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1.6 Continuity CALCULUS 9/17/14 Warm-up The cost (in dollars) of removing p% of the pollutants from the 25,000π water in a small lake is given by πΆ = , 0 ≤ π < 100 100−π 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 − πΆ. Explain your results π₯→100 Warm-up (1.6 Continuity-day 2) 1) Find the limit. 2) lim π π₯ π₯→−1 lim+ π₯→0 1 2+π₯ − 2π₯ 1 2 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 2 π₯ −4 π π₯ = π₯−2 ο§On what interval is this function continuous? ο§ “The function has a discontinuity at c” Removable and nonremovable discontinuities o Removable- if π can be made continuous by defining π π at that point oNonremovable – when the function cannot be made continuous at x=c -Ex. π π₯ = 1 π₯ cannot be redefined at x=0 Continuity on a closed interval ο§ If π is continuous on the open interval (a,b) ο§ lim+ π π₯ = π(π) πππ lim− π π₯ = π(π) π₯→π π₯→π ο§Then π is continuous on the closed interval [a,b] π π₯ = 3−π₯ • Domain: •Graph •Continuous 5−π₯, π π₯ = 2 π₯ − 1, −1 ≤ π₯ ≤ 2 2<π₯≤3 ο±Is π(π₯) continuous on a closed interval ο±Closed endpoints? ο± Continuous on open interval (a,b)? π₯ + 2, π π₯ = 2 14 − π₯ , −1 ≤ π₯ < 3 3≤π₯≤5 Greatest integer function ο§ π₯ = greatest integer less than or equal to x Ex. 5 p. 66 πΆ = 5000 1 + π₯−1 10,000 + 3x -Sketch the graph and analyze the discontinuities