Download Sec 3.1 – Exponential Functions

Sec 3.1 – Exponential Functions
DEFINITION:
An exponential function f is given by f (x)  a x ,
where x is any real number, a > 0, and a ≠ 1. The number a is called
the base.
x
Examples:
x
1
f  x   2 x , f  x     , f  x    0.4 
2
Examples of problems involving the use of exponential functions:
growth or decay,
compound interest,
the statistical "bell curve,“
the shape of a hanging cable (or the Gateway Arch in St. Louis),
some problems of probability,
some counting problems,
the study of the distribution of prime numbers.
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Sec 3.1 – Exponential Functions
−𝑥
𝑦=2
2012 Pearson Education, Inc. All rights reserved
𝑦 = 𝑒𝑥
𝑦 = 2𝑥
Sec 3.1 – Exponential Functions
DEFINITION:
e  lim 1 h   2.718281828459
1h
h0
e is called the natural base.
THEOREM 1
The derivative of the function f given by f  x   e x is:
d x
f   x   f  x  , or
e  ex
dx
2012 Pearson Education, Inc. All rights reserved
Sec 3.1 – Exponential Functions
THEOREM 2
d f (x)
f (x)
e
 e  f (x)
dx
d u
u du
e e 
dx
dx
The derivative of e to some power is the product of e to that power and
the derivative of the power.
2012 Pearson Education, Inc. All rights reserved
Sec 3.1 – Exponential Functions
Find the derivatives:
𝑦=
2 +3𝑥
2𝑥
𝑒
𝑑𝑦 =
𝑦=
𝑑𝑦
2 +3𝑥
2𝑥
𝑒
4𝑥 + 3 𝑑𝑥
5
4
𝑥
𝑥 𝑒
5
= 𝑥 4 𝑒 𝑥 5𝑥 4 𝑑𝑥
𝑑𝑦 =
5
8
𝑥
5𝑥 𝑒
+4𝑥 3
5
3
𝑥
+ 4𝑥 𝑒
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5
𝑥
𝑒 𝑑𝑥
𝑑𝑥
Sec 3.1 – Exponential Functions
Find the derivatives:
𝑥 2 + 3𝑒 𝑥
𝑦=
2𝑒 𝑥 − 𝑥
𝑑𝑦 =
2𝑒 𝑥 − 𝑥 2𝑥 + 3𝑒 𝑥 𝑑𝑥 − 𝑥 2 + 3𝑒 𝑥 2𝑒 𝑥 − 1 𝑑𝑥
2𝑒 𝑥 − 𝑥
2
4𝑥𝑒 𝑥 + 6𝑒 2𝑥 − 2𝑥 2 − 3𝑥𝑒 𝑥 − 2𝑥 2 𝑒 𝑥 + 𝑥 2 − 6𝑒 2𝑥 + 3𝑒 𝑥
𝑑𝑦 =
𝑑𝑥
𝑥
2
2𝑒 − 𝑥
𝑥𝑒 𝑥 − 𝑥 2 − 2𝑥 2 𝑒 𝑥 + 3𝑒 𝑥
𝑑𝑦 =
𝑑𝑥
𝑥
2
2𝑒 − 𝑥
2012 Pearson Education, Inc. All rights reserved
Sec 3.1 – Exponential Functions
Find the critical values, intervals of inc./dec., points of inflection, and
the intervals of concavity.
𝑑𝑦
𝑦 = 𝑒 −2𝑥
= −2𝑒 −2𝑥
𝑑𝑥
𝑑𝑦
−2𝑥
−2
= 𝑒
𝑑2 𝑦
𝑑𝑥
−2𝑥 (−2)
=
−2𝑒
−2𝑥
0 = −2𝑒
𝑑𝑥 2
−2
4
𝑑2𝑦
−2𝑥
0 = 2𝑥
0 = 2𝑥
= 4𝑒
2
𝑒
𝑒
𝑑𝑥
𝑑𝑦
𝑑𝑦
2𝑦
2𝑦
𝑑
𝑑
≠0
≠ 𝑢𝑛𝑑
≠ 𝑢𝑛𝑑.
𝑑𝑥
≠0
𝑑𝑥
2
2
𝑑𝑥
𝑑𝑥
𝑑𝑦
𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑑2 𝑦
𝑑𝑥
𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
2
𝑑𝑥
∴ 𝑑𝑒𝑐: −∞, ∞
∴ 𝐶𝑈: −∞, ∞
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Sec 3.1 – Exponential Functions
𝑦 = 𝑒 −2𝑥
𝑑𝑒𝑐: −∞, ∞
𝐶𝑈: −∞, ∞
2012 Pearson Education, Inc. All rights reserved
Sec 3.1 – Exponential Functions
Find the equation of the line tangent to the graph of 𝑓 𝑥 = 𝑒 2𝑥 at the
point (0, 1).
𝑓(𝑥) = 𝑒 2𝑥
𝑓′(𝑥) = 𝑒 2𝑥 2
𝑓′(𝑥) = 2𝑒 2𝑥
𝑓′(0) = 2𝑒 2(0)
𝑓′ 0 = 2
𝑦 − 1 = 2(𝑥 − 0)
𝑦 − 1 = 2𝑥
𝑦 = 2𝑥 + 1
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Sec 3.1 – Exponential Functions
The amount of money spent (in billions of dollars) by Americans on
organic food and beverages since 1995 can be approximated by:
𝐴 𝑡 = 2.43𝑒 0.18𝑡 .
The variable t represents the number of year after 1995.
a) How much was spent in 2009?
b) Approximate the rate of growth in 2006.
𝐴 𝑡 = 2.43𝑒 0.18𝑡
𝐴 𝑡 = 2.43𝑒 0.18𝑡
𝑡 = 2009 − 1995 = 14
𝐴′ 𝑡 = 2.43𝑒 0.18𝑡 (0.18)
𝐴 14 = 2.43𝑒 0.18(14)
𝐴′ 𝑡 = 0.4374𝑒 0.18𝑡
𝐴 14 = $30.2 𝑏𝑖𝑙𝑙𝑖𝑜𝑛
𝑡 = 2006 − 1995 = 11
𝐴′ 11 = 0.4374𝑒 0.18(11)
𝐴′
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11
= $3.17𝑏𝑖𝑙𝑙𝑖𝑜𝑛/𝑦𝑟
Sec 3.1 – Exponential Functions
2012 Pearson Education, Inc. All rights reserved