Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 5: Marked Homework Assignment
Due: Thu 2010 Oct 14 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Find an equation of the tangent line to the curve
x
y
+
3
y
x
= 2 at (−1, −1).
2. Find the slope of the curve yex−1 + xey = 1 at (1, 0).
3. Find the derivatives of the following functions, simplifying where possible:
(a) x = cos−1 (sin−1 t)
√
(b) ln(x2 + x3 + x)
(c) f (x) = xarctan x
(d) y = (cos x)sin x
4. If f (x) = ln(x/2), find f (101) (2).
5. A function ψ(x) defined on 0 ≤ x ≤ 1 is required to satisfy the differential equation
(an equation involving an unknown function and its derivative(s))
ψ 00 = −
4π 2
ψ,
λ2
where λ is a positive constant, and ψ(x) must also satisfy the two conditions
ψ(0) = 0,
ψ(1) + ψ 0 (1) = 0.
x satisfies the differential equation, and
(a) Show that the function ψ(x) = C sin 2π
λ
it also satisfies the first condition ψ(0) = 0, for any value of constant C.
x also satisfies the second
(b) Do there exist values of λ such that ψ(x) = C sin 2π
λ
0
condition ψ(1) + ψ (1) = 0 for a nonzero value of C? Explain carefully. (Hint:
Show that ψ(1) + ψ 0 (1) = 0 reduces to tan X = −X, where X = 2π
is positive.
λ