MA1102 Calculus Tutorial 9

Topics Covered:

Mean Value Theorem for Integrals. Inverse Functions again. Derivative of inverse function. Definition of natural logarithmic function. More integral formulae. Exponential function as inverse of the natural logarithmic function. Powers a^x . Logarithmic functions to the base a (> 0). Logarithmic differentiation.

Textbook: Chapter 9, Chapter 10.

Without doubt differentiation of inverse function is a deep result. Theorem 10.2.2 in the text book has two parts: the first assertion is about differentiability and the second the formula for the derivative of the inverse function, . To use this formula, you would need to know the derivative of f and also the image f ^-1(x) for the given x. The natural logarithmic function is defined via Riemann integral: . This is such a beautiful mathematical definition. For it at once has at its disposal, the FTC to generate the properties it enjoys. Then the exponential function is defined simply as the inverse of the natural logarithmic function, not the 'intuitive' definition you encounter in school. The power function a^x can now be easily defined using exponential function. Convince yourself that this is the extension of the usual power function when x is a rational number. We have new functions and so we have new integral formulae associated with logarithmic, exponential, power and logarithmic to the base a functions. It is not necessary to commit all these formulae to memory, you just need to know they exist and that you can easily derive them. Logarithmic differentiation is just a device to differentiate a power function like a product by applying the natural logarithmic function before differentiation. We can then use the product rule to differentiate provided we know the derivative of each term or function involved.