M ATHEMATICS 54
First Semester AY 2010 - 2011 15 October 2010
Final Exam Sample 1
Write all necessary solutions on your bluebooks and box all final answers. You have two hours to finish this exam. Good luck! I. True or False. Write TRUE if the statement is true. Otherwise, write FALSE.
(1 point each)
~ = ~B × C, ~ then A ~ is parallel to both vectors ~B and C. ~ 1. If A 2. If ` is the line of intersection of a1 x + a2 y + a3 z + a4 = 0 and b1 x + b2 y + b3 z + b4 = 0, then the vectors < a1 , a2 , a3 > and < b1 , b2 , b3 > are orthogonal to `. 3. The graph of the equation 4x2 + 4y2 − 4z2 + 4x + 8y − 4z − 16 = 0 is an elliptic cone. ( x = sin t is on the parabola y = x2 . 4. Every point on the parametric curve y = sin2 t 5. The graph of the polar equation r2 = 4 sin 2θ is a rose with four petals.
II. Multiple Choice. Choose the letter of the best answer.
(3 points each)
1. Which of the following is an equation of the tangent plane to the paraboloid z = x2 + y2 at the point (1, 2, 5)? (a) 2x + 4y + z − 5 = 0
(c) 4x + 2y + z − 5 = 0
(b) 2x + 4y − z − 5 = 0
(d) 4x + 2y − z − 5 = 0
2. The domain of the function f ( x, y) = p
-3
-2
is
3
3
2
2
1
1
1
-1
(a)
1 1 − x 2 + y2
2
3
-3
-2
-1
-1
-1
-2
-2
(c)
-3
1
2
3
1
2
3
-3
3
-3
(b)
-2
2
2
1
1
1
-1
2
3
-3
-2
-1
-1
-1
-2
-2
(d)
-3
-3
3. A set of symmetric equations for the line through the origin that is parallel to the line through (−1, 3, 1) and (3, −7, 3) is (a)
x y z = = 4 10 2
(b)
x y z = = −4 10 2
(c)
x y = =z 2 5
(d)
x y = =z 2 −5
4. Which of the following is a linear approximation of f ( x, y) = x3 + 2 ln y2 at the point (2, 1)? (a) 12x + 4y − 20
(b) 6x − 4y − 8
(c) 12x + 4y + 20
∇ M ORE AT THE B ACK ∇
(d) 12x + 4y + 8
5. Which of the following is unit tangent vector ~T (t) of the vector-valued function ~R(t) =< t, sinh t, cosh t >? (a)
√2
√
tanh t,
2
√ 2 2 2 , 2 secht
(c)
(b) < tanh t, 1, secht >
√2 2
√
secht,
√ 2 2 2 , 2
tanh t
(d) < secht, 1, tanh t >
6. What is the vector projection of the vector u = i − j onto v = −i − 2j + k? (a)
1 1 2i − 2j
(b) − 16 i − 31 j + 16 k
(c) 2i − 2j
(d) −6i − 3j + 6k
7. What is the length of the curve represented by the vector function x = 3t, y = sin 4t and z = cos 4t from t = 0 to t = π? (a)
2 2 3 (9π
+ 1)
(b)
2 2 3 (9π
+ 16)
(c) 5π
(d)
√
12π
8. The acceleration vector of the particle moving along the curve having parametric equations x = et , y = e−t , z = et at t = 1 is (b) < e, −e, 0 >
(a) < e, e, 0 >
(c) < e, 1/e, 0 > ∂f ? ∂a ∂w ∂g ∂w ∂h (c) + ∂u ∂a ∂v ∂a
(d) < e, −1/e, 0 >
9. If w = f (u, v) and u = g( a, b) and v = h( a), which of the following is (a)
∂w du ∂w ∂v + ∂u da ∂v ∂a
(b)
∂ f ∂u ∂ f dv + ∂u ∂a ∂v da
(d)
∂ f ∂g ∂ f ∂h + ∂u ∂a ∂v ∂a
10. Which of the following points is the focus of the parabola y2 + 2y − 8x + 17 = 0? (b) (0, −1)
(a) (2, 1)
ˆ III. Evaluate 0
+∞
(c) (2, −3)
x dx. ex
IV. Given the plane curve
(d) (4, −1)
(5 points)
y2 z2 − = 1. 9 4
1. Find the equation of the surface of revolution generated by revolving the given plane curve about the y-axis. (3 points) 2. Sketch the graph of the surface generated.
V. Show that
lim
( x,y)→(0,0)
x2 + 3xy + y2 does not exist. x 2 + y2
F E ND OF E XAM F T OTAL : 50 POINTS
(2 points)
(5 points)