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SECTION 5.3. THE NATURAL EXPONENTIAL FUNCTION. THE NATURAL EXPONENTIAL FUNCTION. Since ln is an increasing function, it is one-to-one and, therefore, has an inverse function. We denote this by: exp Thus, according to the definition of an inverse function,. THE NATURAL EXPONENTIAL FUNCTION.

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## SECTION 5.3

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**SECTION 5.3**THE NATURAL EXPONENTIAL FUNCTION**THE NATURAL EXPONENTIAL FUNCTION**• Since ln is an increasing function, it is one-to-one and, therefore, has an inverse function. • We denote this by: exp • Thus, according to the definition of an inverse function, 5.3**THE NATURAL EXPONENTIAL FUNCTION**• The cancellation equations are: • In particular, we have: exp(0) = 1 since ln 1 = 0 exp(0) = e since ln e = 1 5.3**THE NATURAL EXPONENTIAL FUNCTION**• If r is any rational number, the third law of logarithms gives: ln(er ) = r ln e = r • Therefore, by Notation 1, exp(r) = er • Thus, exp(x) = ex whenever x is a rational number. 5.3**THE NATURAL EXPONENTIAL FUNCTION**• This leads us to define ex, even for irrational values of x, by the equationex = exp(x) • In other words, for the reasons given, we define ex to be the inverse of the function ln x. 5.3**THE NATURAL EXPONENTIAL FUNCTION**• In this equation, Notation 1 becomes: • The cancellation Equations 2 become: 5.3**Example 1**• Find x if ln x = 5. • SOLUTION 1 • From Notation 3, we see that: ln x = 5 means e5 = x • Therefore, x = e5. 5.3**Example 1 SOLUTION 2**• Start with the equation ln x = 5 • Then, apply the exponential function to both sides of the equation:eln x = e5 • However, Equation 4 says that: eln x = x • Therefore, x = e5. 5.3**Example 2**• Solve the equation e5–3x = 10. • SOLUTION • We take natural logarithms of both sides of the equation and use Equation 5: 5.3**Example 2 SOLUTION**• Since the natural logarithm is found on scientific calculators, we can approximate the solution to four decimal places: x≈ 0.8991 5.3**PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION**The exponential function f(x) = ex is an increasing continuous function with domain and range (0, ). Thus, e x > 0 for all x. Also,So, the x-axis is a horizontal asymptote of f(x) = ex. 5.3**Example 3**• Find • SOLUTION • We divide numerator and denominator by e2x: • We have used the fact that t = – 2x→ – as x → and so 5.3**LAWS OF EXPONENTS**If x and y are real numbers and r is rational, then 1. e x+y = e xe y 2. e x–y= e x/e y 3. (e x)r = e rx 5.3**PROOF OF LAW 1**• Using the first law of logarithms and Equation 5, we have: • Since ln is a one-to-one function, it follows that: e xe y = e x+y • Laws 2 and 3 are proved similarly. • See Exercises 69 and 70. • As we will see in next section, Law 3 actually holds when r is any real number. 5.3**DIFFERENTIATION**• The natural exponential function has the remarkable property that it is its own derivative. 5.3**PROOF**• Let y = ex. • Then, ln y = x and, differentiating this latter equation implicitly with respect to x, we get: 5.3**Example 4**• Differentiate the function y = etan x. • SOLUTION • To use the Chain Rule, we let u = tan x. • Then, we have y = e u. • Hence, 5.3**DIFFERENTIATION**• In general, if we combine Formula 8 with the Chain Rule, as in Example 4, we get: 5.3**Example 5**• Find y’ if y = e–4x sin 5x. • SOLUTION • Using Formula 9 and the Product Rule, we have: 5.3**Example 6**• Find the absolute maximum value of the function f(x) = xe–x. • SOLUTION • We differentiate to find any critical numbers: f ’(x) = xe–x(– 1) + e–x(1) = e–x(1 –x) 5.3**Example 6 SOLUTION**• Since exponential functions are always positive, we see that f’(x) > 0 when 1 –x > 0, that is, when x < 1. • Similarly, f’(x) < 0 when x > 1. • By the First Derivative Test for Absolute Extreme Values, f has an absolute maximum value when x = 1. • The absolute maximum value is: 5.3**INTEGRATION**• As the exponential function y = ex has a simple derivative, its integral is also simple: 5.3**Example 7**• Evaluate • SOLUTION • We substitute u = x3. • Then, du = 3x2 dx. • So, and 5.3**Example 8**• Find the area under the curve y = e–3x from 0 to 1. • SOLUTION • The area is: 5.3

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