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SECTION 5.3

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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  1. SECTION 5.3 THE NATURAL EXPONENTIAL FUNCTION

  2. 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

  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

  4. 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

  5. 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

  6. THE NATURAL EXPONENTIAL FUNCTION • In this equation, Notation 1 becomes: • The cancellation Equations 2 become: 5.3

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. DIFFERENTIATION • The natural exponential function has the remarkable property that it is its own derivative. 5.3

  16. PROOF • Let y = ex. • Then, ln y = x and, differentiating this latter equation implicitly with respect to x, we get: 5.3

  17. 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

  18. DIFFERENTIATION • In general, if we combine Formula 8 with the Chain Rule, as in Example 4, we get: 5.3

  19. Example 5 • Find y’ if y = e–4x sin 5x. • SOLUTION • Using Formula 9 and the Product Rule, we have: 5.3

  20. 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

  21. 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

  22. INTEGRATION • As the exponential function y = ex has a simple derivative, its integral is also simple: 5.3

  23. Example 7 • Evaluate • SOLUTION • We substitute u = x3. • Then, du = 3x2 dx. • So, and 5.3

  24. 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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