Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Mathematics Exam: Calculus I - Dawson College, Exams of Calculus

The final examination questions for calculus i at dawson college. The exam covers topics such as limits, continuity, derivatives, integrals, and applications of calculus. Students are required to find limits, use the definition of continuity, find derivatives using the definition, find tangent lines, and perform integrations.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

gaggandeep
gaggandeep 🇮🇳

4.6

(40)

100 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
DAWSON COLLEGE
MATHEMATICS DEPARTMENT
Final Examination
Mathematics 201-NYB-05 Date: Friday, December 18, 2009
Calculus I Regular Time: 9:30 - 12:30
Instructor: M´elanie Beck
1. (12 marks) If the given limit exists, find its value; otherwise explain why the limit doesn’t exist.
(a) lim
x→4
√x−2
x−4
(b) lim
x→1+g(x), where g(x) = √1−x, if x≤1,
x+ 1,if x > 1,
(c) lim
x→−1+
x2−2x+ 1
x+ 1
(d) lim
x→∞
x2+ 2
x−1
2. (4 marks) Let f(x) = (x, if x < 1,
2,if x= 1,
2x−1,if x > 1.
Use the three conditions of the definition of continuity to determine whether this function is continuous at
x= 1. Show all your work.
3. (5 marks) Let f(x) = 1
x+ 1. Find f0(x) using the definition of derivative.
4. (4 marks) Find the derivative of f(x) = √xsin x.
5. (4 marks) For what values of xwill y=4x−3
x2+ 1 have a horizontal tangent?
6. (5 marks) Given that f(x)=2x2+ sin 2x, find f00 (x).
7. (4 marks) Find dy
dx for cos x+ 3xy2=y−tan x.
8. (4 marks) Find the slope of y= ln px2−4 at (3,ln √5).
9. (4 marks) Find dy
dx for y=xsin x.
10. (5 marks) Find an equation of the tangent line to y=e1−x2at the point (1,1).
11. (4 marks) Find the derivative of g(x) = 2x
√x2+ 1 and simplify your result. (Your final answer should be in
the form a
(x2+ 1)p/q , where a,pand qare integers.)
12. (8 marks) A boat is pulled into a dock by means of a winch 12 feet above the deck of a boat. The winch pulls
in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. [Note:
a sketch was provided.]
13. (8 marks) If 1200 cm2of material is available to make a box with a square base and an open top, find the
largest possible volume of the box.
14. (5 marks) Find the absolute maximum and absolute minimum values of f(x) = x2+ 2x−4 on [−2,1].
15. (8 marks) Compute the following integrals
(a) Z√x−1
√x+x2dx
pf2

Partial preview of the text

Download Mathematics Exam: Calculus I - Dawson College and more Exams Calculus in PDF only on Docsity!

DAWSON COLLEGE

MATHEMATICS DEPARTMENT

Final Examination

Mathematics 201-NYB-05 Date: Friday, December 18, 2009 Calculus I Regular Time: 9:30 - 12: Instructor: M´elanie Beck

  1. (12 marks) If the given limit exists, find its value; otherwise explain why the limit doesn’t exist.

(a) (^) xlim→ 4

x − 2 x − 4 (b) (^) xlim→ 1 + g(x), where g(x) =

1 − x, if x ≤ 1, x + 1, if x > 1,

(c) (^) x→−lim 1 +^ x

(^2) − 2 x + 1 x + 1 (d) (^) xlim→∞^ x

x − 1

  1. (4 marks) Let f (x) =

{ (^) x, if x < 1, 2 , if x = 1, 2 x − 1 , if x > 1. Use the three conditions of the definition of continuity to determine whether this function is continuous at x = 1. Show all your work.

  1. (5 marks) Let f (x) = (^) x + 1^1. Find f ′(x) using the definition of derivative.
  2. (4 marks) Find the derivative of f (x) = √x sin x.
  3. (4 marks) For what values of x will y =^4 xx 2 −+ 1^3 have a horizontal tangent?
  4. (5 marks) Given that f (x) = 2x^2 + sin 2x, find f ′′(x).
  5. (4 marks) Find dydx for cos x + 3xy^2 = y − tan x.
  6. (4 marks) Find the slope of y = ln

x^2 − 4 at (3, ln

  1. (4 marks) Find dydx for y = xsin^ x.
  2. (5 marks) Find an equation of the tangent line to y = e^1 −x^2 at the point (1, 1).
  3. (4 marks) Find the derivative of g(x) = √^2 x x^2 + 1

and simplify your result. (Your final answer should be in the form (^) (x (^2) + 1)a p/q , where a, p and q are integers.)

  1. (8 marks) A boat is pulled into a dock by means of a winch 12 feet above the deck of a boat. The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. [Note: a sketch was provided.]
  2. (8 marks) If 1200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
  3. (5 marks) Find the absolute maximum and absolute minimum values of f (x) = x^2 + 2x − 4 on [− 2 , 1].
  4. (8 marks) Compute the following integrals

(a)

x − √^1 x + x^2

dx

(b)

x(x^2 + 3)^4 dx

  1. (16 marks) Consider the function f (x) = x

x^3 + 1. The first and second derivatives of^ f^ are

f ′(x) = 6 x

2 (x^3 + 1)^2 and^ f^

′′(x) =^12 x(1^ −^2 x^3 ) (x^3 + 1)^3. (a) Give the domain of f. (b) Find the intercepts. (c) Find the horizontal and/or vertical asymptotes (if any). (d) Find the intervals where f is increasing, and the intervals where f is decreasing. (e) Find the local maxima and local minima. (f) Find the intervals where f is concave up, and the intervals where f is concave down. Give the inflection point(s). (g) Use the information above to sketch the graph of f. Clearly label all the important points on the graph.

Answers:

  1. (a) 1/4 (b) 2 (c) 3 (d) ∞
  2. 4
  3. (^) (x −+ 1)^12
  4. f ′(x) = sin 2 √^ xx +

x cos x

  1. x = 2 and x = − 1 / 2
  2. f ′′(x) = 4 − 4 sin(2x)
  3. sin^ x^ −^3 y

(^2) − sec (^2) x 6 xy − 1

  1. 3/ 5
  2. y′^ = xsin^ x(cos x ln x + sinx^ x)
  3. y = − 2 x + 3
  4. (^) (x (^2) + 1)^23 / 2
  5. 10.4 feet per second
  6. 4000 cm^3
  7. Absolute maximum value is −1, absolute minimum value is −5.
  8. (a)^2 x

2 / 3 3 −^

2 x^1 /^2

x^3 3 +^ C^ (b)

(x^2 + 3)^5 10 +^ C

  1. (a) R{− 1 } (b) x-intercept (1, 0), y-intercept (0, −1) (c)vertical asymptote: x = −1, horizontal asymptote: y = 1 on both sides (d) increasing on (−∞, −1) ∪ (− 1 , ∞) (e) no extrema (f) concave up on (−∞, −1) ∪ (0, 3

1 /2), concade down on (− 1 , 0) ∪ ( 3