University:
California State University, Northridge
Course:
MATH 150B | Calculus II
Academic year:

2023
Views:

357

Pages:

7
Author:

Nyi Nyi Moe

18Exam (4)Math 150B. Calculus IISection # 18056 Fall 2022Instructor: Henrik SarkssianTHURSDAY DECEMBER 8, 12:30 PM - 13:35 PMName: Nyi Nyi Moc Hlet(Please print)-Show all of your work and make the final answers clear. Full credit will NOT be givenfor merely stating the right answer without any appropriate justification.- Cell phones must not be visible or audible at any time during the exam. This applies to all smarttechnology including watches, cameras, and pens.No Calculators!Problem123456TotalMaximum[25][25][25][25][25][25]150Score252525252525150Awesome!1 Problem 1. (25 pts.)Consider a triangle with vertices: A(-2,5,3),B(5,3,4),(-3,1,2). Find all angles of the triangle.8AB = (5+2,3-5,4-3) -(30AC = <-3+2,1-5,2-3 > = <-1,-4,-1>IABI = 49 + 4 + 1=54IACI / 1 16+1 /18cos ( BAC F AB . Ac=. 0LBAC = 1=90IAB/1ACI /54 / 182A=<-2-5,5-3,3-4)=<-7,2,-1)=1 BC 1 . * /64+4+4 = /72cos LABC : BI. BE . 56-4+2 - = 54 x 54 = 54 30 5 - 51BAllBCl 54 F254572 547242LABC = R = 306LBCA = 80 - (90 + 30 ) = 180 -120252 Problem 2. (25 pts.)Consider a triangle with vertices: A(3,-5,2),B(5,-4,4),C(-1,-2,-3)(i)Find the area of the triangle.(ii) Find two unit vectors perpendicular to the plane of triangle ABC.B(i)<5-3,-4+5,4-2> = <2,1,2>DCAAB x AC = ijk = (-5-6)i-(-10+8))+(6+4)k212-43-5I AB X AC I = / 121 . 4 + 100 = = /225= 15=(ii)x= First UnitVectorSecondUnit Vector253 Problem 3. (25 pts.)For the vectors u= (2,-4,6) and v=(2,-3,1) express u as the sum u =p+n, = where p is parallelto V and n is orthogonal tov.Bajo is = J.VXXPV(4+12+6)(22-3,1)4+9+1= =-254 Problem 4. (25 pts.)Show that the following trajectory lies on a sphere in R3. Find the radius of the sphere, and showthat the position vector and the velocity vector are everywhere orthogonal.r(t) =+ 2 superscript(2) = sin 2t + 4 cos 2t + 3 sin22t= 4 sin 2t + 4 cos'21= ( sin' 2t + cos' 2t)= 4 =22Sphere Radius = 22t, -4 sin 2t, 2 13 cos 2t )(t)..r'(t) = 2 sin2t cos 2t - 8 sin 2t cos 2t + 6 sin 2t cos 2t= 8 sin 2t cos 2t - 8 sin 2t ws 2t=0Therefore, pesition vector and velocity vector are everywhere orthogonal.,255 Problem 5. (25 pts.)Find the length of the following curve on the given intervals.r(t)= t - cost, t+ cost, V2sint); 05151sint, 1- - sint, 52 cost>= sin t) +=1 + 2 sint + sin't + 1- 2 sint + sin't + 2 cos't= = /2+2 (sin't+as't) = =2L- P! dt. P' 2 dt = 2t !! = 12-0=2256 Problem 6. (25 pts.)Given an acceleration vector, initial velocity and initial position (xo,yo,zo) find thevelocity and position vector for .4(0)+C1 00 0+C230+C3=6G = 0 C2 33 C3=6=(t) - Svcts dt = < 2t'+C4 , " 1/12(0)++449c53C6 21r =15