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University of GuelphFinal Examination MATH*1200F11
EXAMINER: S. Gismondi, Department of Mathematics & Statistics, University of Guelph
! READ THIS ENTIRE COVER PAGE !
First Name:
Last Name:
I.D. Number:
Signature:
Pre-check list:
~ Ive read the cover page. I understand the rules. ~ Ive written my name & ID.~ Ive checked the examination. It looks complete. ~ Ive signed the examination.
CRITICAL INFORMATION!
1. Use radian measures in the argument of all trigonometric functions.2. This examination consists of this 13 page bookletincluding this cover page, and 12 pages of
questions. CHECK NOW!!
3.NO calculators allowed.NO communication allowed.NO additional aids allowede.g.
notes, books or scrap paper.
4. This examination is 2 hours long, is marked out of 84 and contributes at least 42% to your
final grade..
5. If due to an emergency or illness, you cannot complete this examination, you must 1)seek
immediate help and 2) obtain documentation as per "Undergraduate Degree Regulations:
Illness or Compassionate Reasons".
Post-check list:~ YES! I attempted EVERY QUESTION! ~ YES! I wrote my name & ID!
~ YES! I signed the examination!
Pg. 2. (TT_III) ________ Pg. 8. (TT_I) __________
Pg. 3. (TT_III) ________ Pg. 9. (TT_I) __________
Pg. 4. (TT_III) ________ Pg. 10. (TT_II) _________
Pg. 5. (TT_III) ________ Pg. 11. (TT_II) _________
Pg. 6. (TT_III) ________ Pg. 12. (TT_III) ________
Pg. 7. (Surprise) _______ Pg. 13. (TT_III) ________
Examination Grade Total _______ / 84
Instructor: S. Gismondi Voice: x53104Office: MACN510 Email: [email protected]
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 3
2.(7 marks) Find the right circular cylinder of maximum volume that can be inscribed in a
sphere of fixed radiusR. Be sure to show how/why you have a maximum volume and not a
minimum. Then tell me the radius and height of the cylinder.
Radius of cylinder:
Height of cylinder:
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3.(7 marks)
(a) Approximate the area betweeny = 1 +x +x and thex axis on [1,3] using a Riemann Sum2
i
with four equal sub-intervals and choosingz to be the left endpoint of the ith sub-interval.
(b) Construct the Riemann Sum that approximates the area betweeny =x and thex axis on2
i[a,b], with n equal sub-intervals, choosingz to be the midpoint of the ith sub-interval. Do not
compute the limit.
Answer:
Answer:
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 5
4.(7 marks) Solve the following indefinite integrals.
(a)
(b)
(c)
Answer:
Answer:
Answer:
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5. (7 marks) Compute the following derivatives i.e. findy.
(a)
(b)
Answer:
Answer:
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 7
6.(7 marks) State and prove Fermats Theorem, in the case where (c,f(c) is a maximum.
Proof: (ok - show me one more time!!)
Theorem:
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7.(7 marks)
(a) Use the epsilon-delta proof technique to prove . Assume and please show where
you assume, that || < 1.
(b) Use cases to solve forx, where |x-2| $3x - 4.
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 9
8.(7 marks)
(a) Compute .
(b) Compute the right and left limits of .
Compute .
Answer:
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 10
9.(7 marks)
(a) Car A travels north through an intersection at 12:00pm at 60 km/hr. Car B travels east
through the same intersection at 80 km/hr at 12:30pm. How fast are the cars separating at
1:00pm?
(b) Letf(x) = sin(x), wherex is in radians. Use differentials and show me how to estimate
sin(0.1) and sin(2.0), where we computef(x) at (0,0). You should get 0.1 and 2.0 respectively.
Answer:
Why is the estimate of sin(2.0) .2.0, a bad estimate?
Why is the estimate of sin(0.1) .0.1, a good estimate?
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Examiner: S. Gismondi Final Examination - MATH*1200F11 Page 11
10.(7 marks) Sketch .
HINT: The quadratic term is a perfect square. Factor it first.
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11.(7 marks) Differentiate the following functions and conclude/complete the corresponding
integral.
(a)
(b)
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12.(7 marks)
(a) Compute the average value of the functiony =x on [1,3].2
(b) A tiny bit of fun - you can do this if you think about it. Angela is filling a leaky tub with
water. She fills it at the rate oft litres per minute (i.e. at an increasing rate over time!) BUT after2
a minute, the level of water in the tub reaches the hole and it starts to leak at the rate of 0.1tlitres
per minute (... also increases over time!!) Write the definite integral that describes how much
water is in the tub after five minutes.
Answer:
Answer:
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