8.7. EXERCISES 169
(a) x2y+ siny = 7,(x2 + cosy
)y′+2xy = 0.
(b) x2y3 + sin(y2)= 5, 2xy3 +
(3x2y2 +2
(cos(y2))
y)
y′ = 0.
(c) y2 sin(y)+ xy = 6,(2y(sin(y))+ y2 (cos(y))+ x
)y′+ y = 0.
3. Show that if D(g)⊆U ⊆D( f ) , and if f and g are both one to one, then f ◦g is alsoone to one.
4. The number e is that number such that lne = 1. Prove ex = exp(x) .
5. Find a formula for dydx for y = bx. Prove your formula.
6. Let y = xx for x ∈ (0,∞). Find y′ (x) .
7. The logarithm test states the following. Suppose ak ̸= 0 for large k and that p =
limk→∞
ln(
1|ak|
)lnk exists. If p > 1, then ∑
∞k=1 ak converges absolutely. If p < 1, then
the series, ∑∞k=1 ak does not converge absolutely. Prove this theorem.
8. Suppose f (x+ y) = f (x)+ f (y) and f is continuous at 0. Find all solutions to thisfunctional equation which are continuous at x = 0. Now find all solutions whichare bounded near 0. Next if you want an even more interesting version of this,find all solutions whose graphs are not dense in the plane. (A set S is dense inthe plane if for every (a,b) ∈ R×R and r > 0, there exists (x,y) ∈ S such that√(x−a)2 +(y−b)2 < r. This is called the Cauchy equation.
9. Suppose f (x+ y) = f (x) f (y) and f is continuous and not identically zero. Findall solutions to this functional equation. Hint: First show the functional equationrequires f > 0.
10. Suppose f (xy) = f (x)+ f (y) for x,y > 0. Suppose also f is continuous. Find allsolutions to this functional equation.
11. Using the Cauchy condensation test, determine the convergence of ∑∞k=2
1k lnk . Now
determine the convergence of ∑∞k=2
1k(lnk)1.001 .
12. Find the values of p for which the following series converges and the values of p forwhich it diverges.∑∞
k=41
lnp(ln(k)) ln(k)k
13. For p a positive number, determine the convergence of ∑∞n=2
lnnnp for various values
of p.
14. Determine whether the following series converge absolutely, conditionally, or not atall and give reasons for your answers.
(a) ∑∞n=1 (−1)n ln(k5)
k
(b) ∑∞n=1 (−1)n ln(k5)
k1.01
(c) ∑∞n=1 (−1)n 10n
(1.01)n
(d) ∑∞n=1 (−1)n sin
( 1n
)
(e) ∑∞n=1 (−1)n tan
(1n2
)(f) ∑
∞n=1 (−1)n cos
(1n2
)(g) ∑
∞n=1 (−1)n sin
( √n
n2+1
)