advertisement

2.4 Continuity and its Consequences Thurs Sept 17 Do Now Find the errors in the following and explain why it’s wrong: HW Review: p.80 #5 13 19 27 29 33 35 • • • • • • • 5) 1/2 13) -2/5 19) 1/5 27) 3 29) 1/16 33) let f(x) = 1/x and g(x) = -1/x 35) proof Continuity - What does it mean? A function is said to be continuous on an interval if its graph on that interval can be drawn without interruption, or without lifting your pencil. • Holes and asymptotes are examples of discontinuous functions Definition of continuous • • • • • A function f is continuous at x = a when 1) f(a) is defined 2) exists 3) Otherwise, f is said to be discontinuous at x = a One-Sided Continuity • A function f(x) is called: – Left-continuous at x = c if – Right-continuous at x = c if What kind of functions are continuous? • • • • • • Polynomials Radical Functions on their domains Sin x and cos x Exponential functions Logarithmic functions on their domains Rational functions on their domains Piecewise Functions • These kind of functions are the big AP type of problems More Continuous Functions • Thm- Suppose that f and g are continuous at x = c. Then: – 1) kf(x) for any constant k – 2) ( f ± g) is continuous at x = c – 3) ( f × g) is continuous at x = c – 4) ( f ) is continuous at x = c if g(c) ¹ 0 g and discontinuous if g(c) = 0 More Continuous Functions • Thm- If f(x) is continuous on an interval I with range R and its inverse exists, then its inverse is continuous with domain R Composite Functions • If g(x) is continuous at x = c, and f(x) is continuous at x = g(c), then f(g(x)) is also continuous at x = c 3 Types of Discontinuities • Removable Discontinuity – Limit exists – F(x) is not equal to the limit – Can redefine function at discontinuity • Jump Discontinuity – Left and right side limits do not agree – Cannot redefine • Infinite Discontinuity – One or both of each sided limits is infinite Closure • Journal Entry: What must be true for a function to be continuous? What is an example of a discontinuity? Which are removable or not? • HW: p.88-89 #1, 3-5, 17-33 odds, 55 57 59 63 65 Continuity Cont’d Fri Sept 18 • • Do Now Is the function f (x) = 16 - x 2 continuous at the following points? 1) X = 3 2) X = 4 HW Review: p.88-89 #1, 3-5, 17-33 odds, 55 57 59 63 65 1) [email protected] [email protected] [email protected] LC 27) x = 2, jump, 3) X = 3 redefine g(3) = 4 29) t = (2n+1)pi/4, n = int 4) C = 1, redefine g(1) = 3 31) continuous for all 5) Omgicantfitthishere 33) x = 0, inf, neither 17) X = 0, inf, neither 55) show right lim = left 19) X = 1, inf, neither 57) c = 5/3 21) Even ints, jump, RC 59) a = 2, b = 1 23) X = 1/2, inf, neither 63) graph 25) Continuous for all x 65) graph Classwork • Side 1(p.53) #3, 4 • Side 2(p.153) #21 22 23 24 25 Closure • Exit pass: Find all discontinuities of x -1 f (x) = 2 x -1 For each discontinuity, state the type, whether it is left/right continuous, and if removable, redefine it so it is continuous HW: none or finish worksheet 2.3-2.5 quiz soon