Download presentation

Presentation is loading. Please wait.

Published byLillian Boone Modified over 3 years ago

1
2.3 Continuity Grand Canyon, Arizona Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

2
Most of the techniques of calculus require that functions be continuous. A function is continuous (in a casual way) if you can draw it in one motion without picking up your pencil… a more thorough definition: A function is continuous at a point if the limit is the same as the function value at that point. The function at the left has discontinuities at x=1 and x=2. f(x) is continuous at x=0 and x=4, because the one-sided limit matches the function value at the endpoint. 1234 1 2 lim f(x) = f(c) = lim f(x) xc- xc+

3
jump infinite oscillating Essential (Non-Removable!) Discontinuities: Removable Discontinuity: (You could fill the gap and establish continuity by adding or moving a single point.)

4
To write an extended function that is continuous at x = 1, find the limit of f(x) approaching x = 1… Removing a point discontinuity: … and add a new piece to the original function: the new height is the limit value (note its x-coordinate). The new function is an extended, now-continuous function. has a removable discontinuity at x=1… (Note: The essential discontinuity at x = -1 cannot be removed!)

5
Removing a discontinuity from a graphic perspective: (Also note the discontinuity at x = -1 that cannot be removed.) ° Note that the original hole in the graph has now been filled in!

6
Continuous functions can be added, subtracted, multiplied, divided, multiplied by a constant, or composed, and the new function remains continuous. Composites of continuous functions are continuous! examples: Outer function: y = sin( ) Inner function: y = ( ) 2 Outer function: y = abs( ) Inner function: y = cos( )

7
Intermediate Value Theorem If a function is continuous between a and b, then it takes on every function value between and Because the function is continuous, f reaches every height -value between f(a) and f(b). (Sometimes that height is reached more than once!)

8
Example 5: Is any real number exactly one less than its cube? Since f is a continuous function, by the IVT, f must take on every function value between -1 and 5. Therefore, there must be at least one solution (value of 0) between 1 and 2. Use your calculator to approximate that solution, via 2 nd TRACE zero… f(1) = -1 … … a negative height and f(2)= 5 … … a positive height x 1.32472

9
Graphing calculators can make non-continuous functions appear to be continuous… look out! Graph: MATH 5 int( Note x- resolution. In connected mode, the calculator can connect the dots, and cover up the discontinuities at every integer. NUM

10
Graphing calculators can make non-continuous functions appear continuous … look out! The open and closed endpoints do not show, but we can see the discontinuities more clearly! If we change the x-resolution to 1, then we get a graph that is closer to the actual step graph. Graph:

Similar presentations

OK

4.4 Optimization Buffalo Bills Ranch, North Platte, Nebraska Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin,

4.4 Optimization Buffalo Bills Ranch, North Platte, Nebraska Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin,

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on recycling of waste management Ppt on intelligent manufacturing services Ppt on network switching infrastructure New era of management ppt on communication Ppt on natural resources and conservation job Ppt on varactor diode applications Ppt on animal cell and plant cell Ppt on frame relay and atm Ppt on waxes by andrea Ppt on drugs and alcohol abuse