7-Continuity othhhhhhhhhhhhf a Function.pptx

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1. CONTINUITY OF A FUNCTION OBJECTIVES FOR TODAY 1 2 3

2. NAME Role here NAME Role here NAME Role here NAME Role here REVIEW The limit of a function 𝑓(𝑥) is the value it approaches as the value of 𝑥 approaches a certain value. “As 𝑥 approaches 𝑎, the limit of 𝑓(𝑥) approaches L”. (Mercado, 2016) This is written in symbols as follows;

3. A limit that is indeterminate of type 0/0 may exist. To find the actual value, one should find an expression equivalent to the original, by factoring or by rationalizing. So, the expression that will emerge after factoring or rationalizing will have a computable unit. INSERT YOUR TITLE HERE Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Maecenas porttitor congue massa. Fusce posuere, magna sed pulvinar ultricies, purus lectus malesuada libero, sit amet commodo magna eros quis urna . REVIEW

4. When do we say that a function is continuous?

5. CONTINUITY OF A FUNCTION If one or more of the conditions at the left fails to hold at C, the function is said to be discontinuous .

6. EXAMPLE 1 Determine if the function is continuous or not at x = 1.

7. EXAMPLE 2 Determine if the function is continuous or not at x = 3.

8. EXAMPLE 3 Determine if the function is continuous or not at x = 4.

9. EXAMPLE 4 Determine if the function is continuous or not on a closed interval [-4, -1].

10. EXAMPLE 5 Determine if the function is continuous or not on a closed interval [-4, 1].

11. TYPES OF DISCOUNTINUITY

12. TYPES OF DISCONTINUITY 1. The figure illustrates the function defined at c and that the limit coming from the right and left of c both exist thus the two sided limit exist. But which violates the third condition. This kind of discontinuity is called REMOVABLE DISCONTINUITY .  

13. EXAMPLES Given the function f defined as draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous. SOLUTION

14. GRAPH y x REMOVABLE DISCONTINUITY

15. TYPES OF DISCONTINUITY 2. The figure illustrates that the limit coming from the right and left both exist but are not equal, thus the two-sided limit does not exist which violates the second condition. This kind of discontinuity is called JUMP DISCONTINUITY .

16. EXAMPLES SOLUTION

17. EXAMPLES Given the function f defined as draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous. SOLUTION

18. GRAPH

19. TYPES OF DISCONTINUITY 3. The figure illustrates that the limit coming from the right and left of c are both , thus the two-sided limit does not exist which violates the second condition. This kind of discontinuity is called INFINITE DISCONTINUITY.  

20. EXAMPLES Given the function f defined as draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous. SOLUTION This type of discontinuity exists if a function has one or more infinite limits. Many rational functions exhibit this type of behavior.

21. GRAPH

22. 1. A function is continuous. Which of the following is TRUE about its graph? A. It has a hole or gap. B. It can be drawn without lifting your pen. C. It approaches positive infinity. D. It represents a rational function.

23. 2. Which of the following functions is continuous on all values of x? A. Polynomial functions B. Rational functions C. Exponential functions D. Radical functions

24. 3. Which of the following illustrates removable discontinuity?

25. 4. When can we say that exists?   A. If the left-hand limit approaches infinity. B. If the right-hand limit approaches infinity. C. If the left-hand limit is equal to the right-hand limit. D. If the left-hand and right-hand limits approach the y-axis.

26. 5. What value of the denominator will make a rational function discontinuous? Answer: Zero (0)

27. 6. How to illustrate the discontinuity of a function? Answer: Explain the different types of discontinuity (3)

28. 7. Enumerate six (6) kinds of functions that can be discontinuous. POSSIBLE ANSWERS : rational, radical, exponential, logarithmic, trigonometric, piecewise

29. PERFORMANCE TASK

30. TO DO: Represent a graph showing the voltage readings as y-values and the time as x-values. Is the graph continuous or not? If your answer is not continuous, at what point or interval does the discontinuity happen?

31. In mathematics, a continuous function is a function that does not have any abrupt changes in value. More precisely, sufficiently small changes in the input of continuous functions result in arbitrary small changes in its output. If not continuous, a function is said to be discontinuous.

32. In life, problems are what make life worth living. They help us adapt to become tougher as we adapt to different situations. Just continue to live and focus positively whatever problem you are facing because it has always a solution. Therefore, never allow your challenges to stop you from fulfilling your true potentials in life.

33. Mr. dela Cruz encourages his students to solve Math problems fast and with accuracy. He observed in his math class that the time it takes a student to solve x word problems is defined by the function where f(x) is in minutes. Solve for the instantaneous rate of change in solving time at 2-word problems.  

34. Which of the following does NOT define the slope of the tangent line to the curve? A. It is constant. B. It is not constant and must be determined by a point. C. It is equal the derivative of the function. D. It is derived from the concept of the slope of a second line.