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Homework 6 Calculus 1
1. The function ex has two identical definitions (shown in class):
ex =
∞
X
k=0
xk
k! = lim
n→∞
1 + x
n
n
.
Without using differentiation, do the followings:
(a) Show the series above converges absolutely for all x.
(b) Using either definition, show that for all x, y ∈ R, ex+y = exey. (c) Show that for all x ∈ R, e−x= e1x, and therefore ex−y = eexy. (d) Show that ex > 0 for all x ∈ R.
(e) Show that for all x, y ∈ R, (ex)y = exy. (Note that you can’t raise 1 + xn above to the power ny, since that is what we are proving here.
(f) Show that ex is strictly increasing. That is, x > y ⇒ ex > ey. (It might be useful to first show that ex > 1 ∀x > 0.)
Note that you are NOT allowed to use any ”power rule” you have learnt before. You need to derive them.
2. Salas 12.5: 10, 18, 20, 35, 42.
3. Salas 12.8: 8, 20, 30, 35.
