Mastering Calculus: Practice Final Exam for MATH 2ZZ3
School
McMaster University**We aren't endorsed by this school
Course
ENGINEER 2ZZ3
Subject
Mathematics
Date
Dec 11, 2024
Pages
1
Uploaded by JusticeAtomWalrus54
MATH 2ZZ3 Winter 2024 – practice final examNote that I paid no attention to total duration of this exam which islikely to be longer than 2h and did not try too hard to make calculationseasy.1. LetSbe the surface in Section 9.14 - problem 10 of the textbook andCbe its boundary. LetF(x, y, z) =xyi+2yzj+xzk. Compute curlF.ComputeRRScurlF·n dSdirectly and using Stoke’s theorem.2. LetDbe the region of space under the graph ofx+2y+3z= 1 insidethe first octant (i.e.such thatx≥0,y≥0, andz≥0). LetG(x, y, z)be a given function. WriteRRRDf(x, y, z)dVas a triple integral in sixdifferent ways. Do not try to compute this integral.3. LetF(x, y, z) = (xyexz+y)i+(exz+x)j+xyexzk. LetCbe the curveparameterized byx(t) = cosπt,y(t) = 2 sin 2πt,z(t) =t20≤t≤1.ComputeRCF·dr.4. LetRbe the triangle with vertices (-2,0), (2,0), and (1,4) andCitsboundary positively oriented. Compute the work of the forceF(x, y) =x2i−yjusing Green’s theorem.5. Consider a particle moving along a curveCparameterized byr(t) =t3/3i+t2j+ 2tk, 0≤t≤1. Compute the velocity and accelerationvectors. LetG(x, y, z) = 4yzi+ 4xzj+ 4xyk. Compute the work ofGalongC.6. Letz= ln(u2+v2), whereu=t2+sandv=t−1/s.Compute∂z/∂tand∂z/∂s.7. Letf(x) =e−x. Compute the complex Fourier series iffon (−1,1).8. Compute the Fourier series expansion off(x) =|sinx|on−π, π.9. Find the volume of the solid bounded byx2+y2= 1,x2+y2= 9,z= 0, andz=p16−x2−y2.10. ComputeRRR2xy dAwhenRis bounded by the graphs ofy=x2andy= 2x3.11. Find the position, velocity, and the normal and tangential componentsof the acceleration of a moving particle whose position is parameterizedbyr(t) =t2i=t3j+t4k.12. Find the curvature att=πof the curver(t) = (t−sint)i+(1−cost)j.1
