Download presentation
Presentation is loading. Please wait.
1
1.6 Continuity Calculus 9/17/14
2
Warm-up The cost (in dollars) of removing p% of the pollutants from the water in a small lake is given by πΆ= 25,000π 100βπ , 0β€π<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 π₯β 100 β πΆ. Explain your results
3
1.6 Continuity
5
What are some examples of continuous functions?
Polynomials β continuous at every real number Rational functions β continuous at every number in its domain
7
π π₯ = π₯ 2 β4 π₯β2 On what interval is this function continuous?
π π₯ = π₯ 2 β4 π₯β2 On what interval is this function continuous? βThe function has a discontinuity at cβ
8
Removable and nonremovable discontinuities
Removable- if π can be made continuous by defining π π at that point Nonremovable β when the function cannot be made continuous at x=c -Ex. π π₯ = 1 π₯ cannot be redefined at x=0 Use example from slide before for removable and slide 6 graph a for nonremovable
9
Continuity on a closed interval
If π is continuous on the open interval (a,b) lim π₯β π + π π₯ =π(π) πππ lim π₯β π β π π₯ =π(π) Then π is continuous on the closed interval [a,b]
10
π π₯ = 3βπ₯ Domain: Graph Continuous
11
π π₯ = 5βπ₯ , β1β€π₯β€2 π₯ 2 β1, 2<π₯β€3 Is π(π₯) continuous on a closed interval Closed endpoints? Continuous on open interval (a,b)? Limits from all sides
12
π π₯ = π₯+2, β1β€π₯<3 14β π₯ 2 , 3β€π₯β€5
14
Greatest integer function
π₯ = greatest integer less than or equal to x
15
Ex. 5 p. 66 πΆ= π₯β1 10, x -Sketch the graph and analyze the discontinuities
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.