lesson 9: the derivative as a function
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Section 2.8The Derivative as a Function
Math 1a
October 15, 2007
Announcements
I Midterm I review session 10/22, 7:30pm in Hall D
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
Page4of14
Last time: Worksheet problems 3 and 4
ProblemLet f (x) = x1/3. Find f ′(x) and its domain.
Answer
f ′(x) =1
3x−2/3. The domain is all numbers except 0.
ProblemLet f (x) = x2/3. Find f ′(x) and its domain.
Answer
f ′(x) =2
3x−1/3. The domain is all numbers except 0.
Super-continuity
TheoremIf f is differentiable at a, then f is continuous at a.
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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How can a function fail to be continuous?
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Notation
I Newtonian notation
f ′(x) y ′(x) y ′
I Leibnizian notation
dy
dx
d
dxf (x)
df
dx
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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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:
d2y
dx2
d2
dx2f (x)
d2f
dx2
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:
d2y
dx2
d2
dx2f (x)
d2f
dx2
Worksheet #1
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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Worksheet #2
Math 1a - October 15, 2007.GWBMonday, Oct 15, 2007
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