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Unformatted text preview: Math 241 Final Exam Spring 2010 This exam has 8 questions. Instructions: Number the answer sheets from 1 to 8. Fill out all the informations at the top
of each sheet (write and sign the Honor Pledge on page 1 only). Answer one question on each
sheet in the correct order. (Do not answer one question on more than one sheet. Use the back of
the correct sheet if you need more space). None of the following are allowed: lecture notes, book, electronic devices of any kind
(including calculators, cell phones, etc.) You may keep with you one sheet of handwritten notes. 1. [25 points] Consider the line 60 with symmetric equations x+2 y—4
2—, 22—1 2 3 and the line 61 with symmetric equations x—1_y—2 z—2 3 ~2 ,3 (a) Show that 60 and E1 are perpendicular.
(b) Find the point of intersection P0 of the lines 60 and 61.
(c) Find an equation of the plane containing both £0 and £1. 2. [25 points] Consider the curve C parametrized by 77(15):: 2ti+t2j+lntl€, 1 g t < 4. (a) Explain why C is a smooth curve (b) Find parametric equations for the line passing through P0 := F(2) in the direction of
the tangent to C at P0. (c) Find the length of the curve C. 3. [25 points] Consider the function
ﬂay) == 96231 - $2 - 21/2 + 3 (a) Find all critical points of the function f (b) Determine whether each critical point yields a relative maximum value, a relative
minimum value or a saddle point. 4. [20 points] Let
f($,y,z) I: wry—1’2 (a) Find the directional derivative of f at the point (1, 1, 1) in the direction of
a :2 2 i2 5' — 2 1'5. (b) Find the unit vector for which the directional derivative is maximal at the point (1, 1, 1). 5. [30 points] Compute the double integral 1
//Rl+m4dA where R is the triangle with vertices (0,0), (1,0) and (1, 1). Hint: Integrate ﬁrst in the y
variable. 6. [25 points] Compute the triple integral (x2 +y2)dV where D is the region bounded above by the paraboloid z = 25 — x2 —— y2 and below by the
x—y plane. 7. [25 points] Use Green’s theorem to compute the line integral /F~d7” C where 13 := y3i+ x3; and C is the triangle with vertices (0,0), (1,0) (0, 1) oriented counter—
clockwise. 8. [25 points] Use Gauss’s theorem (also known as the divergence theorem) to compute the ﬂux
//8 (13 - fi)clS where F := x2§+ 1423+ 22]; and S is the boundary of the region D bounded above by the
sphere 9:2 + y2 + 22 = 9 and below by the 92-3; plane. 71’ is the unit normal vector directed
outward from the region D. ...
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- Spring '08