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Calculus I Examination, Dawson College, May 2012, Exams of Calculus

West Bengal State UniversityCalculus

A final examination for calculus i students at dawson college, mathematics department. The exam consists of 13 questions covering limits, derivatives, integrals, and related topics. Students are required to answer all questions directly on the exam paper and are allowed to use small calculators and dictionaries.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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1
Dawson College
Mathematics Department
Final Examination
201-NYA-05, Calculus I- Chem. Tech/Lab Tech.
Section: 00005
Tuesday, May 22, 2012
Student Name: _____________________________________________
Student I.D. #:______________________________________________
Instructor: O. Veres
Time: 9:30 AM – 12:30 PM
INSTRUCTIONS:
Print your name and student number in the space
provided above.
Attempt all questions. Show all your work.
All questions are to be answered directly on the
examination paper.
Small, noiseless, NON-PROGRAMMABLE
calculators without text storage or graphic
capabilities are permitted.
Translation and regular dictionaries are permitted.
This examination consists of 13 questions on 13
pages.
Please ensure that you have a complete exam package
before starting.
The examination must be returned intact.
Question# Marks
obtained/available
1 /6
2 /16
3 /5
4 /5
5 /5
6 /12
7 /4
8 /4
9 /16
10 /4
11 /5
12 /10
13 /8
Total/100
pf3
pf4
pf5

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Dawson College

Mathematics Department

Final Examination

201-NYA-05, Calculus I- Chem. Tech/Lab Tech.

Section: 00005

Tuesday, May 22, 2012

Student Name: _____________________________________________

Student I.D. #:______________________________________________

Instructor: O. Veres

Time: 9:30 AM – 12:30 PM

INSTRUCTIONS:

 Print your name and student number in the space

provided above.

 Attempt all questions. Show all your work.

 All questions are to be answered directly on the

examination paper.

 Small, noiseless, NON-PROGRAMMABLE

calculators without text storage or graphic

capabilities are permitted.

 Translation and regular dictionaries are permitted.

 This examination consists of 13 questions on 13

pages.

 Please ensure that you have a complete exam package

before starting.

 The examination must be returned intact.

Question# Marks

obtained/available

Total/

  1. [6 marks] Find the following limits:

a. lim ݔ→ ∞

ହ௫ రି^ ଶ௫ య^ ାଷ ଶ௫ ర^ ା௫ మ^ ାସ௫

b. lim (^) ௫→ିଵ

ଶ௫ మି^ ଶ ௫ మି^ ௫ିଶ

  1. [16 marks] Find the derivative of each function.

a. ൌ ݕ

ଵା௖௢௦ଷ௫ ௘ మೣ^ ା௫ మ

b. ݂ ሺݔሻ ൌ ln ଶ^ ሺsin 2ݔሻ

c. ݕൌ ඥsecሺݔ ଶ^ ൅ 3 ݔ൅ 1 ሻ

d. ݂ ሺݐሻ ൌ tanି ଵ^ ሺ1 ൅ 4ݐ ଶ^ ሻ

  1. [5 marks] Find the derivative of ݂ ሺݔሻ ൌ ଶ௫ using the definition of the derivative.
  2. [5 marks] The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of ݉ܿ 2.5 ଶ^ ݊݅݉/. At what rate is the base of the triangle changing when the altitude is ݉ܿ 10 and the area is ݉ܿ 100 ଶ?
  3. [5 marks] A farmer plans to fence a rectangular field that borders a river. The area of this field is 1440 square meters. What dimension would require the least amount of fencing if no fencing is needed along the river?
  4. [12 marks] Let ݂ ሺݔሻ ൌ 2ݔ ଷ^ െ 12ݔ ଶ^ ൅ 18ݔ.

a. Find the x and y intercepts.

b. Find the intervals where f(x) is increasing or decreasing.

c. Find all relative maxima and minima.

d. Find the intervals where f(x) is concave upward or concave downward.

e. Find all points of inflection.

f. Using the above information sketch the graph of f(x). Clearly label the important points on the graph.

Answers:

  1. a. 5/ b. 4/
  2. a. ݕ ᇱ^ ൌ െ

ଷ൫௘ మೣ^ ା௫ మ^ ൯ ୱ୧୬ ଷ௫ାଶ൫௘ మೣ^ ା௫൯ሺଵାୡ୭ୱ ଷ௫ሻ ሺ௘ మೣ^ ା௫ మ^ ሻమ^ b.^ ݂

ᇱ (^) ሺݔሻ ൌ 4 cot 2ݔ lnሺcos 2ݔሻ

c. ଵଶ ሺ2 ݔ൅ 3ሻ tanሺݔ ଶ^ ൅ 3 ݔ൅ 1ሻ ඥsecሺݔ ଶ^ ൅ 3 ݔ൅ 1ሻ

d. ݂ ᇱ^ ሺݐሻ ൌ ଼

௧ ଵାሺଵାସ௧ మ^ ሻమ

  1. ݂ ᇱ^ ሺݔሻ ൌ െ

ଶ ௫ మ

  1. ݊݅݉/ ݉ܿ 1.
  2. 53.67x26.
  3. a. (0,0) , (3,0) b. f is increasing on ሺെ∞, 1ሻ ∪ ሺ3, ∞ሻ, f is decreasing on ሺ1,3ሻ c. relative maximum (1,8), relative minimum (3,0) d. f is concave upward on ሺ2, ∞ሻ and f is concave downward on ሺെ∞, 2ሻ e. e. inflection point (2,4)
  1. a. ݁ ୟ୰ୡୱ୧୬ ௫^ ܥ ൅ b. 145/6 c. െ

௖௢௦ ర^ ௫ ସ ܥ ൅^ d.^

ଷ ଶ ݔ|ln^

  1. x=2, x=‐3/
  2. ൌ ܣ

ଷଶ ଷ ݑ^

ଶ

  1. a. ܸൌ

ସగ ହ ݑ^

ଷ (^) b. ܸݑ ߨ ൌ ଷ

  1. a. ௗ௬ௗ௫ ൌ (^) ଷ௬ସ௫ିଷ మି (^) ସ௬ b. y= 3x+