Mathematics 1225A/B Study Guide - Final Guide: Saddle Point, Maxima And Minima, Lagrange Multiplier
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8 Feb 2013
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from 2011 tests
Page 1 CODE 111 Mathematics 1225B
Practice Final Exam
PART A (35 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET. Only the (scantron) answer sheet will be marked for Part
A. Extra time will NOT be given for coding answers at the end of the exam.
A1. If log3x−log3x2= 2, find x.
A: 6 B: 3 C:1
3D: 9 E:1
9
A2. If f(x) = 4e√x, find f′(4).
A:1
2e2B:e2C: 2e2D: 4e2E: 8e2
A3. If f(x) = 2−x, find f′(0).
A: 0 B: 1 C:−1D:−
1
ln 2 E:−ln 2
A4. Find the slope of the tangent line to the graph of y=e2xat the point where x= ln 3.
A: 8 B: 6 C: 12 D: 18 E: 9
A5. If 8x−11
(x−2)(x−1) =A
x−2+B
x−1, find B.
A: 1 B: 2 C: 3 D: 4 E: 5
A6. Find Z12x3ln x dx.
A:x43 ln x−
1
4+CB: 3x3(xln x−1) + CC: 12x4(ln x−1) + C
D: 3x4ln x−
1
4+CE:3x4(ln x)2
2+C
A7. Evaluate Ze3
e
1
xln xdx.
A: ln 3 B:e3−eC:1
3D:−
2
3E: 0
Mathematics 1225B
Practice Final Exam CODE 111 from 2011 tests
Page 2
A8. Evaluate Z∞
1
−1
x5dx.
A:1
4B:−
1
4C: 1 D:−1E: diverges
A9. Determine which one of the following integrals represents the area of the region bounded
by y=x2and y= 4.
A:Z2
0
(4 −x2)dx B:Z4
0
(4 −x2)dx C:Z2
0
√y dy D:Z4
0
√y dy E:Z4
0
2√y dy
A10. Find the area of the region bounded by y=x2−xand y=x.
A:8
3B:−
8
3C:4
3D:−
4
3E:16
3
A11. Find the volume of the solid of revolution obtained by rotating the region bounded by
y=√x,y= 2 and x= 0 about the x-axis.
A: 6πB: 8πC:8π
3D:16π
3E:32π
5
A12. Find the volume of the solid of revolution obtained by rotating the region bounded by
y=√x,y= 2 and x= 0 about the y-axis.
A: 6πB: 8πC:8π
3D:16π
3E:32π
5
A13. If f(x, y) = x2−3xy +y3, find fy(3,1).
A:−7B: 3 C: 24 D:−6E: 1
A14. If f(x, y) = yx, find fyx(x, y).
A:yxln yB:xyx−1C:yx−1D:yx−1(1 + xln y)E:yx
A15. Let f(x, y) = yex−3x−y. Find the only critical point of f(x, y).
A: (0,0) B: (0,3) C: (1,1) D: (3,1) E: (−3,−1)
from 2011 tests
Page 3 CODE 111 Mathematics 1225B
Practice Final Exam
A16. Find all the critical points of the function f(x, y) = 3x−x3−3xy2.
A: (0,0) B: (1,1),(−1,−1) C: (1,0),(−1,0),(0,1),(0,−1)
D: (0,0),(−1,1),(1,1) E: (1,−1),(1,1),(−1,1),(−1,−1)
Use the following information for questions 17, 18 and 19.
f(x, y) = x3−6xy −y2
fx(x, y) = 3x2−6y
fy(x, y) = −6x−2y
fxx(x, y) = 6x
fyy(x, y) = −2
fxy(x, y) = −6
A17. Which one of the following is true for the point (1,−3)?
A: (1,−3) is not a critical point of f(x, y).
B:f(x, y) has a saddle point at (1,−3).
C:f(x, y) has a local minimum at (1,−3).
D:f(x, y) has a local maximum at (1,−3).
E: The second partials test yields no information.
A18. Which one of the following is true for the point (−6,18)?
A: (−6,18) is not a critical point of f(x, y).
B:f(x, y) has a saddle point at (−6,18).
C:f(x, y) has a local minimum at (−6,18).
D:f(x, y) has a local maximum at (−6,18).
E: The second partials test yields no information.
A19. Which one of the following is true for the point (0,0)?
A: (0,0) is not a critical point of f(x, y).
B:f(x, y) has a saddle point at (0,0).
C:f(x, y) has a local minimum at (0,0).
D:f(x, y) has a local maximum at (0,0).
E: The second partials test yields no information.
A20. If the method of Lagrange multipliers is used to maximize the function f(x, y) = xy + 10
subject to the constraint x2+ 9y2= 18, what system of equations must be solved?
A:xy + 10 + λ(x2+ 9y2−18) = 0 B: 2x+λy = 0
18y+λx = 0
xy + 10 = 0
C:y+ 2xλ = 0
x+ 18yλ = 0
x2+ 9y2−18 = 0
D:xy + 2λx = 0
xy + 2λy = 0
x2+ 9y2−18 = 0
E:xy + 10 = 0
x2+ 9y2−18 = 0