Differentiable Function - Advanced Calculus - Exam, Exams of Advanced Calculus

This is the Exam of Advanced Calculus which includes Line, Parametric Equations, Plane, Distance, Plane, Line, Surface, Tangent Plane, Parallel, Vectors etc. Key important points are: Differentiable Function, Directional Derivative, Point, Move Towards, Express, Quantity, Terms, Surface, Determine, Value

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The University of British Columbia
Final Examination - December 9, 2006
Mathematics 226
Section 101
Instructor: C. Lamb
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
Special Instructions:
No notes, books or calculators are to be used. No credit will be given for the correct answer
without the (correct) accompanying work. Use the back of the pages if you need extra space.
Rules governing examinations
Each candidate must be prepared to produce, upon request, a
UBCcard for identification.
Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
Candidates suspected of any of the following, or similar, dishon-
est practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Having at the place of writing any books, pap ers
or memoranda, calculators, computers, sound or image play-
ers/recorders/transmitters (including telephones), or other mem-
ory aid devices, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other can-
didates or imaging devices. The plea of accident or forgetfulness
shall not be received.
Candidates must not destroy or mutilate any examination mate-
rial; must hand in all examination papers; and must not take any
examination material from the examination room without permis-
sion of the invigilator.
Candidates must follow any additional examination rules or di-
rections communicated by the instructor or invigilator.
1 15
2 10
3 10
4 10
5 10
6 15
7 15
8 15
Total 100
Page 1 of 17 pages
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The University of British Columbia Final Examination - December 9, 2006 Mathematics 226 Section 101 Instructor: C. Lamb

Closed book examination Time: 2.5 hours

Last Name First Signature

Student Number

Special Instructions:

No notes, books or calculators are to be used. No credit will be given for the correct answer without the (correct) accompanying work. Use the back of the pages if you need extra space.

Rules governing examinations

  • Each candidate must be prepared to produce, upon request, a UBCcard for identification.
  • Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
  • No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
  • Candidates suspected of any of the following, or similar, dishon- est practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other mem- ory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
  • Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
  • Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.

Total 100

Page 1 of 17 pages

[15] 1. Let z = f (x, y) be a differentiable function on R^2 such that f (1, 2) = 3, f (1. 2 , 2 .3) = 3.4 and f (0. 9 , 2 .1) = 3.2.

(a) Estimate ∂z ∂x (1, 2) and ∂z ∂y (1, 2). [10pt]

(b) Estimate the value of the directional derivative of z = f (x, y) at the point (1, 2) as you move towards the point (2, 3). [5pt]

[10] 2. Let z = f (x, y) be a differentiable function on R^2 , x = (s^2 + t^2 )/2 and y = (s^2 − t^2 )/2. Express the quantity (^) ( ∂z ∂x

∂z ∂y

in terms of s, t, ∂z ∂s and ∂z ∂t

[10] 4. Let the temperature at a point (x, y, z) be given by w = x^3 y^2 z. Find the point on the plane 2x + 2y + z = 24 where the temperature is a maximum. You do not need to justify that your answer actually gives the maximum. Hint: You may assume in your calculations that x 6 = 0, y 6 = 0 and z 6 = 0 since if x, y or z equals 0, then w = 0 and this will not be the maximum temperature.

[10] 5. Evaluate the interated double integral

∫ (^) y=√π

y=

∫ (^) x=√π

x=y

sin(x^2 ) dx dy.

[15] 7. Let W be the 3-dimensional solid defined by the inequalities x^2 + y^2 ≤ 1 and 0 ≤ z ≤ 2 − x^2 − y^2.

(a) Draw a sketch of W. Be sure to show the units on the axes. [5pt]

(b) Use cylindrical coordinates to evaluate

W

zdV. [10pt]

[15] 8. Let z = f (x, y) be defined by

z =

y^5 x^2 + y^2 if (x, y) 6 = (0, 0)

0 if (x, y) = (0, 0).

(a) Use the definition of partial derivatives as limits to calculate ∂z ∂x (0, 0) and ∂z ∂y (0, 0). [5pt]

(b) Is z = f (x, y) differentiable at (0, 0)? If you think that this is the case, then give a formal  − δ proof to justify your belief. Otherwise, indicate clearly why you believe that z = f (x, y) is not differentiable at (0, 0). [10pt]

The End