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Math 115 Exam 2: Limits and Functions - Prof. Jennifer R. Mcneilly, Exams of Mathematics

University of Illinois - Urbana-ChampaignMathematics
Prof. Jennifer R. Mcneilly

The instructions and problems for exam 2 of math 115, focusing on limits and functions. Students are required to remove hats, place book bags inaccessibly, and turn off cell phones and calculators during the exam. The exam consists of several limit problems, where students are asked to determine the function, evaluate the limit using a sequence, and identify the property that allows arbitrary sequence choice. The document also includes problems involving function graphs, inverse functions, and solving equations.

Typology: Exams

2010/2011

Uploaded on 06/28/2011

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Math 115 – Exam 2 – 12PM V1 Name:
April 7, 2010
50 points possible
1. No hats or dark sunglasses. All hats are to be removed.
2. All book bags are to be closed and placed in a way that makes them inaccessible. Do
not reach into your bag for anything during the exam. If you need extra pencils, pull
them out now.
3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during
the exam, it will be construed as cheating and you will be asked to leave. This includes
using it as a time-piece.
4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell
phones.
5. If you have a question, raise your hand and a proctor will come to you. Once you stand
up, you are done with the exam. If you have to use the facilities, do so now. You will
not be permitted to leave the room and return during the exam.
6. Every exam is worth a total of 50 points. Check to see that you have all of the pages.
Including the cover sheet, each exam has 9 pages.
7. Be sure to print your proper name clearly.
8. If you finish early, quietly and respectfully get up and hand in your exam. You need
to show your student ID when you hand in the exam. (Drivers license, passport, etc.
will work also.) No exam will be accepted without ID.
9. When time is up, you will be instructed to put down your writing utensil, close the
exam and remain seated. Anyone seen continuing to write after this announcement
will have their exam marked and lose all points on the page they are writing on. We
will come and collect the exams from you. Have your ID ready.
10. Good luck. You have fifty minutes to complete the exam.
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Download Math 115 Exam 2: Limits and Functions - Prof. Jennifer R. Mcneilly and more Exams Mathematics in PDF only on Docsity!

Math 115 – Exam 2 – 12PM V1 Name:

April 7, 2010

50 points possible

  1. No hats or dark sunglasses. All hats are to be removed.
  2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now.
  3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time-piece.
  4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell phones.
  5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam.
  6. Every exam is worth a total of 50 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 9 pages.
  7. Be sure to print your proper name clearly.
  8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID.
  9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready.
  10. Good luck. You have fifty minutes to complete the exam.
  1. Consider the limit (^) x→−lim 3 +^ x x^ −+ 3^3. (a) (1pt) State the function f (x) associated with this limit and determine the function’s domain.

(b) (2pts) Determine a sequence a(n) that can be used to evaluate this limit.

(c) (3pts) Use your sequence in (b) to evaluate the limit.

(d) (1pt) What property of the function f (x) allows an arbitrary choice of the sequence a(n) to evaluate this limit?

  1. For the following limits:
    1. Determine the limit’s indeterminate form and state that form.
    2. Using any method, evaluate the limit. (You are not required to appeal to a sequence to evaluate the limit.)

(a) (3pts) lim x→ 2 2 x

3 x^2 − 4 x − 4

(b) (3pts) (^) x→−∞lim^2 −^ x^ +^ x

4 x^3 − 5 x + 2

  1. (Continued) For the following limits:
    1. Determine the limit’s indeterminate form and state that form.
    2. Using any method, evaluate the limit. (You are not required to appeal to a sequence to evaluate the limit.)

(c) (3pts) lim h→ 0

√1 + h − 1 h

(d) (3pts) (^) xlim→ 2 −

x − 2 −^

x^2 − 4

(a) (2pts) Define what it means for two functions f (x) and g(x) to be inverse functions.

(b) (3pts) Show that the function f (x) := − x^2 x + 2^ −^3 is its own inverse.

  1. Consider the following graph of a function f (x):
    • 4 - 2 2 4 x
      • 4
      • 2

2

4^ y

(a) (2pts) State the domain and the range of f (x).

(b) (2pts) Explain why f (x) is not one-to-one.

(c) (2pts) Find an interval on which f is one-to-one.

(d) (2pts) On the graph above, sketch the inverse of f corresponding to the restricted domain you chose in (c).

Math 115 – Exam 2 – 12PM V2 Name:

April 7, 2010

50 points possible

  1. No hats or dark sunglasses. All hats are to be removed.
  2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now.
  3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time-piece.
  4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell phones.
  5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam.
  6. Every exam is worth a total of 50 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 9 pages.
  7. Be sure to print your proper name clearly.
  8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID.
  9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready.
  10. Good luck. You have fifty minutes to complete the exam.
  1. Consider the limit (^) x→−lim 3 −^ x x^ −+ 3^3. (a) (1pt) State the function f (x) associated with this limit and determine the function’s domain.

(b) (2pts) Determine a sequence a(n) that can be used to evaluate this limit.

(c) (3pts) Use your sequence in (b) to evaluate the limit.

(d) (1pt) What property of the function f (x) allows an arbitrary choice of the sequence a(n) to evaluate this limit?

  1. For the following limits:
    1. Determine the limit’s indeterminate form and state that form.
    2. Using any method, evaluate the limit. (You are not required to appeal to a sequence to evaluate the limit.)

(a) (3pts) lim x→ 23 x

(^2) − 4 x − 4 2 x^2 − 8

(b) (3pts) (^) x→−∞lim^2 −^ x^ +^ x

2 x^3 − 5 x + 2

  1. (Continued) For the following limits:
    1. Determine the limit’s indeterminate form and state that form.
    2. Using any method, evaluate the limit. (You are not required to appeal to a sequence to evaluate the limit.)

(c) (3pts) lim h→ 0

√9 + h − 3 h

(d) (3pts) (^) xlim→ 2 +

x − 2 −^

x^2 − 4

(a) (2pts) Define what it means for two functions f (x) and g(x) to be inverse functions.

(b) (3pts) Show that the function f (x) := − x^2 x + 2^ −^3 is its own inverse.

  1. Consider the following graph of a function f (x):
    • 4 - 2 2 4 x
      • 4
      • 2

2

4^ y

(a) (2pts) State the domain and the range of f (x).

(b) (2pts) Explain why f (x) is not one-to-one.

(c) (2pts) Find an interval on which f is one-to-one.

(d) (2pts) On the graph above, sketch the inverse of f corresponding to the restricted domain you chose in (c).