Calc 5.5a
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Calc 5.5a

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Calc 5.5a Calc 5.5a Presentation Transcript

  • Define exponential functions that have bases other than e Differentiate and integrate functions that have other bases Use Exponential functions to model compound interest and exponential growth
  • The base of the natural exponential function is e . This base can be used to assign meaning to a general base a
    • The laws of exponents apply here as well:
    • a 0 = 1
    • a x a y = a x+y
    • a x /a y = a x – y
    • (a x ) y = a xy
  • Ex 1 p. 360 Radioactive Half-life Model The half-life of carbon-14 is about 5715 years. A sample contains 1 gram of carbon-14. How much will be present in 10,000 years? Solution: Let t = 0 represent the present time and y represent the amount (in grams) of carbon-14. Using a base of ½, you can model y by the equation If t = 0, y = 1 gram. If t = 5715, then y = ½, which would be correct.
  • Remember, this is just the change of base rule you’ve seen before, just in a new setting!
  • Logarithmic Properties still apply Exponential functions and logarithmic functions are inverse functions
  •  
  • So as review, we’ll work with these properties with bases other than base e Ex. 2a, p. 361 Take the log base 3 to each side
  • Ex. 2b, p. 361 Exponentiate each side using base 2
  • When thinking of these derivatives, it is often helpful to think of them as natural exponential things or as natural log things.
  • Ex 3 p. 362 Differentiating functions to other bases. Find the derivative of each function. log with no base shown is a common log, base 10
  • Sometimes an integrand will work with a exponential function involving another base than e. When this occurs, we can do one of two things – convert to base e using the formula and integrate, or integrate directly, using the following formula: Ex 4 p. 363 Integrating an exponential function with base 3
  • When the power rule was introduced in Ch. 2, we limited it to rational exponents. Now the rule is extended to cover all real exponents.
  • This next example compares the derivatives of four different functions involving exponents. Be CAREFUL! Logarithmic differentiation required!
  • 5.5a p.366/ 3-60 mult 3