Page 1 | © 2011 by Janice L. Epstein 4.1 Exponentials Page 2 | © 2011 by Janice L. Epstein Exponential Functions and Their Derivatives
(Section 4.1)
EXAMPLE 1
Find the following limits:
x
æp ö
÷
ç
(a) lim ç ÷÷
x-¥ ç
è4ø
f ( x) = a x , a > 0, a ¹ 1 is called an exponential function
If a > 1, then this is exponential growth function where
8
y
7
 The domain is (-¥, ¥)
 The range is (0, ¥)
 lim a x = ¥
3
6
2x
x
æ 2p ö
(b) lim çç ÷÷÷
x¥ ç
è7ø
x
5
x
4
3
2
x¥
1
 lim a x = 0
x
x-¥
-2
-1
1
2
æ 1 ö 2-x
(c) lim+ çç ÷÷÷
x 2 ç
è 4ø
3
-1
2
(d) lim- 5 x-1
If 0 < a < 1, then this is exponential decay function where
8
0.25x
x1
y
7
 The domain is (-¥, ¥)
 The range is (0, ¥)
 lim a x = 0
6
5
0.5
x
4
(e) lim+ 5
x1
3
x¥
2
x-1
2
 lim a x = ¥
1
x
x-¥
-3
-2
-1
1
-1
2
p x - p -3 x
x¥ p 3 x + p -3 x
(f) lim
Properties to remember: Given that a and b are positive,
a x+ y = a x a y
(a x )
y
= a xy
(ab) = a xb x
x
2- x + 2 x
x-¥ 4- x + 3 x
(g) lim
4.1 Exponentials Page 3 | © 2011 by Janice L. Epstein 4.1 Exponentials Page 4 | © 2011 by Janice L. Epstein EXAMPLE 2
Given f ( x) = 2 x and g ( x ) = 3x , estimate f ¢(0) and g ¢(0) using a
table of values.
EXAMPLE 3
Find the following derivatives
(a) y = e x + x
(b) f ( x ) = e-5 x cos(3 x )
(c) g ( x ) = e x sin x
e x +h - e x
=1
h0
h
Definition: e is the number such that lim
Therefore:
d x
d g ( x)
e = e x and so
e = g ¢ ( x )e g ( x )
dx
dx
4.1 Exponentials Page 5 | © 2011 by Janice L. Epstein 4.1 Exponentials Page 6 | © 2011 by Janice L. Epstein 4.1 Exponentials EXAMPLE 4
Find the equation of the tangent line to the graph of 2e xy = x + y at
the point (0, 2)
EXAMPLE 6
Find the equation of the tangent line to the parametric curve
x = e-t , y = te 2t at t=0.
EXAMPLE 5
For what value(s) of r does y = e rx satisfy the differential equation
y + y ¢ = y ¢¢ ?
EXAMPLE 7
Given y = g (e x ) + e g (sin x ) , find y ¢ .