讲解Math 253编程、辅导Java，c++程序语言、CS，Java程序设计辅导讲解SPSS|辅导Web开发

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Math 253 Final Exam December 2018 Duration: 150 minutes
This test has 8 questions on 8 pages, for a total of 80 points.
• Read the questions carefully.
• Where appropriate, give complete arguments and explanations for all your calculations.
Answers without proper justification may not be marked.
• Continue on the closest blank page if you run out of space, and indicate this clearly
on the original page.
• Attempt to answer all questions for partial credit.
• This is a closed-book examination. No aids of any kind are allowed, including:
notes, cheat sheets, electronic devices of any kind (including calculators, phones, etc.)
First Name: Solutions Last Name:
Student-No: Section:
Signature:
Question: 1 2 3 4 5 6 7 8 Total
Points: 12 12 9 8 8 8 8 15 80
Score:
Student Conduct during Examinations
1. Each examination candidate must be prepared to produce,
upon the request of the invigilator or examiner, his
or her UBC id card for identification.
2. Examination candidates are not permitted to ask questions
of the examiners or invigilators, except in cases of
supposed errors or ambiguities in examination questions,
illegible or missing material, or the like.
3. No examination 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. Should the examination
run forty-five (45) minutes or less, no examination
candidate shall be permitted to enter the examination
room once the examination has begun.
4. Examination candidates must conduct themselves honestly
and in accordance with established rules for a given
examination, which will be articulated by the examiner or
invigilator prior to the examination commencing. Should
dishonest behaviour be observed by the examiner(s) or
invigilator(s), pleas of accident or forgetfulness shall not
be received.
5. Examination candidates suspected of any of the following,
or any other similar practices, may be immediately dismissed
from the examination by the examiner/invigilator,
and may be subject to disciplinary action:
i. speaking or communicating with other examination
candidates, unless otherwise authorized;
ii. purposely exposing written papers to the view of other
examination candidates or imaging devices;
iii. purposely viewing the written papers of other examination
candidates;
iv. using or having visible at the place of writing any
books, papers or other memory aid devices other than
those authorized by the examiner(s); and,
v. using or operating electronic devices including but
not limited to telephones, calculators, computers, or
similar devices other than those authorized by the
examiner(s)—(electronic devices other than those authorized
by the examiner(s) must be completely powered
down if present at the place of writing).
6. Examination candidates must not destroy or damage any
examination material, must hand in all examination papers,
and must not take any examination material from
the examination room without permission of the examiner
or invigilator.
7. Notwithstanding the above, for any mode of examination
that does not fall into the traditional, paper-based
method, examination candidates shall adhere to any special
rules for conduct as established and articulated by
the examiner.
8. Examination candidates must follow any additional examination
rules or directions communicated by the examiner(s)
or invigilator(s).
Math 253 Final Exam pg 2 of 8 Student-No.
12 marks 1. Consider the following contour plot of a function f(x, y). The values of the contours are
equally spaced and other reasonable assumptions can be made.1
The gradient of f at (9.5, 5.5) is indicated by the black arrow; it is (∇f)(9.5, 5.5) = h1, 0i.
(a) Find the approximate coordinates of the critical points of f on the domain shown.
Classify each as a local maximum, a local minimum, a saddle point, or “other”.
Solution: (12, 6) local max (3, 3) local max (6.5, 7.5) local min
(7, 4.5) saddle (3.5, 8.5) saddle
(b) Find the coordinates of the global maximum of f and the global minimum of f.
Solution: (12, 6) is global max, by counting contours. (0, 7.5) is the global min.
(c) Is fy(11, 10) negative or positive? (circle the correct answer)
Solution: negative: as we move upwards, the contour values are decreasing.
(d) Is fyy(11, 10) negative or positive? (circle the correct answer)
Solution: negative: as we move upwards, the contour values are decreasing and
they are also becoming closer together—they are decreasing at a faster rate.
(e) Suppose f(9.5, 5.5) = 10. Give the approximate value of f(13, 6). Hint: use the
gradient to estimate the contour spacing.
Solution: First note the magnitude of the gradient is 1. This is a slope = rise
over run. Specifically, over a run of 1 we have a rise of 1.
1Reasonable assumptions include: the gradient does not vanish along an entire contour; and the function
does not fluctuate wildly on a scale smaller that shown by the contours.
Page 2
Math 253 Final Exam pg 3 of 8 Student-No.
Now we look at the diagram. Over a run of 1 (from 9.5 to 10.5) we crossed two
contours. Thus the contour spacing must be 0.5. So f(13, 6) ≈ 11.5 (its three
contours above 10).
(f) What is the direction of the gradient at the point (6.5, 1)? Circle your answer.
(g) An angry fire ant is at (2, 10). It starts walking and with each small step, it moves
in the direction of maximum increase of f. On the contour plot above, draw the
path followed by the ant. Hint: practice on the spare rough copy to the left.
12 marks 2.
2 and consider the implicit surface defined by F(x, y, z) = 0.
(a) Find all points on the surface where x = 1 and y = 2
Solution: There are two, with z = ±2.
(b) Let a be a parameter. Consider two planes given by
a(x − 1) − 4(y − 2) + 4(z − 2) = 0,
a(x − 1) − 4(y − 2) − 4(z + 2) = 0.
What is the cosine of the angle between their normal vectors?
Solution: The normal vectors are u = ha, −4, 4i and v = ha, −4, 4i.
(c) Find the value of a such that both planes are tangent to the implicit surface at
x = 1 and y = 2. We see that
a = −16 (noting the last component matches using our answer from (a)).
(d) Set up (but do not evaluate) an integral for the total surface area of the surface
where (x, y) is in the rectangular region R = {(x, y) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 2}.
Solution: Solve for z
2 = x
4
y
2
. We’ll need to multiply by 2 because there is a
top and bottom to the surface,
Math 253 Final Exam pg 4 of 8 Student-No.
9 marks 3. In the contour plots below, the values of the contours are evenly spaced. Nine of these
twelve plots correspond to graphs on the next page.
Math 253 Final Exam pg 5 of 8 Student-No.
Put the letter of the corresponding contour plot from the previous page in the box below
each graph. (The axes of the nine graphs below are all oriented in the standard way: the
positive x-axis is on the left, the positive y-axis is on the right, and the positive z-axis is
upward.)
Math 253 Final Exam pg 6 of 8 Student-No.
8 marks 4. Consider the integral Z 8
(a) Sketch the domain of integration.
Solution: The domain of integration is the bounded region under the curve
y = x
3 over the interval 0 ≤ x ≤ 2.
(b) Evaluate the integral.
Solution: We cannot find a closed form for the antiderivative of e
x
4
, so our
strategy is to switch the order of integration. Upon switching the order of
integration,
(b) Indicate whether each statement about a lamina is true or false (circle your answer).
i. True/ False : If ρ(x, y) = ρ0, then the mass is independent of ρ0.
ii. True /False: If ρ(x, y) = ρ0, then the centre of mass is independent of ρ0.
iii. True/ False : If ρ(x, y) = f(x) then ¯y = 0 (because y is an odd function).
(c) Sketch a shape D of uniform density whose centre of mass is not contained in D.
Solution: Lots of non-convex shapes have this property, an annulus or part of
an annulus for example.
3. Fill in the boxes below to complete the integrals for the volume of C.

讲解Math 253编程、辅导Java，c++程序语言、CS，Java程序设计辅导 讲解SPSS|辅导Web开发

讲解Math 253编程、辅导Java，c++程序语言、CS，Java程序设计辅导讲解SPSS|辅导Web开发