MATH 148, SPRING 2016
COMMON EXAM III (PART 1) - VERSION A
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Question
1–13
Points Awarded
Points
52
1
PART I: Multiple Choice (4 points each)
1. Find the directional derivative of f (x, y) = ln(x2 + y 2 ) at (2, 1) in the direction of ~v = h3, −1i.
√
10
(a) −
5
(b) 2
√
10
(c)
5
(d) −2
(e) None of these.
2. Find
∂f
x3 + y 3
for f (x, y) = 2
.
∂x
x + y2
(a)
x4 + 3x2 y 2 − 2xy 3
(x2 + y 2 )2
(b)
2xy 3 − x4 − 3x2 y 2
(x2 + y 2 )2
(c)
x4 + 3x2 y 2 − 2xy 3
x4 + y 4
(d)
2xy 3 − x4 − 3x2 y 2
x4 + y 4
(e)
3x2 y 2 + y 4 − 2x2 y
(x2 + y 2 )2
2
3. Find the maximum rate of increase of f (x, y) = sin(xy) at (1, 0) and the direction in which it occurs.
(a) Maximum change: 1 Direction: ~v = h0, 1i
√
(b) Maximum change: 2 Direction: ~v = h1, 1i
(c) Maximum change: 1
Direction: ~v = h1, 0i
(d) Maximum change: 1 Direction: ~v = h−1, 0i
√
(e) Maximum change: 2 Direction: ~v = h−1, −1i
4. Consider the system of linear difference equations
x1 (t + 1)
=
−1.6x1 (t) + 0.3x2 (t)
x2 (t + 1)
=
0.6x2 (t).
Which of the following statements regarding the stability of the equilibrium (0, 0) is true for this system?
(a) Since λ1 < 1 and λ2 < 1, (0, 0) is locally stable.
(b) Since |λ1 | < 1 and |λ2 | < 1, (0, 0) is unstable.
(c) Since |λ1 | < 1 and |λ2 | < 1, (0, 0) is locally stable.
(d) Since λ1 < 1 and λ2 < 1, (0, 0) is unstable.
(e) None of these.
3
5. Find the domain and range of f (x, y) =
(a) Domain: {(x, y)|2x2 + y 2 ≤ 4}
p
4 − 2x2 − y 2 .
Range: [0, ∞)
2
2
Range: [0, 2]
2
2
Range: [0, ∞)
2
2
(b) Domain: {(x, y)|2x + y ≤ 4}
(c) Domain: {(x, y)|2x + y ≥ 4}
(d) Domain: {(x, y)|2x + y ≥ 4}
Range: [0, 2]
(e) Domain: {(x, y)|2x2 + y 2 ≤ 4}
Range: [0, 4]
6. Find a vector which is orthogonal (perpendicular) to the level curve of f (x, y) = xe−y + 3y at (1, 0).
(a) h1, 3i
(b) h1, 4i
(c) h1, 2i
(d) h2, 1i
(e) None of these.
4
7. Evaluate the limit
lim
y cos
(x,y)→(π,−π)
x+y
4
(a) 0
π
(b) − √
2
(c) π
π
(d) √
2
(e) −π
8. Let z = xex/y , where x(t) = cos t and y(t) = e2t . Find
(a) 2e
(b) 0
(c) −2e
(d) 2
(e) −2
5
dz
at t = 0.
dt
.
9. Let f (x, y) = x3 y + 12x2 − 8y. Which of the following statements is true?
(a) The function has a saddle point at (2, −4).
(b) The function has a local maximum at (2, −4).
(c) The function has a local minimum at (2, −4).
(d) The function has saddle points at (0, −4) and (2, −4)
(e) None of these.
10. Consider the system of nonlinear difference equations
x1 (t + 1)
=
x2 (t + 1)
=
1
x2 (t)
4
ax1 (t) − cos(x2 (t)) + 1.
Find all values of a > 0 for which the equilibrium (0, 0) is locally stable.
(a) 0 < a < 2
1
(b) 0 < a <
2
(c) 0 < a < 4
1
(d) 0 < a <
4
(e) None of these.
6
11. Find an equation of the tangent plane to the surface z = 2xe3y at the point (1, 0, 2).
(a) 2x + 6y − z = 0
(b) 2x + 2y − z = 0
(c) 6x + 2y − z = 0
(d) 2x + 6y − z = −4
(e) 2x + 2y − z = −4
12. Describe the level curves of f (x, y) = ln(x2 + y).
(a) The level curves are ellipses.
(b) The level curves are circles.
(c) The level curves are hyperbolas.
(d) The level curves are lines.
(e) The level curves are parabolas.
7
13. Find the linearization of f (x, y) = ln(x − 3y) at (7, 2).
(a) L(x, y) = x − 3y − 13
(b) L(x, y) = x + y − 9
(c) L(x, y) = x − 3y
(d) L(x, y) = x − 3y − 1
(e) L(x, y) = x + y − 1
8