Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 3: Marked Homework Assignment
Due: Thu 2010 Sep 30 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Let f be the same function defined in Question 1 of the Week 2 Marked Homework,


Ax2 + Bx + C if −∞ < x ≤ 0
f (x) = 
x3/2 cos(1/x)
if 0 < x < ∞,
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.).
2. Vampires love the dark. Assume that in mid-summer, the sun moves up and down the
y-axis, reaching a maximum height of y = 15 around noon each day. The Transylvanian
landscape for x ≥ 0 has the elevation
y=



0
if 0 ≤ x < 6,
−x2 + 17x − 66 if 6 ≤ x ≤ 11,


0
if 11 < x < ∞.
At what points on the mountain and the nearby plain can a blood-sucking real estate
promoter truthfully promise, “Life in the Shadows, All Year Round”?
√
4
3. Find f 0 (16), if f (x) = x3 g(x), g(16) = 8 and g 0 (16) = −1.
4. Find f 00 (x), if f (x) = xex .
5. If f (x) = cot x, use the quotient rule to find f 0 (x).
6. Find all points on the curve y = x − sin x where the tangent line is horizontal.
7. Find the limit, or determine that the limit does not exist. Give justification.
sin4 2s
s→0
s4
(a) lim
tan 3(θ − π/2)
θ→π/2
θ − π/2
(b) lim
h sin h
h→0 1 − cos h
(d) lim
(c) lim
(e) lim sin(1/t)
t→∞
x→0
sin 8x
tan 4x