Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 2: Marked Homework Assignment
Due: Thu 2010 Sep 23 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Let
f (x) =


Ax2 + Bx + C if −∞ < x ≤ 0

x3/2 cos(1/x)
if 0 < x < ∞,
and determine all values of the constants q
A, B and C such that f is continuous (i.e. is
√
3/2
continuous on (−∞, ∞)). (Note: x = (x3 ) = x x for x > 0; remember to justify
your answer, don’t just give the answer!)
2. Prove that tan x = −x has a solution between 4.72 and 6.28. Hint. First sketch the
graphs of y = tan x and y = −x, then find an appropriate function and apply the
Intermediate Value Theorem to it. You may use a calculator.
3. Find the limit, or state that the limit does not exist. If you find the limit, show
intermediate steps (explicit detailed justification in terms of limit laws is not required).
If you state that the limit does not exist, give justification.
√
(a) limx→∞
2x2 +5x+1
4x+7
3
5
(b) limx→∞ (x − x )
(c) limt→∞ cos(t)
(d) limt→∞ cos(1/t)
(e) limx→∞
√1
x− x2 −4x
(f) limx→−∞
√1
x+ x2 −4x
4. Use the definition of the derivative to find f 0 (5), if f (x) =
will be given for using only differentiation formulas.
√1
x−1
for x > 1. No marks
5. Let f (x) = x|x|, and use the definition of the derivative to determine whether f 0 (0)
exists. If it exists, find its value. Sketch the graph of y = f (x). No marks will be given
for using only differentiation formulas.
6. Let f be the same function defined in Question 1, and use the definition of the derivative
to determine all possible values of the constants A, B and C such that f 0 (0) exists.
No marks will be given for using only differentiation formulas (i.e. f 0 (x) = 2Ax + B if
−∞ < x ≤ 0, etc.).