Download §7.3* –The Natural Exponential Function

§7.3* –The Natural Exponential Function
Mark Woodard
Furman U
Fall 2010
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
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Outline
1
The definition of the exponential function
2
exp is an exponential function
3
The derivative of e x
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§7.3* –The Natural Exponential Function
Fall 2010
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The definition of the exponential function
Definition
ln, the natural logarithm, is an increasing function with domain (0, ∞) and
range R. Let exp denote the inverse of ln; thus, exp has domain R and
range (0, ∞).
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§7.3* –The Natural Exponential Function
Fall 2010
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The definition of the exponential function
Definition
ln, the natural logarithm, is an increasing function with domain (0, ∞) and
range R. Let exp denote the inverse of ln; thus, exp has domain R and
range (0, ∞).
Elementary properties of exp
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
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The definition of the exponential function
Definition
ln, the natural logarithm, is an increasing function with domain (0, ∞) and
range R. Let exp denote the inverse of ln; thus, exp has domain R and
range (0, ∞).
Elementary properties of exp
The graph of exp can be obtained directly from the graph of ln.
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
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The definition of the exponential function
Definition
ln, the natural logarithm, is an increasing function with domain (0, ∞) and
range R. Let exp denote the inverse of ln; thus, exp has domain R and
range (0, ∞).
Elementary properties of exp
The graph of exp can be obtained directly from the graph of ln.
We have the important inverse relationships:
ln(exp(x)) = x
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and
exp(ln(x)) = x.
§7.3* –The Natural Exponential Function
Fall 2010
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exp is an exponential function
Euler’s number, e
Recall the definition of e : exp is the number such that ln(e) = 1.
(e ≈ 2.71828)
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
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exp is an exponential function
Euler’s number, e
Recall the definition of e : exp is the number such that ln(e) = 1.
(e ≈ 2.71828)
Theorem
exp(r ) = e r for all rational numbers r .
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
4/7
exp is an exponential function
Euler’s number, e
Recall the definition of e : exp is the number such that ln(e) = 1.
(e ≈ 2.71828)
Theorem
exp(r ) = e r for all rational numbers r .
Definition
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
4/7
exp is an exponential function
Euler’s number, e
Recall the definition of e : exp is the number such that ln(e) = 1.
(e ≈ 2.71828)
Theorem
exp(r ) = e r for all rational numbers r .
Definition
Because exp(r ) = e r for all rational numbers, we will define e x by
exp(x) for all real numbers x.
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
4/7
exp is an exponential function
Euler’s number, e
Recall the definition of e : exp is the number such that ln(e) = 1.
(e ≈ 2.71828)
Theorem
exp(r ) = e r for all rational numbers r .
Definition
Because exp(r ) = e r for all rational numbers, we will define e x by
exp(x) for all real numbers x.
In particular,
ln e x = x
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and e ln(x) = x.
§7.3* –The Natural Exponential Function
Fall 2010
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exp is an exponential function
Theorem (The laws of exponents)
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
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exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
5/7
exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
e x−y = e x /e y (for homework)
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
5/7
exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
e x−y = e x /e y (for homework)
r
e x = e rx (for homework)
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
5/7
exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
e x−y = e x /e y (for homework)
r
e x = e rx (for homework)
Problem
Solve the following equations:
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
5/7
exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
e x−y = e x /e y (for homework)
r
e x = e rx (for homework)
Problem
Solve the following equations:
e 2x−3 = 8
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
5/7
exp is an exponential function
Theorem (The laws of exponents)
e x+y = e x e y
e x−y = e x /e y (for homework)
r
e x = e rx (for homework)
Problem
Solve the following equations:
e 2x−3 = 8
e 2x + 2e x − 8 = 0
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§7.3* –The Natural Exponential Function
Fall 2010
5/7
The derivative of e x
Theorem
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§7.3* –The Natural Exponential Function
Fall 2010
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The derivative of e x
Theorem
d x
e = ex
dx
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Theorem
d x
e = ex
dx
Z
e x dx = e x + C .
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Theorem
d x
e = ex
dx
Z
e x dx = e x + C .
Problem
Find y 0 in each case:
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Theorem
d x
e = ex
dx
Z
e x dx = e x + C .
Problem
Find y 0 in each case:
y = ex
2
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Theorem
d x
e = ex
dx
Z
e x dx = e x + C .
Problem
Find y 0 in each case:
y = ex
2
3
y = x exp
x +1
x +2
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Theorem
d x
e = ex
dx
Z
e x dx = e x + C .
Problem
Find y 0 in each case:
y = ex
2
3
y = x exp
y=
x +1
x +2
e x − e −x
e x + e −x
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
6/7
The derivative of e x
Problem
Solve the following integrals:
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
7/7
The derivative of e x
Problem
Solve the following integrals:
Z
ex
dx
ex + 1
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
7/7
The derivative of e x
Problem
Solve the following integrals:
Z
ex
dx
ex + 1
Z
2
xe x dx
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
7/7
The derivative of e x
Problem
Solve the following integrals:
Z
ex
dx
ex + 1
Z
2
xe x dx
Z 3
e x+5 dx
0
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
7/7
The derivative of e x
Problem
Solve the following integrals:
Z
ex
dx
ex + 1
Z
2
xe x dx
Z 3
e x+5 dx
0
Problem
Sketch the graph of y = xe −x .
Mark Woodard (Furman U)
§7.3* –The Natural Exponential Function
Fall 2010
7/7