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CHAPTER 6. TRANSCENDENTAL FUNCTIONS. 6.1 Natural Logarithm Function. The natural logarith function, denoted by ln, is defined by The domaian of the natural logarithm function is the set of positive real numbers. Geometric Meaning.

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  2. 6.1 Natural Logarithm Function • The natural logarith function, denoted by ln, is defined by • The domaian of the natural logarithm function is the set of positive real numbers.

  3. Geometric Meaning • Consider the graph of f(x) = 1/x. The ln(x) represents the area under f(x) between 1 and x.

  4. Use First Fundamental Theorem to find the Derivative of the Natural Logarithm Function

  5. Use u-substitution within natural logarithm function • Example:

  6. Properties of natural logarithms • If a and b are positive numbers and r is any rational number, then • A) ln 1 = 0 • B) ln ab = ln a + ln b • C) ln(a/b) = ln a – ln b • D)

  7. Logarithmic differentiation • Using properties of logarithms, a complicated expression can be rewritten as a sum or difference of less complicated expressions. Then, the function can be differentiated more easily. • Example next slide.

  8. Find g’(x)

  9. 6.2 Inverse Functions & Their Derivatives • If f is strictly monotonic on its domain, f has an inverse.

  10. Inverse Function Theorem • Let f be differentiable and strictly monotonic on an interval I. If f’(x) does NOT euqal 0 at a certain x in I, then the inverse of f is differentiable at the corresponding point y = f(x) in the range of f and

  11. Graphical interpretation • The slope of the tangent to a curve at point (x,y) is the reciprocal of the slope of the tangent to the curve of the inverse function at (y,x).

  12. 6.3 The Natural Exponential Function • The inverse of ln is called the natural exponential function and is denoted by exp. Thus x = exp y, and y = ln x. • The letter “e” denotest he unique positive real number such than ln e = 1.

  13. The natural exponential function and the natural logarithmic function are inverses of each other. • Properties that apply to inverse functions apply to these 2 functions.

  14. The derivative of the natural exponential function is itself

  15. Integrate the following:

  16. 6.4 General Exponential & Logarithmic Functions

  17. Logarithms could have a base other than e.

  18. Example:

  19. 6.5 Exponential Growth & Decay • Functions modeling exponential growth (or decay) are of this form:

  20. Compound InterestA = amountr = interest raten = # times compoundedt=time in years

  21. 6.6 1st-Order Linear Differential Equations • Sometimes it is not possible to separate an equation such that all expressions with x and dx are on one side and y and dy are on the other. • General form of a first-order linear differential equation:

  22. Example: Solve the differential equation:xy’(x) – 2y(x) = 2

  23. 6.7 Approximations for Differential Equations Slope fields: Consider a first-order differential equation of the form y’ = f(x,y) At the point (x,y) the slope of a solution is given by f(x,y). Example:y’ = 3xy, at (2,4), y’=24, at (-2,1), y’=-6; at (0,5), y’ = 0; at (2,0), y’=0, etc. If all the slopes (y’) were graphed on a coordinate axes at those specific points, the resulting graph would be a “slope field”.

  24. Approximating solutions of a differential equation • Euler’s Method: To approximate the folution of y’ = f(x,y) with initial condition y(x-not)=y-not, choose a step size ha nd repeat the following steps for n = 1,2,3,…

  25. Applying Euler’s Method • Use your calculator and the table function to evaluate the function until the solution is found with the desired error.

  26. 6.8 Inverse Trigonometric Functions & Their Derivatives • If the domain of the trigonometric functions is restricted, a portion of the curve is monotonic and has an inverse.

  27. Using triangles, some useful identities are established.

  28. Derivatives of 4 Inverse Trigonometric Functions

  29. Corresponding integral formulas follow from these derivatives

  30. Example

  31. 6.9 Hyperbolic Functions & Their Inverses

  32. Derivatives of Hyperbolic Functions

  33. Example

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