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These properties are verified just by exploiting the fact that the exponential is the inverse of the logarithm, as we saw in Example 67
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[exp( a) ]2 [exp( b) ]3 [exp( c) ]4
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Use the basic properties to simplify the expression
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We calculate that [exp( a) ] [exp( a) ] [exp( b) ] [exp( b) ] [exp( b) ] [exp( a) ]2 [exp( b) ]3 = 4 [exp( c) ] [exp( c) ] [exp( c) ] [exp( c) ] [exp( c) ] = exp( a + a + b + b + b) = exp( a + a + b + b + b c c c c) exp( c + c + c + c)
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= exp( 2a + 3b 4c)
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YOU TRY IT Simplify the expression ( exp a) 3 ( exp b) 2 /( exp c) 5
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6 T R A N S C E N D E N T A L F U N C T I O N S
622 Calculus Properties of the Exponential
Now we want to learn some calculus properties of our new function exp(x) These are derived from the standard formula for the derivative of an inverse, as in Section 251 For all x we have d (exp(x)) = exp(x) dx In other words, exp(x) dx = exp(x) More generally, du d exp(u) = exp(u) dx dx and exp(u) du dx = exp(u) + C dx
We note for the record that the exponential function is the only function (up to constant multiples) that is its own derivative This fact will come up later in our applications of the exponential
EXAMPLE
Compute the derivatives: d d d exp( 4x) , ( exp( cos x) ) , ( [exp( x) ] [cot x]) dx dx dx
SOLUTION
For the first problem, notice that u = 4x hence du/dx = 4 Therefore we have d d exp( 4x) = [exp( 4x) ] ( 4x) = 4 exp( 4x) dx dx
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Similarly, d ( exp( cos x) ) = [exp( cos x) ] dx d cos x = [exp( cos x) ] ( sin x) , dx d cot x dx
d d ( [exp( x) ] [cot x]) = exp( x) ( cot x) + [exp( x) ] dx dx
= [exp( x) ] ( cot x) + [exp( x) ] ( csc2 x)
YOU TRY IT Calculate ( d/dx) exp( x sin x)
EXAMPLE
exp( 5x) dx, [exp( x) ]3 dx, exp( 2x + 7) dx
Calculate the integrals:
SOLUTION
We have exp( 5x) dx = [exp( x) ]3 dx = = 1 2 1 exp( 5x) + C 5 [exp( x) ] [exp( x) ] [exp( x) ] dx exp( 3x) dx = 1 exp( 3x) + C 3 1 exp( 2x + 7) + C 2
exp( 2x + 7) dx =
exp( 2x + 7) 2 dx =
EXAMPLE
[exp( cos3 x) ] sin x cos2 x dx
Evaluate the integral
6 T R A N S C E N D E N T A L F U N C T I O N S
SOLUTION
For clarity, we let ( x) = exp( cos3 x) , ( x) = 3 cos2 x ( sin x) Then the integral becomes 1 3 exp( ( x) ) ( x) dx = 1 exp( ( x) ) + C 3
Resubstituting the expression for ( x) gives [exp( cos3 x) ] sin x cos2 x dx = 1 exp( cos3 x) + C 3
EXAMPLE
exp( x) + exp( x) dx exp( x) exp( x)
Evaluate the integral
SOLUTION
For clarity, we set ( x) = exp( x) exp( x) , ( x) = exp( x) + exp( x) Then our integral becomes ( x) dx = ln | ( x) | + C ( x) Resubstituting the expression for ( x) gives exp( x) + exp( x) dx = ln exp( x) exp( x) + C exp( x) exp( x)
YOU TRY IT Calculate x exp( x2 3) dx
623 The Number e
The number exp(1) is a special constant which arises in many mathematical and physical contexts It is denoted by the symbol e in honor of the Swiss mathematician Leonhard Euler (1707--1783) who first studied this constant We next see how to calculate the decimal expansion for the number e
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In fact, as can be proved in a more advanced course, Euler s constant e satisfies the identity 1 1+ n
n +
= e
[Refer to the You Try It following Example 59 in Subsection 523 for a consideration of this limit] This formula tells us that, for large values of n, the expression 1+ 1 n
gives a good approximation to the value of e Use your calculator or computer to check that the following calculations are correct: n = 10 n = 50 n = 100 n = 1000 n = 10000000 1+ 1+ 1+ 1+ 1+
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