Final exam / 2005.12.8 / Calculus for Applications / MAT 3243
Name:
Please show all work and justify your statements. Label sketches, draw conclusions using complete sentences
including units, and box your final answers as appropriate.
1. Compute the following linear approximations.
(a) Find an equation for the plane tangent to the surface πyln z+xcos(xz) = 0 at the
point [π, 1,1].
(b) Find a parametric formula for the line tangent to [cos(πt),sin(πt),2t] at [0,−1,3].
2. Consider the vector field F(x, y, z) = [x+y z, y +z x, z +xy].
(a) Compute DF and ∇ · F.
(b) Find a scalar potential for F. What conclusion can you make about ∇ ×F? Explain.
3. Suppose F=f(u, v, w), where u= 2y+z,v= 3z+x,w=x+ 2y. Express the partial
derivatives of Fwith respect to x,y, and zin terms of the partial derivatives of fwith
respect to u,v, and w.
4. For the scalar field f(x, y) = x+ 2y+ ln(xy) find all critical points and classify them
using the Hessian criterion.
5. Consider the triangle formed by the coordinate axes in the plane and a line with intercepts
aand b(a, b > 0). Use Lagrange multipliers to determine the dimensions of the rectangle
with two sides along the coordinate axes, inscribed in the triangle, and having the largest
possible area.
6. Integrate η=y dx +x dy +z dz along the straight line segment from [0,1,1] to [1,2,3].
Compute dη. Had we chosen a different path between the two points, would the integral
remain the same? Explain.
7. Find an equation and a parametric formula for the plane tangent to the surface
[s2t, st2, st] at [−2,4,−2].
8. Compute the flux of F= [x, y, 2−z] through the unit disc in the x-yplane. Then, use
the divergence theorem to find the flux of Fthrough the top half of the unit sphere.
[Hint: the hemisphere and the disc together, with appropriate orientation, form the b oundary of a solid.]
[Formulas: surface area of a sphere of radius ρis 4πρ2and its volume is 4
3πρ3]
1 2 3 4 5 6 7 8 total (80)
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