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

Document file for examinations, Cheat Sheet of Mathematics

Mathematics

67676767666767676767676767676767

Typology: Cheat Sheet

2025/2026

Uploaded on 12/14/2025

ly-sokun-virakbot
ly-sokun-virakbot 🇰🇭

1 document

1 / 10

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Calculus I (UN1101) Practice Final A Instructor: Robin Zhang
Student Name (UNI):
Instructions:
This exam contains 10 pages (including this cover page) and 6 questions. The total number of possible
points is 80 points. You will have 150 minutes to complete this exam.
•Print your name and UNI in the space above.
•Answer the questions in the space provided on
the question sheets. You may use extra paper.
•Clearly identify and simplify your answers.
You will not receive full credit if there are multiple
apparent answers, even if one of them is correct.
•Write legibly and show your work. You may
receive partial credit for intermediate steps. Correct
answers without any reasoning or work will not re-
ceive full credit.
•No calculators, computational devices, or con-
sulting other people during the duration of this
exam. Any cheating will result in an automatic fail-
ing grade in the course and potential administrative
action.
•You may consult your notes and textbook
for this exam. This does not include WebAssign,
Courseworks, or other online resources.
•Upload your exam to Gradescope via PDF or
images at the end of the time allotted. At the end of
the exam, you have 15 minutes to upload your exam.
Any exams uploaded after the end of the upload pe-
riod will not be accepted.
•Remain in the Zoom call with your camera.
If you need to leave your work area for any reason,
please inform the instructor beforehand.
Do not write in the table to the right.
Question Points Score
1 12
2 16
3 21
4 9
5 16
6 6
Total: 80
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Document file for examinations and more Cheat Sheet Mathematics in PDF only on Docsity!

Calculus I (UN1101) Practice Final A Instructor: Robin Zhang

Student Name (UNI):

Instructions:

This exam contains 10 pages (including this cover page) and 6 questions. The total number of possible points is 80 points. You will have 150 minutes to complete this exam.

  • Print your name and UNI in the space above.
  • Answer the questions in the space provided on the question sheets. You may use extra paper.
  • Clearly identify and simplify your answers. You will not receive full credit if there are multiple apparent answers, even if one of them is correct.
  • Write legibly and show your work. You may receive partial credit for intermediate steps. Correct answers without any reasoning or work will not re- ceive full credit.
  • No calculators, computational devices, or con- sulting other people during the duration of this exam. Any cheating will result in an automatic fail- ing grade in the course and potential administrative action.
  • You may consult your notes and textbook for this exam. This does not include WebAssign, Courseworks, or other online resources.
  • Upload your exam to Gradescope via PDF or images at the end of the time allotted. At the end of the exam, you have 15 minutes to upload your exam. Any exams uploaded after the end of the upload pe- riod will not be accepted.
  • Remain in the Zoom call with your camera. If you need to leave your work area for any reason, please inform the instructor beforehand.

Do not write in the table to the right.

Question Points Score

1 12

2 16

3 21

Total: 80

  1. Consider the function

f (x) =

sin(x) if x > ⇡ x ⇡ if x  ⇡

(a) (4 points) Identify the real numbers at which f (x) is discontinuous. Hint: You should justify why f (x) is discontinuous at certain values of x and why f (x) is continuous everywhere else.

(b) (4 points) Identify the horizontal and vertical asymptotes of f (x).

(c) (4 points) What does the Mean Value Theorem say about f (x) on the interval [2⇡, 3 ⇡]?

  1. Consider the function

f (x) = 2x 3 + 9x 2 + 12x + 1.

(a) (3 points) State the domain and range of the function f (x).

(b) (4 points) Find f 0 (x) and f 00 (x).

(c) (5 points) Find the local extrema of f (x).

(d) (4 points) Find all of the values of x where f (x) has an inflection point.

  1. The spread of a rumor over time within a town with a population of 12000 people can be modeled by a logistic function R(t) =

1 + e t^

The logistic function R(t) gives the number of people at time t who have heard the rumor. (a) (6 points) Find the linearization of R(t) at t = 0.

(b) (3 points) Use linear approximation at t = 0 to estimate the number of people in the town who have heard the rumor at time t = 1.

  1. Consider the curve y = x 3 + x.

(a) (5 points) Approximate the area under the curve between x = 0 and x = 4 by a Riemann sum of four rectangles using right endpoints.

(b) (4 points) Express the area under the curve as the limit of a Riemann sum. You do not need to evaluate the area for this part. You can leave your answer as a limit.

  1. (6 points) Evaluate Z 2 cos 2 (t)

s 6 +

sin(t) cos(t)

  • t

dt.