General Math (I) Lecture 7 PDF

General Math (I) Lecture (7)  Always Do Ur Best What U Plant Now U Will Harvest Later  Chapter 2 Functions & Limits Continuity of Functions 1. Continuity at a Point 2. Functions are Continuous on Their Domains 3. Continuity Characteris...

General Math (I) Lecture (7)  Always Do Ur Best What U Plant Now U Will Harvest Later  Chapter 2 Functions & Limits Continuity of Functions 1. Continuity at a Point 2. Functions are Continuous on Their Domains 3. Continuity Characteristics 4. Continuity of a Function on an Interval 1. Continuity at a Point  1. Continuity at a Point Examples Discuss the Continuity of the Following Functions at the Indicates Points 1. f(x) is Continuous at x=-1 2. 1. g(0) (Not Defined) g(x) is Not Continuous at x= 0  1. Continuity at a Point Examples Discuss the Continuity of the Following Functions at the Indicates Points 3. 1. h(1) (Not Defined) h(x) is Not Continuous at x=1 4. k(x) is Not Continuous at x= 3 k(x) is Continuous at x= 0  1. Continuity at a Point Example Discuss the Continuity of f(x) at x=0 −x f ( x) = e −x  e , x if x  0 f(x)=e =  −x  e , if x  0 1. f(0) = e0 = 1 −x 2. lim f ( x) = lim e = 1, lim f ( x) = lim e = 1  lim f ( x) = 1 x − − + + x →0 x →0 x →0 x →0 x →0 3. lim f ( x ) =1= f (0 ) x →0 f(x) is Continuous at x= 0  2. Functions are Continuous on Their Domains Every Where 3. Continuity Characteristics  3. Continuity Characteristics Example Discuss the Continuity of the Following Function  x 2 − x − 12  if x  −3 f ( x) =  x + 3 − 5 if x = −3  (1) For x ≠ -3, f(x) is a quotient of two polynomials, so it is continuous for all x ≠ -3. (2) At x = -3, We examine the three conditions of continuity at a point: 1. f(-3) = -5 by the definition of the function. 2. We study the limit of the function as x approaches -3 as follows: x 2 − x − 12 ( x − 4 )( x + 3 ) lim f ( x ) = lim = lim = lim ( x − 4 ) = −7 x →−3 x →−3 x+3 x →−3 x+3 x →−3 3.  lim f ( x) = −7  f (−3) = −5 x → −3  f(x) is discontinuous at x = −3  4. Continuity on an Interval Definition A function is continuous on a closed interval if is continuous at all interior points as well as continuous at the two end points  4. Continuity on an Interval Example Discuss the Continuity of 𝒇 𝒙 = 𝟏 − 𝒙𝟐 on the Interval [-1,1]  4. Continuity on an Interval Solution x→ -1+ x→ 1-  Dr/ Soaad Abd El-Badie Attia El- Afefy ( S. Abd El-Badie) Don't 4 Get & Do All Ur Best 2 B Z Best