Download presentation
Presentation is loading. Please wait.
1
The Derivative (cont.) 3.1
2
A Differential Function is Continuous
If y = f(x) has a derivative at x = c, then f(x) is continuous at x = c. When the Derivative Fails to Exist The derivative fails to exist when The graph of the function has a corner. The graph of the function has a vertical tangent. The graph of the function has a break (discontinuity).
3
To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at: corner cusp discontinuity vertical tangent
4
Continuity does not imply Differentiability
Differentiability implies Continuity
5
Find the interval where the function is differentiable.
Limit does not exist, so the function is not continuous and not differentiable at x = 0 Possible point of discontinuity at x = 0
6
Using your calculator Graph |x| + 1 Zoom in on “corner”
Notice the corner does not change A differentiable curve will “straighten out”
7
Find at x = 2. Example: 1. Graph the function
2. Press 2nd TRACE to enter the CALC Menu 3. Select 6. dy/dx 4. Press 2 and ENTER 5. BE CAREFUL! The calculator gave an answer of The answer is 12! This should only be used as a check for your homework. You cannot use this method on a test/quiz!!
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.