Week 4 Calculus 1 - Derivatives as a Function.pptx

Derivatives as a Function Lecture (Based on OpenStax - Volume 1)

1. Week 4 In this module we cover 2.2 Derivative as a function 2.2.1 Define the derivative function of a given function. 2.2.2 Graph a derivative function from the graph of a given function. 2.2.3 Derivatives and continuity. Describe three conditions for when a function does not have a derivative. 2.2.4 Higher-order derivatives

2. In previous sections we have seen that the derivative of a function at a given point gives us; the rate of change or slope of the tangent line to the function at that point. To obtain the derivative of a function at each point in the domain of the function, we define the Derivative Function. 2.2 The Derivative as a Function

3. 2.2.1 The Derivative function of a given function Definition: Let be a given function. The derivative function, denoted by , is the function whose domain consists of those values of such that the following limit exists: ……. (i) Where is a small change in In some versions of the equation (i), is denoted by i.e.  

4. Using in limit definition of derivative function   The equation , can also be expressed as Note  

5. Differentiable function A function is said to be differentiable at if exists. i.e exists at More generally, a function is said to be differentiable on S if it is differentiable at every point in an open set S, and a differentiable function is one in which ) exists on its domain.  

6. Example Find the derivative of the function (Using equation ( i ) above) Solution:  

7. Class Activity 1. Find the derivative of the function using limit definition 2. View video linked below https://www.youtube.com/watch?v=71DwCbpdaOo to find the derivative of the following functions, and derivative values at given points at at at Note the examples in the video uses the limit definition  

8. Derivative notations For a function the common alternative notations for derivatives are; The symbol is not a ratio; it is a symbol simply for derivative of , This notation is called Leibniz notation of derivative. So, the symbol simply means  

9. Derivative notations cont. For example, if , then and can be equivalently expressed as Also, if we write , then and can be equivalently expressed as  

10. Derivative notations cont. The derivative of , which can be thought of as the instantaneous rate of change of with respect to , is expressed as , Where is the difference in the values corresponding to the difference in the values, which are expressed as  

11. 2.2.2 Graph a derivative function from the graph of a given function Geogebra /Desmos Activity Obtain the graphs of the function +1 and its derivative function a) What happens to the derivative when is increasing at a decreasing rate? b) What happens to the derivative when is increasing at an increasing rate? c) What happens to the derivative at the points of the graph of ?  

12. Graphically: Graph of a function and its derivative function

13. Class Activity Use each of the graphs below of to sketch the graph of its derivative  

14. Class Activity cont. Consider the function a) Use the limit definition of the derivative to determine a formula for . b) Use a graphing utility( Geogebra or Desmos) to plot both and your result for ; does your formula for generate the graph you expected? c) Use the limit definition of the derivative to find a formula for where g d) Compare and contrast the formulas for and you have found. How do the constants 5, 4, 12, and 3 affect the results?  

15. 2.2.3 Derivatives and Continuity Differentiability implies continuity: Let be a function and be in its domain. If is differentiable at ( i.e exists), then is continuous at . Explanation: If a function is differentiable at a point in its domain, the is continuous. i.e., if exists, then Conversely, if is not continuous at , then is not differentiable at i.e., if then does not exist.  

16. Types of discontinuities The four types of functions that are not differentiable are: 1) Corners Ex: If , the derivative is -1. If the derivative is 1. What is the derivative at ? Since the left derivative does not equal the right derivative, the derivative does not exist. 2) Cusps Ex: Derivative is negative when and derivative is positive when . Since the left derivative does not equal the right derivative, the derivative does not exist.  

17. Types of Discontinuity Cont. 3) Vertical tangents Derivative is positive when and derivative is positive when When the derivative looks like a vertical line which is undefined.  

18. Example 4) Any discontinuities

19. In Summary: From examples above; We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable. 2. We saw that failed to be differentiable at 0 because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at 0. From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point. Algebraically, Consider the function This function is continuous everywhere; however, is defined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further.  

20. Example: Continuity does not imply differentiability For The derivative does not exist at because However, the function is continuous at Thus, Continuity does not imply differentiability. i.e., a function could be continuous but not necessarily differentiable at a point.  

21. 2.2.4 Higher-order derivatives The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of can be expressed in any of the following forms:  

22. Example: Derivative of first derivative Find if Solution: Now Note: The derivative of the (first) derivative is called the second derivative of denoted by  

23. Finding Higher Order Derivatives: For Find  

24. Solution Cont.

25. Class Activity For the function find  

26. Class Activity Exercise: The graph in the following figure models the number of people who have come down with the flu weeks after its initial outbreak in a town with a population of 50,000 citizens. a. Describe what represents and how it behaves as t increases. b. What does the derivative tell us about how this town is affected by the flu outbreak?