Note: You may consult your classmates or other resources (including tutors) for ideas on
the problems; however, the solutions you turn in must be in your own words and must reflect
your own understanding. Your solutions and write-ups will be checked for textual similarities.
You may not copy from, reword, or paraphrase another student’s work or any other resource
material; such conduct will be treated as a violation of academic integrity. Remember that you
will not learn anything by simply copying, rewording or paraphrasing another person’s work.
You will receive no credit for solely writing the final answer when explanation is necessary. The
below homework specifications will be enforced. If the specifications are not respected, points
might be deducted, or the homework assignment may not be accepted for grading.
Each homework assignment that you upload and submit must:
• be legible, with each problem clearly labeled. Only one problem should appear per page;
for instance, parts (a), (b), and (c) of the same problem can be on the same page, but
Problems 1 and 2 should be placed on separate pages.
• be properly oriented;
• contain only your final version (write drafts of homework solutions on scratch paper);
• not have anything crossed out or contain notes in the margins;
• have proofs and solutions in which all steps are clearly shown and explained.
• have grammatically correct (including punctuation and spelling) complete sentences;
• be written using mathematical terminology and notation correctly;
• have final answers in exact forms (do not approximate unless otherwise stated);
• have a box or circle around your final answer for each question.
• be uploaded in a PDF file on Gradescope with pages matched to the right exercises.
Unless stated otherwise, these problems are intended to be solved without a calculator. While
you are free to use a calculator while completing this assignment, please note that calculators
are not allowed during exams. It may be beneficial to practice solving the problems without
one.MATH-UA 121:
Calculus I HW 1
Written Homework 1
1. Find the domain of the function f (x) =
pp
4 − x −
p
3 + x.
2. In each of the following, describe the elementary transformations required to transform
f (x) into g(x). Then, sketch both f (x) and g(x) on the same axes.
For example, to get −sin(2x) + 1 from sin(x) we take the graph of sin(x) and:
• Reflect the graph with respect to x axis.
• Shift it up by 1 unit.
• Shrink it horizontally by a factor of 2.
(a) f (x) = −x
2
, g(x) = 3(x − 1)
2
(b) f (x) = |x|, g(x) = |
x
2 + 1|
(c) f (x) =
p
x, g(x) = 4 − 3
p
−x
3. Sketch the graph of an example of an even function f that is defined on the interval [−3, 3]
and satisfies all of the given conditions:
(a) lim
x→0+
f (x) = 1
(b) lim
x→1−
f (x) = −1
(c) lim
x→1+
f (x) = −2
(d) limx→−2
f (x) = f (−2)
(e) f (0) = 0
(f) f (2) = 2
y
x
-4 -3 -2 -1 1 2 3 4
-1
-2
-3
-4
1
2
3
4
4. The graphs of the functions g(x) and h(x) are shown below. Determine the following limits.
If a limit does not exist, write "DNE" and explain why.
1 of 2MATH-UA 121:
Calculus I HW 1
g(x)
-3 -2 -1 1 2 3
-1
1
2
3
4
-3 -2 -1 1 2 3
-2
-1
1
2
3
h(x)
-3 -2 -1 1 2 3
-1
1
2
3
4
-3 -2 -1 1 2 3
-2
-1
1
2
3
(a) limx→−2
h(x)
(b) limx→−2
g(x)
(c) limx→1
g(h(x))
(d) limx→−2
h(g(x))
5. Calculate the following limits. If the limit does not exist, write "DNE".
Note: Do not use advanced techniques such as L’Hôpital’s Rule for this problem.
(a) limx→2
x
2 − 4x + 4
x
3 − 2x
2 + 2x − 4
(b) limx→9
3 −
p
x
9x − x
2
(c) limx→0
1
x
−
1
x + x
2
(d) limx→0
1 − cos x
x
2
(e) limx→0
tan x − sin x
x
3
(f) limx→2
x − 2
p
x
2 + 5 − 3
(g) limx→0
sin|x|
x
6. Use the Squeeze Theorem to show that limx→0
x
4
sin
1
x
3
= 0
2 of 2
