Basic Calculus (Module 8.1) - Extreme Value Theorem

Basic Calculus

1. The Equation of the Tangent Line Since the tangent line is the limiting position of the secant lines as Q approaches P, it follows that the slope of the tangent line (TL) at the point P is the limit of the slopes of the secant lines as x approaches . In symbols Finally, since the tangent line passes through then its equation is given by  

2. Example 1: Find the equation of the tangent line to the graph of          

3. Example 2: Find the equation of the tangent line to the graph of         -  

4. Introduction to Derivatives

5. Definition of Derivative Module Content The derivative of a function at at tells us that is the slope of the tangent line to the graph of the function at point .     provided the limit exists. The derivative of a function denoted at any in the domain of the given function is defined as    

6. Provided the limit exists. The following are the different notations of derivatives when )   Note that the process of solving the derivative is called differentiation . The techniques in differentiating functions is the same with evaluating limits and in indeterminate form , factoring and rationalization process could be utilized.  

7. Example 1:  

8. Example 2:  

9. Example 3:  

10. Example 4:  

11. Example 3:  

12. Example 5:  

13. Derivative Of Algebraic Functions

14. Polynomial, Rational and Radical Functions DIFFERENTIATION RULES

15. I. DERIVATIVE OF A CONSTANT FUNCTION The graph of a horizontal function is a horizontal line, and a horizontal line has zero slope. Recall that the derivative measures the slope of the tangent line , and so the derivative of a constant term is zero . Example : then the 1st derivative is  

16. I. DERIVATIVE OF A POWER FUNCTION A function of the form : where n is a real number, is called a power function. In general, this is called the POWER RULE If then,  

17. POLYNOMIAL FUNCTION 1. 2. If Then ;  

18. RATIONAL FUNCTION Find , where can be written as:  

19. RADICAL FUNCTION Find the derivative of SOLUTIONS: rewrite the expression to exponential form apply the power rule subtract the exponents simplify to radical form  

20. RADICAL FUNCTION Given ; find SOLUTIONS: rewrite to exponential form apply to power rule subtract the exponents simplified radical expression  

21. CONSTANT MULTIPLE RULE States that the derivative of a constant times a differentiable function is the constant times the derivative of the function. If where k is constant ( k is the numerical coefficient of the function of x ) ; then :  

22. CONSTANT MULTIPLE RULE Given: SOLUTION : Let rewrite in the form apply constant multiple and power rule   subtract exponents and combine similar terms expressed in positive exponent simplest radical form  

23. CONSTANT MULTIPLE RULE Given: SOLUTION: rewrite in the form apply constant multiple and power rle subtract exponents and combine similar terms in positive exponent to simplest radical form  

24. SUM / DIFFERENCE RULE Given two differentiable functions g and h, if EXAMPLES: Given: From the given functions above find the following :  

25. SUM / DIFFERENCE RULE combine the given functions by addition as indicated apply the constant multiple rule for each function use the power subtract the exponents expressed the terms with positive exponents simplified radical expression  

26. SUM / DIFFERENCE RULE combine the functions by addition as indicated a pply the constant multiple rule for each function use the power rule subtract the exponents expressed the terms with positive exponents simplest radical form  

27. DERIVATIVE OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS In the previous study of the different function graphs you have learned that exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions. Just as when we found the derivatives of algebraic functions, we can also find the derivatives of exponential and logarithmic functions using formulas.

28. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS Derivative of Exponential Functions : If , then If , then Derivative of Logarithmic Functions : If , then Derivative of Natural Logarithm : If , then  

29. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXAMPLES : Determine the derivative of the following functions : 1. SOLUTIONS: expressed with positive exponent apply the quotient rul simplify the terms derivative in simplest form  

30. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXAMPLES : Determine the derivative of the following functions : 1. SOLUTIONS: expressed with positive exponent apply the quotient rul simplify the terms derivative in simplest form  

31. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS EXAMPLES : Determine the derivative of the following functions : 1. SOLUTIONS: expressed with positive exponent apply the quotient rul simplify the terms derivative in simplest form  

32. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS 2. SOLUTIONS: Let and given apply the product formula combine similar terms factor and simplify  

33. DIFFERENTIATION FORMULAS : EXPONENTIAL AND LOGARITHMIC FUNCTIONS 3. SOLUTIONS: clear off parenthesis apply the differentiation formula combine similar terms simplified into a single expression