WEEKLY LEARNING ACTIVITY SHEET Basic Calculus, Grade 11, Quarter 3, Week 3 CONTINUITY OF FUNCTIONS Most Essential Learning Competencies 1. 2. 3. 4. illustrate continuity of a function at a number (STEM_BC11LCIIIc-1); determine whether a function is continuous at a number or not (STEM_BC11LCIIIc-2); illustrate continuity of a function on an interval (STEM_BC11LCIIIc-3); and solve problems involving continuity of a function (STEM_BC11LCIIId-3) Specific Objectives At the end of the lesson, the learner shall be able to: 1. 2. 3. 4. illustrate continuity of a function at a point; determine whether a function is continuous at a point or not; illustrate continuity of a function on an interval; and determine whether a function is continuous on an interval or not. Time Allotment: 4 hours Key Concepts CONTINUITY AT A POINT Definition A function a) b) c) is said to be continuous at exists; exists; and if the following three conditions are satisfied: . If at least one of these conditions is not met, is said to be discontinuous at Consider the graph of the function. • The function is continuous at . • The function is discontinuous at . • The function is discontinuous at . • The function is continuous at . . Illustration 1. Consider the function . Is the function continuous at Sol uti on. • 𝑓 • 𝑥 • 𝑥 Since 𝑓 𝑥 𝑥 continuous at 𝑥 Illustration 2. Consider the function ? , then 𝑓 is . . Is the function continuous at ? Sol uti on. • • 𝟑 𝟏 𝟐 𝟒 𝟏 𝟏 𝟎 𝒇 𝟏 𝟏 𝟏 𝟎 Si nce 𝒇 𝟏 does not exist, then 𝒇 i s di sconti nuous at 𝒙 𝟏. CONTINUITY ON AN INTERVAL Consider the graph of the function below. Determine whether the function is continuous on the following intervals: a) b) c) Consider the graph of the function below. Determine whether the function is continuous on the following intervals: a) ] b) [ One-sided Continuity a) A function is said to be continuous from the left at b) A function is said to be continuous from the right at if if Theorem 3.2.1 a) Polynomial functions are continuous everywhere. | | is continuous everywhere. b) The absolute value function c) Rational functions are continuous on their respective domains. d) The square root function . √ is continuous on [ Theorem 3.2.2. A function is said to be continuous… a) Everywhere if is continuous at every real number. In this case, we also say continuous on . b) on if is continuous at every point in . c) on [ if is continuous on and from the right at . ] if is continuous on d) on and from the left at . ] if is continuous on [ ]. e) on [ and on f) on if is continuous at all . g) on [ if is continuous on and from the right at . h) on if is continuous at all ] if is continuous on i) on and from the left at . is Illustration 1. Determine the largest interval over which is continuous. Solution. Observe that is not defined when . That is, . Hence, the domain of is . From Theorem 3.2.1, a rational function is continuous on its domain. Thus, is continuous over . Illustration 2. Determine the largest interval over which the function is √ continuous. Solution. Observe that is defined only if . That is, . Moreover, for all √ , and . Hence is continuous over . √ √ √ Also, when , the right at . Therefore, for all and √ , [ √ √ . Hence is continuous from is continuous. Activity No. 1 (Investigate Me!) What you need: Paper and Ball pen What to do: Determine whether the following functions are continuous at the given value of . 1. 2. at at 3. at √ 4. ⁄ { at 5. at Activity No. 2 (Solve Me!) What you need: Paper and Ball pen What to do: Solve the following problems involving continuity on functions. Show your complete , clear, and neat solution. 1. Determine the largest interval where is continuous. 2. Determine the largest interval where 3. Determine if 4. Let is continuous. is continuous or not at Find the value of c so that { 5. Consider √ is continuous at Find the value of c so that { is continuous at Reflection On a separate sheet of paper, write a short reflective essay (one to two paragraphs) detailing your experiences in completing the activities. You may summarize the things that you have learned, their applications in our daily lives, and the things that you enjoy or dislike. RUBRICS 10 – 9 points 8 – 6 points 5 – 3 points 2 – 0 points The reflection explains the student’s own thinking and learning experiences, as well as implications for future learning. The reflection explains the student’s thinking about his/her own learning experiences. The reflection attempts to demonstrate thinking about learning but is vague and/or unclear about the personal learning experiences. The reflection does not address the student’s thinking and/or learning. Writer: Reviewers: CHRISTIAN JAY M. BUSA ELMER R. ANDEBOR AMALIA B. RINGOR, DevEdD Special Science Teacher I Agusan National High School STEM Group Head Agusan National High School Academic Track Head Agusan National High School RUTH A. CASTROMAYOR ISRAEL B. REVECHE, PhD Principal IV SHS Assistant Principal Agusan National High School Education Program Supervisor Butuan City Division SHS Coordinator KEY TO CORRECTION Activity No. 1 (Investigate Me!) 1. Continuous 2. Continuous 3. Continuous 4. Not continuous 5. Continuous Activity No. 2 (Investigate me!) 1. Solut ion: Note that 𝑔 𝑥 is undefined at 𝑥 . Thus, if 𝑥 ≠ , 𝑔 𝑥 𝑥 , which is a polynomial 𝑥 2. 3. function. Therefore, 𝑔 𝑥 is continuous for all 𝑥 𝑥 Solut ion: Since 𝑥 , for any 𝑥 ℎ is defined on . Moreover, for any 𝑐 ℎ 𝑐 √𝑐 𝑥 𝑐ℎ 𝑥 Hence, ℎ is continuous at . Solut ion: We have to check the three conditions for continuity of a function. 1. If 𝑥 , then 𝑓 2. 3. 4. 𝑥 0 𝑓 𝑥 𝑥 𝑥 𝑥 𝑥 0 𝑥 𝑥 𝑥 𝑥 0 𝑓 𝑥 . 𝑓 𝑥 0 , 𝑥 𝑥 0 Solut ion: 1. The function is defined at 𝑥 Since 𝑦 3. 𝑥 2. 𝑥 𝑓 𝑥 and its value is 𝑓 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑐 𝑐 . . is continuous at 𝑥 , we have: 𝑓 𝑥 𝑐𝑥 𝑐 𝑐𝑥 𝑥 5. 𝑥 In order to make all three of these the same, we need 𝑐 Thus, 𝑐 . Solut ion: 1. The function is defined at 𝑥 and its value is 𝑓 𝑐 2. Since 𝑦 𝑥 𝑐 is continuous at 𝑥 , we have: 𝑓 𝑥 𝑥 𝑐 𝑐. 𝑥 𝑥 3. Since 𝑦 𝑥 is continuous at 𝑥 , we have: 𝑓 𝑥 𝑥 𝑥 𝑥 In order to make all three of these the same, we need Thus, 𝑐 . 𝑐 Balmecada, J. P. et. al. (2016). Basic Calculus Teacher's Guide (1st ed.). Philippines: Department of Education. Leithold, L. (1976). The Calculus with Analytic Geometry (3rd ed.). New York: Harper & Row. References