MATH 110, Assignment 2
Make sure you justify all your work and include complete arguments and explanations. Please include your name and student number on the top of the first
page of your assignment.
Problem 1. Let f (x) =
x
.
1+x
(i) Find the rates of change of f (x) in the following intervals:
[0.5, 1],
[0.9, 1],
[0.99, 1],
[1, 1.01] [1, 1.1],
[1, 1.5].
(You can use calculator for this equation.)
(ii) Using the result of (i), guess the rate of change of f (x) at x = 1.
(iii) Sketch the graph of f (x) and interpret the geometric meanings of the result of (i) and
(ii) by describing on the graph.
(iv) Express the rate of change of f (x) by using limit and find the limit.
Problem 2. Find each limit, or explain why it does not exist(ex. the limit is infinity.).
(a) limx→3 (2x + |x − 3|).
(b) limx→2
x2 −x−2
.
x2 −4x+4
Problem 3. Is there a number a such that
3x2 + ax + a + 3
x→−2
x2 + 3x + 2
lim
exists? If so, find the value of a and the value of the limit.
Problem 4. Sketch the graph of a function f (x) defined on [−5, 5] which satisfies the
following conditions:
• f (−5) = − 51 and f (5) = −15.
• limx→0− f (x) = ∞.
• The x-intercepts of f (x) are 0 and 2.
• The rates of change of f (x) are −1 and 0 at -1 and 1, respectively.
1
• f is continuous on the domain except at x = 0.
Problem 5. For what value of the constant c is the function f continuous on (−∞, ∞)?
(
cx2 + 2x x < 2
f (x) =
x3 − c|x| x ≥ 2.
Problem 6. Prove that there is a root of x2 =
2
√
x + 1 in (1, 2).