,4 of the booklet) Evaluate the following integrals: (5 pts each) a. ( ) dx x xe x + 1 2 ∫ b. dx x x ∫ + 2 3 1 c. ( )( ) dx x x x x ∫ + − − + 1 1 3 2 2 2 d. ( ) dx x / ∫ + ∞ 1 5 3 9 100 4 (Test for convergence or divergence only). Explain Problem 2. (Answer on page 5 of the booklet Find the following limits: (5 pts each) 1)       →∞ ln x x x lim 3 2) x x x x lim 3 2       − →∞ Problem 3. (Answer on page 6 of the booklet) (5 pts) Find the length of the curve 2π 0 5 5 5 5 ≤ ≤ − = − = t sin t sint y cos t cost x Problem 4 (Answer on pages 7, 8, 9, 10 of the booklet) a. (5 pts) Find the distance from the point S(1,1,3) to the plane 6 6 2 3 = + + z y x b. (5 pts) Find the distance from the point S(1,1,5) to the line L: t z ,t y ,t x 2 3 1 = − = = + c. (5 pts) Find an equation for the plane through the point (2,1,-1) and perpendicular to the line of intersection of the planes 2 2 3 2 z y , x z y x = + + = − + d. (5 pts) Find an equation for the plane through the point (2,4,5) perpendicular to the line L: t z ,t y ,t x 4 3 1 5 = = + + = Problem 5. (Answer on pages 11, 12 of the booklet). a. (5 pts) Find the area of a parallelogram determined by the vectors u = -3i -2j +2k and v = -2i +2j+3k b. (5 pts) Write u = 2i -j +3k as a sum of a vector parallel to v = 4i –j+2k and a vector orthogonal to v. Problem 6. (Answer on pages 13, 14, 15 of the booklet) (25 pts) Determine the unit tangent vector T, the principal unit normal vector N, the binormal vector B, the curvature κ, and the torsionτ for the curve r(t) = ( ) ( ) t cost sint k j i 4 3 3 + + Problem 7. (Answer on page 16 of the booklet) (10 pts) Without finding T and N, write the acceleration of the curve r(t) = ( ) ( ) bt a sint acost k j i + + in the form a = aTT + aN N