MATH1200F11 Final Examination

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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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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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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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8.(7 marks)

(a) Compute .

(b) Compute the right and left limits of .

Compute .

Answer:


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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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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:

MATH1200F11 Final Examination

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