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Math 215 — Second Midterm
November 19, 2009
Name:
Instructor: Section:
- Do not open this exam until you are told to do so.
- This exam has 9 pages including this cover. There are 8 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
- Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam.
- Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam.
- Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate.
- You may use no aids (e.g., calculators or notecards) on this exam.
- If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use.
- Turn off all cell phones and pagers, and remove all headphones.
- Because no calculators or notecards are allowed, you are not required to evaluate expressions you may obtain in the solution of these problems. In some cases it may be to your advantage to simplify expressions, but it will not be required for full credit.
- Note that problems 1–4 will be graded giving very little partial credit.
- There is a list of possibly useful formulas included as the second page of this exam.
Problem Points Score
Total 100
Some possibly useful formulas
Areas and Volumes ◦ Area of a triangle: A = 12 b h, where b is the length of one side and h the length of the perpendicular from that side to the opposite angle. ◦ Area of a sector of a circle: A = 12 r^2 θ, where r is the radius of the circle and θ the angle of the sector. ◦ Volume and surface area of a sphere: V = 43 π r^3 ; S = 4 π r^2. ◦ Volume of right circular cylinder: V = π r^2 h, where r is the radius and h the height of the cylinder. ◦ Volume of a cone: V = 13 π r^2 h, where r is the radius of the base of the cone and h its height.
Trigonometry ◦ cos^2 x + sin^2 x = 1 ◦ cos(x + y) = cos(x) cos(y) − sin(x) sin(y); sin(x + y) = cos(x) sin(y) + cos(y) sin(x) ◦ cos(2x) = cos^2 x − sin^2 x; sin(2x) = 2 sin x cos x ◦ cos^2 x = 12 (1 + cos(2x)); sin^2 x = 12 (1 − cos(2x))
Vector Formulas ◦ a × b = −b × a; ca × b = c(a × b) = a × (cb) ◦ a × (b + c) = a × b + a × c; (a + b) × c = a × c + a × b ◦ The unit tangent T, normal N and binormal B vectors for a space curve r(t) are T(t) = r′(t)/|r′(t)|, N(t) = T′(t)/|T′(t)|, and B(t) = T(t) × N(t).
Other Formulas, Almost At Random ◦ Average value of a function f (x) on a ≤ x ≤ b: (^) b−^1 a
∫ (^) b a f^ (x)^ dx; average value of a function f (x, y) on a region D: 1 Area(D)
D f^ (x, y)^ dA; average value of a function^ f^ (x, y, z) on a volume E: 1 Volume(E)
E f^ (x, y, z)^ dV^.
◦ Moment of inertia of a lamina on a region D, around the x-axis:
D y (^2) ρ(x, y) dA. Similarly, around the y-axis:
D x
(^2) ρ(x, y) dA. In either case ρ(x, y) is the density of the lamina.
[12 points] Consider the circular region x^2 + y^2 ≤ a (where a is a positive constant). Set up a double integral that gives the average distance from any point (x, y) in this region to the origin. Evaluate your integral to find the average distance.
[12 points] Let F = a i + b j + c k, where a, b and c are constants. Let the curve C be the line segment from (1, 2 , 3) to (3, 7 , 10). For what values of the contants a, b and c will
C F^ ·^ dr^ = 0 (other than a = b = c = 0, of course)?
PSfrag replacements
C 1
C 2
C 3
C 4
x
y
- [12 points] Consider the graph shown to the right, of the vector field F and the line segments C 1 , C 2 , C 3 and C 4. The length of the vectors at each point is the strength of the vector field there. a. [8 points] Place in order, from smallest to largest, the six values
C 1 F^ ·^ dr,^
C 2 F^ ·^ dr,^
∫^ C^3 F^ ·^ dr, C 4 F^ ·^ dr,^
C 1 F^ ·^ j^ dy, and the number 2. In one or two sentences, explain your ordering.
b. [4 points] Is this vector field F a gradient vector field? Explain in one or two sentences.
Math 215 / Exam 2 (November 19, 2009) page 7
C 1
C 2
C 3
C 4
x y
0
0
0
1
22
− 2
x
y
z
- [16 points] Consider the three-dimensional solid shown in the figure to the right. In this, the side that ap- pears circular is, and the face along the x-axis is flat. Let the density of the solid be a constant, δ: then the mass of the solid is M = 8 πδ 3. a. [12 points] Write two triple integrals, one in cylindrical and one in spherical coordinates, that both give the z-coordinate of the center of mass of the region.
b. [4 points] Evaluate one of your integrals from (a) to find the z-coordinate of the center of mass.
- [12 points] Radium 223 decays with a half-life of 11.43 days; Radium 234, with a half life of 3.632 days. As a result, the probability that an atom of Radium 223 will decay at a time x days has a density function p(x) = m e−mx, where m = 0.06064, and the probability that an atom of Radium 224 will decay at a time y days has a density function q(y) = n e−ny, where n = 0.1908. Assuming that the decay times of the two atoms is independent, find the probability that an atom of Radium 223 will decay before an atom of Radium 224. You may leave your answer in terms of m and n (and this may be to your advantage!).
