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Lesson 9: The Derivative as a function
1. Section 2.8
The Derivative as a Function
Math 1a
October 15, 2007
Announcements
Midterm I review session 10/22, 7:30pm in Hall D
2.
3. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.
4.
5.
6. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Answer
1
f (x) = x −2/3 . The domain is all numbers except 0.
3
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.
7.
8. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Answer
1
f (x) = x −2/3 . The domain is all numbers except 0.
3
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.
Answer
2
f (x) = x −1/3 . The domain is all numbers except 0.
3
9. Super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
16. Notation
Newtonian notation
f (x) y (x) y
Leibnizian notation
dy d df
f (x)
dx dx dx
17.
18. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!
19. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!
Leibnizian notation:
d 2y d2 d 2f
f (x)
dx 2 dx 2 dx 2
