Lesson: Derivative Applications 3 ï¶ Objective – Logarithms,...

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Lesson: Derivative Applications 3 Objective Logarithms, Eulers Number, & Differentiation, oh my! Slide 2 Slide 3 1. What is a Logarithm? A logarithm can be defined as an exponent Consider the following logarithmic expression It represents the exponent that the base of 7 must be raised to, in order to get the value 49. Slide 4 2. What is Eulers Number? e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation.Euler's Equation Slide 5 The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant.Leonhard EulerJohn Napier, Slide 6 3. What is the Natural Logarithm (LN)? A logarithm with a base of e 4. Properties of Logs Slide 7 Formulas: Slide 8 Basically, think of derivatives of e and Ln as chain rule problems. Slide 9 Lets try a few: Slide 10 Slide 11 Whats up with e x ? How can it be the derivative of itself? Proof: Lets start with: LN both sides: Use log property: Simplify: Slide 12 Derive both sides with respect to x: Multiply both sides by y: Simplify: Replace y with the original equation: Q.E.D. Slide 13 EX. 1: Find Slide 14 EX. 2: Find Slide 15 EX. 3: Find Now isnt that special? If the mere sight of this problem makes you want to break stuff & cry dont worry. You are not alone. But fear not, it is very workable with the use of the logarithmic properties. Slide 16 Slide 17 Slide 18 Slide 19 EX. 4: Find y, if Slide 20 Slide 21 Slide 22 EX. 5: Find Slide 23 EX. 6: Find Slide 24 Deriving the exponential function Remember: What about Stuff Slide 25 Ex. 7: Differentiate Slide 26 HW 4.2: Log Differentiation Worksheet