WORKSHEET ON THE EXPONENTIAL FUNCTION
MARCH 8, 2017
We are going to explore the world of exponential functions. Our starting
point is the basic algebraic property of exponential functions, namely that
they convert addition into multiplication. (To make the problem tractable, we
also assume some continuity and differentiability properties.)
Definition. For the purpose of this worksheet, an exponential function is
any function f from the real numbers to the real numbers, with the following
properties:
(A) For all real numbers x and y, we have f (x + y) = f (x)f (y).
(B) f is defined and continuous for all real numbers.
(C) f is not constant: there are two numbers x1 and x2 with f (x1 ) 6= f (x2 ).
(D) f is differentiable at zero.
The goal is to see what these four properties can tell us about f .
(1) Show that f (0) = 1.
(2) Show that f (x)f (−x) = 1 for all real numbers x.
(3) Show that f (x) 6= 0 for all real numbers x.
(4) Show that f (x) > 0 for all real numbers x.
(5) Show that f (−x) = 1/f (x) for all real numbers x.
(6) Show that f (2x) = (f (x))2 for all real numbers x.
(7) Using mathematical induction, show that f (nx) = (f (x))n for all real
numbers x and all natural numbers n ≥ 1.
(8) Show that f (x/n) = (f (x))1/n for all real numbers x and all natural
numbers n ≥ 1.
(9) Show that f (rx) = (f (x))r for all real numbers x and all rational
numbers r.
(10) Using the definition of the derivative, show that f is differentiable at
every real number x. Further, prove that f 0 (x) = f 0 (0)f (x).
(11) Show that, because f is not constant, we must have f 0 (0) 6= 0. (Hint:
use the mean value theorem from calculus.)
(12) Show that f has derivatives of all orders at every point.
(13) Show that if f 0 (0) > 0, then f is increasing at every point; show that
if f 0 (0) < 0, then f is decreasing at every point.
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MARCH 8, 2017
Let us write a for the real number f (1).
(14) Show that if f 0 (0) > 0, then a > 1. Similarly, show that if f 0 (0) < 0,
then a < 1.
(15) Show that f (r) = ar for every rational number r.
Since f (x) has derivatives of all orders, we can write down its Taylor series at
the origin; call it
∞
X
(∗)
cn x n .
n=0
We already know that c0 = f (0) = 1.
(16) Differentiate term by term in (∗), and then use (10), to show that
cn−1
.
cn = f 0 (0)
n
Use mathematical induction to deduce that cn = (f 0 (0))n /n! for all
n ≥ 0.
Let us write b for the real number f 0 (0); recall that b 6= 0.
(17) Show that the power series
∞
X
bn n
x
n!
n=0
converges for all real numbers x; conclude that
∞
X
bn n
x .
f (x) =
n!
n=0
(18) Show that a and b are related by the formula
∞
X
bn
a=
.
n!
n=0
When b = 1, we get the “natural” exponential function ex . This function is
defined mathematically as the convergent power series
∞
X
1 n
x
x ,
e =
n!
n=0
and e is an abbreviation for the real number
∞
X
1
e=
.
n!
n=0
(19) By adding the first few terms of the series for e, show that e ≈ 2.718.