1.6 Continuity Calculus 9/17/14

Warm-up The cost (in dollars) of removing p% of the pollutants from the water in a small lake is given by 𝐶= 25,000𝑝 100−𝑝 , 0≤𝑝<100 Where C is the cost and p is the percent of pollutants. A) find the cost of removing 50% of the pollutants B) What percent of the pollutants can be removed for $100,000? C) Evaluate lim 𝑥→ 100 − 𝐶. Explain your results

1.6 Continuity

What are some examples of continuous functions? Polynomials – continuous at every real number Rational functions – continuous at every number in its domain

𝑓 𝑥 = 𝑥 2 −4 𝑥−2 On what interval is this function continuous? 𝑓 𝑥 = 𝑥 2 −4 𝑥−2 On what interval is this function continuous? “The function has a discontinuity at c”

Removable and nonremovable discontinuities Removable- if 𝑓 can be made continuous by defining 𝑓 𝑐 at that point Nonremovable – when the function cannot be made continuous at x=c -Ex. 𝑓 𝑥 = 1 𝑥 cannot be redefined at x=0 Use example from slide before for removable and slide 6 graph a for nonremovable

Continuity on a closed interval If 𝑓 is continuous on the open interval (a,b) lim 𝑥→ 𝑎 + 𝑓 𝑥 =𝑓(𝑎) 𝑎𝑛𝑑 lim 𝑥→ 𝑏 − 𝑓 𝑥 =𝑓(𝑏) Then 𝑓 is continuous on the closed interval [a,b]

𝑓 𝑥 = 3−𝑥 Domain: Graph Continuous

𝑔 𝑥 = 5−𝑥 , −1≤𝑥≤2 𝑥 2 −1, 2<𝑥≤3 Is 𝑔(𝑥) continuous on a closed interval Closed endpoints? Continuous on open interval (a,b)? Limits from all sides

𝑓 𝑥 = 𝑥+2, −1≤𝑥<3 14− 𝑥 2 , 3≤𝑥≤5

Greatest integer function 𝑥 = greatest integer less than or equal to x

Ex. 5 p. 66 𝐶=5000 1+ 𝑥−1 10,000 +3x -Sketch the graph and analyze the discontinuities