MATH-UA 121:
Calculus I HW 1
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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
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