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C H A P T E R 0EXPONENTIALS,LOGARITHMS AND OTHERTRANSCENDENTALFUNCTIONS

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Proof see p. 483-484

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Proof see p. 491

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Proof see p. 497

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For any

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In this age, we are all keenly aware of how infection by food-borne microorganisms causes disease. Certain strains of the bacterium Escherichia coli (E. coli) have become particularly notorious.

If we let y(t) represent the number of bacteria in a culture at time t, then the rate of change of the population with respect to time is y(t). Thus, since y(t) is proportional to y(t), we have

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In general, sup

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Inverse sine function:

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Inverse cosine function:

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Inverse tangent function:

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Inverse secant function:

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for -1 < x < 1.

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for -1 < x < 1.

, for | x | > 1.

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8.1 Finding the Derivative of an Inverse Trigonometric Function

Compute the derivative of (a) cos -1(3x2) and (b) (sec -1x)2.

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(a) From the chain rule, we have

(b) Also from the chain rule, we have

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Integrals Involving the Inverse Trigonometric Functions

,

(8.1)

(8.2)

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8.3 An Integral Related to tan-1x

Evaluate

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Notice that the integrand is nearly the derivative of tan -1x. However, the constant in the denominator is 9, instead of the 1 we need. With this in mind, we rewrite the integral as

If we let then du = dx and hence,

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We leave it as an exercise to prove the more general formula:

8.4 An Integral Requiring a Simple Substitution

Evaluate

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Think about how you might approach this. You probably won't recognize an antiderivative immediately. Remember that it often helps to look for terms that are derivatives of other terms. You should also recognize that e2x = (ex)2. With this in mind, we let u = ex, so that du = ex dx. We then have

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The hyperbolic sine function is defined by

for all x (- , ). The hyperbolic cosine function is defined by

,

6.9 Hyperbolic Function

again for all x (- , ) .

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You can easily use the preceding definitions to verify the important identity

, (9.1) for any value of u. (We leave this as an exercise.) In light of this identity, notice that if x = cosh u and y = sinh u, then

which you should recognize as the equation of a hyperbola.

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We define the hyperbolic tangent function the hyperbolic cotangent function the hyperbolic secant function sech x and the hyperbolic cosecant function csch x as follows:

First, note that

Similarly, we can establish the remaining derivative formulas:

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9.1 Computing the Derivative of a Hyperbolic Function

Compute the derivative of f (x) = sinh2(3x) .

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From the chain rule, we have

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For any x (- , ), we define the inverse hyperbolic sine by

For any x 1, we define the inverse hyperbolic cosine by

. Finally, for any x (-1, 1), we define the inverse hyperbolic tangent by

Inverses for the remaining three hyperbolic functions can be defined similarly and are left to the exercises. We show the graphs of and in Figures 6.48a, 6.48b and 6.48c, respectively. (As usual, you can obtain these by reflecting the graph of the original function through the line y = x. )

INVERSE HYPERBOLIC

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we have shown that

Note the similarity with the derivative formula for sin -1x. We can likewise establish derivative formulas for the other five inverse hyperbolic functions. We list these below for the sake of completeness.

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