MATH 211 Midterm: MATH 211 Amherst F14M211 2803 29Benedetto
15 Feb 2019
School
Department
Course
Professor
Math 211, Section 03, Fall 2014
Final Exam, Monday, December 15, 2014
Instructions: Do all twelve numbered problems. If you wish, you may also attempt the three
optional bonus questions. Show all work, including scratch work. Little or no credit may
be awarded, even when your answer is correct, if you fail to follow instructions for a
problem or fail to justify your answer.
If your answer for a given problem is a sum of fractions with different denominators, you
may leave it that way. Otherwise, simplify your answers whenever possible.
If you have time, check your answers.
WRITE LEGIBLY. NO CALCULATORS.
1. (12 points) A particle is travelling in such a way that its velocity vector at time tis
given by ~r ′(t) = ht2,√2t, 1i.
(1a) How far does the particle travel from time t= 1 to time t= 2?
(1b) What is the curvature of the path the particle traces out at the point it passes
through at time t= 2?
2. (18 points) Let f(x, y) =
2x3+y3
2x2+y2if (x, y)6= (0,0)
0 if (x, y) = (0,0).
(2a) Compute fx(0,0) and fy(0,0).
(2b) Compute D~uf(0,0), where ~u is the unit vector pointing in the direction of h1,−1i.
(2c) Based on your answers to parts (a) and (b), explain (briefly) why fcannot be
differentiable at (0,0).
3. (18 points) Find and classify (as local minimum, local maximum, or saddle point) every
critical point of the function f(x, y) = x2y−3x2−6y2+ 2.
4. (12 points) Find the point on the ellipse x2+6y2+3xy = 40 with the largest x-coordinate.
5. (20 points) Find the volume of the region that is inside the sphere x2+y2+z2= 9 and
also inside the cylinder x2+y2= 3x.
(Note that the cylinder is centered around a vertical line that is not the z-axis.)
6. (20 points) Let Ebe the solid lying
•inside the sphere x2+y2+z2= 9,
•outside the sphere x2+y2+z2= 1,
•below the cone z=px2+y2, and
•in the first octant.
Compute ZZZE
y dV .
7. (20 points) Let Ebe the solid bounded by the surfaces y=√x,x= 2y,z= 4, and
x+z= 4. Compute the volume of E.
8. (15 points) Let Cbe the quarter of the circle x2+y2= 9 in the second quadrant, i.e.,
the quarter-circle arc from (0,3) to (−3,0). Compute ZC
x2y ds.
Document Summary
If you wish, you may also attempt the three optional bonus questions. Little or no credit may be awarded, even when your answer is correct, if you fail to follow instructions for a problem or fail to justify your answer. If your answer for a given problem is a sum of fractions with di erent denominators, you may leave it that way. Compute zzze: (20 points) let e be the solid bounded by the surfaces y = x, x = 2y, z = 4, and x + z = 4. Compute the volume of e: (15 points) let c be the quarter of the circle x2 + y2 = 9 in the second quadrant, i. e. , the quarter-circle arc from (0, 3) to ( 3, 0). Compute zc x2y ds: (15 points) let c be the boundary of the triangle with vertices (0, 0), (1, 0), and (1, 2), oriented counterclockwise. Let ~f (x, y) = h3y2, x2y + cos8 yi.
