OF DEPARTMENT MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 EXAM 1 Spring 2006
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In problems that require reasoning or algebraic calculation, it is not sufficient just to write the answers. You must explain how you arrived at your answers, and show your algebraic calculations. You can leave answers in terms of fractions and square roots, but if approximate numerical answers are used, they should be rounded off to 4 significant figures. x x < x, y, z >, [x, y, z], y , y ; are all permissible notations for the vector xi + yj + zk. z z
1. 2. 3. 4. 5. 6. Total
(30) (21) (12) (18) (9) (10)
Perfect Paper - 100 Points. There are nine pages, including this one, in this exam and six problems. Make sure you have them all before you begin!
2 FORMULAS a b = a1 b1 + a2 b2 + a3 b3 = |a||b| cos |a| = aa= a2 + a2 + a2 . 2 3 1
i j k a b = a1 a2 a3 b1 b2 b3 where
=
a2 a3 b2 b3
i -
a1 a3 b1 b3
j +
a1 a2 k b1 b2
a b = ad - bc. c d
|a b| = |a||b| sin = area of parallelogram with sides a, b. The vector projection of b onto (in the direction of) a is proj a b = The scalar projection (component) of b onto a is comp a b = ab |a| ab a aa
The volume of the "box" determined by a, b and c is |a (b c)| Arc length of parametrized curve:
b
L=
a
|r (t)| dt.
3 1. Given three points: P = (1, -3, 7), Q = (-2, -6, 5), R = (-2, -6, 9) a) (6 points) Show that the triangle P QR has two sides of the same length.
b) (6 points) Find the angle between sides P Q and P R.
c) (6 points) find the point where the line containing P and Q intersects the plane z - x = 4.
4 d) (6 points) Give an equation for the plane containing P , Q, and R.
e) (6 points) Find the area of the triangle P QR.
5 2. (3 points each) For each label picture, it with the matching equation (1) through (6). It is enough to check that it has the right sort of traces when the surface is sliced by planes parallel to the xy, xz, and yz planes. You will not use all the equations, but every picture will match one of the equations. You do not need to give reasons for your choices, or name the surfaces. (1) x2 - z = 0 (2) x2 + 2y 2 = 4 (3) 9x2 + 4y 2 + z 2 = 1 (4) x2 - y 2 + z 2 = 4 (5) 2x2 - y + z 2 = 0 (6) -x2 + y 2 - z 2 = 4
6 3. Consider the curve C given by the vector function: r(t) = <
1 2
t2 ,
4 3
t3/2 , 2t >.
a) (7 points) Give parametric equations for the tangent line to C at t = 1.
b) (8 points) What is the total length of the part of C between (0, 0, 0) and (8, 32 , 8)? 3
c) (6 points) Find the point or points where the curve meets the plane x - z = 0.
7 4. (18 points total) A person is standing on a 122.5 meter tall cliff overlooking the ocean. She hits a golf ball off the cliff, sending it perfectly horizontally at a speed of 2 m/sec. Since she is standing on the surface of Earth, we can ta...