lesson 7: the derivative (slides)
DESCRIPTION
Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.TRANSCRIPT
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.... NYUMathematics
Sec on 2.1–2.2The Deriva ve
V63.0121.011: Calculus IProfessor Ma hew Leingang
New York University
February 14, 2011
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Announcements
I Quiz this week onSec ons 1.1–1.4
I No class Monday,February 21
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ObjectivesThe Derivative
I Understand and state the defini on ofthe deriva ve of a func on at a point.
I Given a func on and a point in itsdomain, decide if the func on isdifferen able at the point and find thevalue of the deriva ve at that point.
I Understand and give several examplesof deriva ves modeling rates of changein science.
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ObjectivesThe Derivative as a Function
I Given a func on f, use the defini on ofthe deriva ve to find the deriva vefunc on f’.
I Given a func on, find its secondderiva ve.
I Given the graph of a func on, sketchthe graph of its deriva ve.
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OutlineRates of Change
Tangent LinesVelocityPopula on growthMarginal costs
The deriva ve, definedDeriva ves of (some) power func onsWhat does f tell you about f′?
How can a func on fail to be differen able?Other nota onsThe second deriva ve
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The tangent problemA geometric rate of change
ProblemGiven a curve and a point on the curve, find the slope of the linetangent to the curve at that point.
Solu onIf the curve is given by y = f(x), and the point on the curve is(a, f(a)), then the slope of the tangent line is given by
mtangent = limx→a
f(x)− f(a)x− a
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A tangent problem
Example
Find the slope of the line tangent to the curve y = x2 at the point(2, 4).
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Graphically and numerically
.. x.
y
..2
..
4
.
x m =x2 − 22
x− 2
3 52.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....3
..
9
x m =x2 − 22
x− 23
52.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....3
..
9
x m =x2 − 22
x− 23 5
2.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.5
..
6.25
x m =x2 − 22
x− 23 52.5
4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.5
..
6.25
x m =x2 − 22
x− 23 52.5 4.5
2.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.1
..
4.41
x m =x2 − 22
x− 23 52.5 4.52.1
4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.1
..
4.41
x m =x2 − 22
x− 23 52.5 4.52.1 4.1
2.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.01
..
4.0401
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01
4.01limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....2.01
..
4.0401
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....1
..1
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.991.9 3.91.5 3.5
1
3
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Graphically and numerically
.. x.
y
..2
..
4
....1
..1
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.991.9 3.91.5 3.5
1 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.5
..
2.25
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.991.9 3.9
1.5
3.5
1 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.5
..
2.25
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.991.9 3.9
1.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.9
..
3.61
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.99
1.9
3.9
1.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.9
..
3.61
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 41.99 3.99
1.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.99
..
3.9601
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 4
1.99
3.99
1.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....1.99
..
3.9601
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 4
1.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
.
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01
limit 4
1.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....3
..
9
...2.5
..
6.25
...2.1
..
4.41
...2.01
..
4.0401
...1
..1 ...1.5
..
2.25
...1.9
..
3.61
...1.99
..
3.9601
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01limit
4
1.99 3.991.9 3.91.5 3.51 3
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Graphically and numerically
.. x.
y
..2
..
4
....3
..
9
...2.5
..
6.25
...2.1
..
4.41
...2.01
..
4.0401
...1
..1 ...1.5
..
2.25
...1.9
..
3.61
...1.99
..
3.9601
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
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The tangent problemA geometric rate of change
ProblemGiven a curve and a point on the curve, find the slope of the linetangent to the curve at that point.
Solu onIf the curve is given by y = f(x), and the point on the curve is(a, f(a)), then the slope of the tangent line is given by
mtangent = limx→a
f(x)− f(a)x− a
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The velocity problemKinematics—Physical rates of change
ProblemGiven the posi on func on of a moving object, find the velocity ofthe object at a certain instant in me.
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A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 15
1.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5
− 12.51.1 − 10.51.01 − 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.5
1.1 − 10.51.01 − 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1
− 10.51.01 − 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1 − 10.5
1.01 − 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01
− 10.051.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.05
1.001 − 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001
− 10.005
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Numerical evidenceh(t) = 50− 5t2
Fill in the table:
t vave =h(t)− h(1)
t− 12 − 151.5 − 12.51.1 − 10.51.01 − 10.051.001 − 10.005
![Page 40: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/40.jpg)
A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
![Page 41: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/41.jpg)
A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
![Page 42: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/42.jpg)
A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
![Page 43: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/43.jpg)
A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
![Page 44: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/44.jpg)
A velocity problemExample
Drop a ball off the roof of theSilver Center so that its height canbe described by
h(t) = 50− 5t2
where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
Solu onThe answer is
v = limt→1
(50− 5t2)− 45t− 1
= limt→1
5− 5t2
t− 1
= limt→1
5(1− t)(1+ t)t− 1
= (−5) limt→1
(1+ t)
= −5 · 2 = −10
![Page 45: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/45.jpg)
Velocity in generalUpshot
If the height func on is givenby h(t), the instantaneousvelocity at me t0 is given by
v = limt→t0
h(t)− h(t0)t− t0
= lim∆t→0
h(t0 +∆t)− h(t0)∆t ... t..
y = h(t)
....t0
..t
..
h(t0)
..
h(t0 +∆t)
. ∆t.
∆h
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Population growthBiological Rates of Change
ProblemGiven the popula on func on of a group of organisms, find the rateof growth of the popula on at a par cular instant.
Solu onThe instantaneous popula on growth is given by
lim∆t→0
P(t+∆t)− P(t)∆t
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Population growth exampleExample
Suppose the popula on of fish in the East River is given by thefunc on
P(t) =3et
1+ etwhere t is in years since 2000 and P is in millions of fish. Is the fishpopula on growing fastest in 1990, 2000, or 2010? (Es matenumerically)
AnswerWe es mate the rates of growth to be 0.000143229, 0.749376, and0.0001296. So the popula on is growing fastest in 2000.
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DerivationSolu onLet∆t be an increment in me and∆P the corresponding change inpopula on:
∆P = P(t+∆t)− P(t)
This depends on∆t, so ideally we would want
lim∆t→0
∆P∆t
= lim∆t→0
1∆t
(3et+∆t
1+ et+∆t −3et
1+ et
)
But rather than compute a complicated limit analy cally, let usapproximate numerically. We will try a small∆t, for instance 0.1.
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DerivationSolu onLet∆t be an increment in me and∆P the corresponding change inpopula on:
∆P = P(t+∆t)− P(t)
This depends on∆t, so ideally we would want
lim∆t→0
∆P∆t
= lim∆t→0
1∆t
(3et+∆t
1+ et+∆t −3et
1+ et
)But rather than compute a complicated limit analy cally, let usapproximate numerically. We will try a small∆t, for instance 0.1.
![Page 50: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/50.jpg)
Numerical evidenceSolu on (Con nued)
To approximate the popula on change in year n, use the difference
quo entP(t+∆t)− P(t)
∆t, where∆t = 0.1 and t = n− 2000.
r1990
≈ P(−10+ 0.1)− P(−10)0.1
=10.1
(3e−9.9
1+ e−9.9 −3e−10
1+ e−10
)= 0.000143229
r2000
≈ P(0.1)− P(0)0.1
=10.1
(3e0.1
1+ e0.1− 3e0
1+ e0
)= 0.749376
![Page 51: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/51.jpg)
Numerical evidenceSolu on (Con nued)
To approximate the popula on change in year n, use the difference
quo entP(t+∆t)− P(t)
∆t, where∆t = 0.1 and t = n− 2000.
r1990 ≈P(−10+ 0.1)− P(−10)
0.1
=10.1
(3e−9.9
1+ e−9.9 −3e−10
1+ e−10
)= 0.000143229
r2000 ≈P(0.1)− P(0)
0.1
=10.1
(3e0.1
1+ e0.1− 3e0
1+ e0
)= 0.749376
![Page 52: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/52.jpg)
Numerical evidenceSolu on (Con nued)
To approximate the popula on change in year n, use the difference
quo entP(t+∆t)− P(t)
∆t, where∆t = 0.1 and t = n− 2000.
r1990 ≈P(−10+ 0.1)− P(−10)
0.1=
10.1
(3e−9.9
1+ e−9.9 −3e−10
1+ e−10
)
= 0.000143229
r2000 ≈P(0.1)− P(0)
0.1=
10.1
(3e0.1
1+ e0.1− 3e0
1+ e0
)
= 0.749376
![Page 53: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/53.jpg)
Numerical evidenceSolu on (Con nued)
To approximate the popula on change in year n, use the difference
quo entP(t+∆t)− P(t)
∆t, where∆t = 0.1 and t = n− 2000.
r1990 ≈P(−10+ 0.1)− P(−10)
0.1=
10.1
(3e−9.9
1+ e−9.9 −3e−10
1+ e−10
)= 0.000143229
r2000 ≈P(0.1)− P(0)
0.1=
10.1
(3e0.1
1+ e0.1− 3e0
1+ e0
)
= 0.749376
![Page 54: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/54.jpg)
Numerical evidenceSolu on (Con nued)
To approximate the popula on change in year n, use the difference
quo entP(t+∆t)− P(t)
∆t, where∆t = 0.1 and t = n− 2000.
r1990 ≈P(−10+ 0.1)− P(−10)
0.1=
10.1
(3e−9.9
1+ e−9.9 −3e−10
1+ e−10
)= 0.000143229
r2000 ≈P(0.1)− P(0)
0.1=
10.1
(3e0.1
1+ e0.1− 3e0
1+ e0
)= 0.749376
![Page 55: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/55.jpg)
Solu on (Con nued)
r2010
≈ P(10+ 0.1)− P(10)0.1
=10.1
(3e10.1
1+ e10.1− 3e10
1+ e10
)= 0.0001296
![Page 56: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/56.jpg)
Solu on (Con nued)
r2010 ≈P(10+ 0.1)− P(10)
0.1
=10.1
(3e10.1
1+ e10.1− 3e10
1+ e10
)= 0.0001296
![Page 57: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/57.jpg)
Solu on (Con nued)
r2010 ≈P(10+ 0.1)− P(10)
0.1=
10.1
(3e10.1
1+ e10.1− 3e10
1+ e10
)
= 0.0001296
![Page 58: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/58.jpg)
Solu on (Con nued)
r2010 ≈P(10+ 0.1)− P(10)
0.1=
10.1
(3e10.1
1+ e10.1− 3e10
1+ e10
)= 0.0001296
![Page 59: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/59.jpg)
Population growth exampleExample
Suppose the popula on of fish in the East River is given by thefunc on
P(t) =3et
1+ etwhere t is in years since 2000 and P is in millions of fish. Is the fishpopula on growing fastest in 1990, 2000, or 2010? (Es matenumerically)
AnswerWe es mate the rates of growth to be 0.000143229, 0.749376, and0.0001296. So the popula on is growing fastest in 2000.
![Page 60: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/60.jpg)
Population growthBiological Rates of Change
ProblemGiven the popula on func on of a group of organisms, find the rateof growth of the popula on at a par cular instant.
Solu onThe instantaneous popula on growth is given by
lim∆t→0
P(t+∆t)− P(t)∆t
![Page 61: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/61.jpg)
Marginal costsRates of change in economics
ProblemGiven the produc on cost of a good, find the marginal cost ofproduc on a er having produced a certain quan ty.
Solu onThe marginal cost a er producing q is given by
MC = lim∆q→0
C(q+∆q)− C(q)∆q
![Page 62: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/62.jpg)
Marginal Cost ExampleExample
Suppose the cost of producing q tons of rice on our paddy in a year is
C(q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
AnswerIf q = 5, then C = 125,∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
![Page 63: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/63.jpg)
Marginal Cost ExampleExample
Suppose the cost of producing q tons of rice on our paddy in a year is
C(q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
AnswerIf q = 5, then C = 125,∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
![Page 64: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/64.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q)
AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)
4
112 28 13
5
125 25 19
6
144 24 31
![Page 65: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/65.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q)
AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)
4 112
28 13
5
125 25 19
6
144 24 31
![Page 66: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/66.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q)
AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)
4 112
28 13
5 125
25 19
6
144 24 31
![Page 67: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/67.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q)
AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)
4 112
28 13
5 125
25 19
6 144
24 31
![Page 68: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/68.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q
∆C = C(q+ 1)− C(q)
4 112
28 13
5 125
25 19
6 144
24 31
![Page 69: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/69.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q
∆C = C(q+ 1)− C(q)
4 112 28
13
5 125
25 19
6 144
24 31
![Page 70: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/70.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q
∆C = C(q+ 1)− C(q)
4 112 28
13
5 125 25
19
6 144
24 31
![Page 71: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/71.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q
∆C = C(q+ 1)− C(q)
4 112 28
13
5 125 25
19
6 144 24
31
![Page 72: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/72.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)4 112 28
13
5 125 25
19
6 144 24
31
![Page 73: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/73.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)4 112 28 135 125 25
19
6 144 24
31
![Page 74: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/74.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)4 112 28 135 125 25 196 144 24
31
![Page 75: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/75.jpg)
ComparisonsSolu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q ∆C = C(q+ 1)− C(q)4 112 28 135 125 25 196 144 24 31
![Page 76: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/76.jpg)
Marginal Cost ExampleExample
Suppose the cost of producing q tons of rice on our paddy in a year is
C(q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
AnswerIf q = 5, then C = 125,∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
![Page 77: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/77.jpg)
Marginal costsRates of change in economics
ProblemGiven the produc on cost of a good, find the marginal cost ofproduc on a er having produced a certain quan ty.
Solu onThe marginal cost a er producing q is given by
MC = lim∆q→0
C(q+∆q)− C(q)∆q
![Page 78: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/78.jpg)
OutlineRates of Change
Tangent LinesVelocityPopula on growthMarginal costs
The deriva ve, definedDeriva ves of (some) power func onsWhat does f tell you about f′?
How can a func on fail to be differen able?Other nota onsThe second deriva ve
![Page 79: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/79.jpg)
The definitionAll of these rates of change are found the same way!
Defini onLet f be a func on and a a point in the domain of f. If the limit
f′(a) = limh→0
f(a+ h)− f(a)h
= limx→a
f(x)− f(a)x− a
exists, the func on is said to be differen able at a and f′(a) is thederiva ve of f at a.
![Page 80: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/80.jpg)
The definitionAll of these rates of change are found the same way!
Defini onLet f be a func on and a a point in the domain of f. If the limit
f′(a) = limh→0
f(a+ h)− f(a)h
= limx→a
f(x)− f(a)x− a
exists, the func on is said to be differen able at a and f′(a) is thederiva ve of f at a.
![Page 81: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/81.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h= lim
h→0(2a+ h) = 2a
![Page 82: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/82.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h= lim
h→0(2a+ h) = 2a
![Page 83: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/83.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h= lim
h→0(2a+ h) = 2a
![Page 84: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/84.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h
= limh→0
2ah+ h2
h= lim
h→0(2a+ h) = 2a
![Page 85: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/85.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h
= limh→0
(2a+ h) = 2a
![Page 86: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/86.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h= lim
h→0(2a+ h)
= 2a
![Page 87: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/87.jpg)
Derivative of the squaring functionExample
Suppose f(x) = x2. Use the defini on of deriva ve to find f′(a).
Solu on
f′(a) = limh→0
f(a+ h)− f(a)h
= limh→0
(a+ h)2 − a2
h
= limh→0
(a2 + 2ah+ h2)− a2
h= lim
h→0
2ah+ h2
h= lim
h→0(2a+ h) = 2a
![Page 88: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/88.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 89: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/89.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 90: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/90.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 91: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/91.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 92: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/92.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 93: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/93.jpg)
“Can you do it the other way?”Same limit, different form
Solu on
f′(2) = limh→0
f(2+ h)− f(2)h
= limh→0
12+h −
12
h
= limh→0
2− (2+ h)2h(2+ h)
= limh→0
−h2h(2+ h)
= limh→0
−12(2+ h)
= −14
![Page 94: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/94.jpg)
“Can you do it the other way?”Same limit, different form
Solu on
f′(2) = limh→0
f(2+ h)− f(2)h
= limh→0
12+h −
12
h
= limh→0
2− (2+ h)2h(2+ h)
= limh→0
−h2h(2+ h)
= limh→0
−12(2+ h)
= −14
![Page 95: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/95.jpg)
“Can you do it the other way?”Same limit, different form
Solu on
f′(2) = limh→0
f(2+ h)− f(2)h
= limh→0
12+h −
12
h
= limh→0
2− (2+ h)2h(2+ h)
= limh→0
−h2h(2+ h)
= limh→0
−12(2+ h)
= −14
![Page 96: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/96.jpg)
“Can you do it the other way?”Same limit, different form
Solu on
f′(2) = limh→0
f(2+ h)− f(2)h
= limh→0
12+h −
12
h
= limh→0
2− (2+ h)2h(2+ h)
= limh→0
−h2h(2+ h)
= limh→0
−12(2+ h)
= −14
![Page 97: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/97.jpg)
“Can you do it the other way?”Same limit, different form
Solu on
f′(2) = limh→0
f(2+ h)− f(2)h
= limh→0
12+h −
12
h
= limh→0
2− (2+ h)2h(2+ h)
= limh→0
−h2h(2+ h)
= limh→0
−12(2+ h)
= −14
![Page 98: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/98.jpg)
“How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions
Fact
ab± c
d=
ad± bcbd
1x− 1
2x− 2
=
2− x2x
x− 2=
2− x2x(x− 2)
Paul Sally
![Page 99: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/99.jpg)
“How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions
Fact
ab± c
d=
ad± bcbd
1x− 1
2x− 2
=
2− x2x
x− 2=
2− x2x(x− 2) Paul Sally
![Page 100: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/100.jpg)
What does f tell you about f′?
I If f is a func on, we can compute the deriva ve f′(x) at eachpoint x where f is differen able, and come up with anotherfunc on, the deriva ve func on.
I What can we say about this func on f′?
I If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve)on that interval
I If f is increasing on an interval, f′ is posi ve (technically, nonnega ve)on that interval
![Page 101: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/101.jpg)
What does f tell you about f′?
I If f is a func on, we can compute the deriva ve f′(x) at eachpoint x where f is differen able, and come up with anotherfunc on, the deriva ve func on.
I What can we say about this func on f′?I If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve)
on that interval
I If f is increasing on an interval, f′ is posi ve (technically, nonnega ve)on that interval
![Page 102: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/102.jpg)
Derivative of the reciprocalExample
Suppose f(x) =1x. Use the defini on of the deriva ve to find f′(2).
Solu on
f′(2) = limx→2
1/x− 1/2x− 2
= limx→2
2− x2x(x− 2)
= limx→2
−12x
= −14
..x
.
y
.
![Page 103: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/103.jpg)
What does f tell you about f′?
I If f is a func on, we can compute the deriva ve f′(x) at eachpoint x where f is differen able, and come up with anotherfunc on, the deriva ve func on.
I What can we say about this func on f′?I If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve)
on that intervalI If f is increasing on an interval, f′ is posi ve (technically, nonnega ve)
on that interval
![Page 104: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/104.jpg)
Graphically and numerically
.. x.
y
..2
..
4
....3
..
9
...2.5
..
6.25
...2.1
..
4.41
...2.01
..
4.0401
...1
..1 ...1.5
..
2.25
...1.9
..
3.61
...1.99
..
3.9601
x m =x2 − 22
x− 23 52.5 4.52.1 4.12.01 4.01limit 41.99 3.991.9 3.91.5 3.51 3
![Page 105: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/105.jpg)
What does f tell you about f′?FactIf f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b).
Picture Proof.
If f is decreasing, then all secant linespoint downward, hence havenega ve slope. The deriva ve is alimit of slopes of secant lines, whichare all nega ve, so the limit must be≤ 0. ..
x.
y
......
![Page 106: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/106.jpg)
What does f tell you about f′?FactIf f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b).
Picture Proof.
If f is decreasing, then all secant linespoint downward, hence havenega ve slope. The deriva ve is alimit of slopes of secant lines, whichare all nega ve, so the limit must be≤ 0. ..
x.
y
......
![Page 107: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/107.jpg)
What does f tell you about f′?FactIf f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b).
The Real Proof.
I If∆x > 0, then
f(x+∆x) < f(x) =⇒ f(x+∆x)− f(x)∆x
< 0
![Page 108: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/108.jpg)
What does f tell you about f′?FactIf f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b).
The Real Proof.
I If∆x > 0, then
f(x+∆x) < f(x) =⇒ f(x+∆x)− f(x)∆x
< 0
I If∆x < 0, then x+∆x < x, and
f(x+∆x) > f(x) =⇒ f(x+∆x)− f(x)∆x
< 0
![Page 109: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/109.jpg)
What does f tell you about f′?FactIf f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b).
The Real Proof.
I Either way,f(x+∆x)− f(x)
∆x< 0, so
f′(x) = lim∆x→0
f(x+∆x)− f(x)∆x
≤ 0
![Page 110: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/110.jpg)
Going the Other Way?
Ques on
If a func on has a nega ve deriva ve on an interval, must it bedecreasing on that interval?
AnswerMaybe.
![Page 111: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/111.jpg)
Going the Other Way?
Ques on
If a func on has a nega ve deriva ve on an interval, must it bedecreasing on that interval?
AnswerMaybe.
![Page 112: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/112.jpg)
OutlineRates of Change
Tangent LinesVelocityPopula on growthMarginal costs
The deriva ve, definedDeriva ves of (some) power func onsWhat does f tell you about f′?
How can a func on fail to be differen able?Other nota onsThe second deriva ve
![Page 113: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/113.jpg)
Differentiability is super-continuityTheoremIf f is differen able at a, then f is con nuous at a.
Proof.We have
limx→a
(f(x)− f(a)) = limx→a
f(x)− f(a)x− a
· (x− a)
= limx→a
f(x)− f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
![Page 114: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/114.jpg)
Differentiability is super-continuityTheoremIf f is differen able at a, then f is con nuous at a.
Proof.We have
limx→a
(f(x)− f(a)) = limx→a
f(x)− f(a)x− a
· (x− a)
= limx→a
f(x)− f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
![Page 115: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/115.jpg)
Differentiability is super-continuityTheoremIf f is differen able at a, then f is con nuous at a.
Proof.We have
limx→a
(f(x)− f(a)) = limx→a
f(x)− f(a)x− a
· (x− a)
= limx→a
f(x)− f(a)x− a
· limx→a
(x− a)
= f′(a) · 0
= 0
![Page 116: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/116.jpg)
Differentiability is super-continuityTheoremIf f is differen able at a, then f is con nuous at a.
Proof.We have
limx→a
(f(x)− f(a)) = limx→a
f(x)− f(a)x− a
· (x− a)
= limx→a
f(x)− f(a)x− a
· limx→a
(x− a)
= f′(a) · 0 = 0
![Page 117: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/117.jpg)
Differentiability FAILKinks
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
.
.
![Page 118: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/118.jpg)
Differentiability FAILKinks
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
.
.
![Page 119: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/119.jpg)
Differentiability FAILKinks
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
..
![Page 120: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/120.jpg)
Differentiability FAILCusps
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 121: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/121.jpg)
Differentiability FAILCusps
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 122: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/122.jpg)
Differentiability FAILCusps
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 123: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/123.jpg)
Differentiability FAILCusps
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 124: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/124.jpg)
Differentiability FAILVertical Tangents
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 125: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/125.jpg)
Differentiability FAILVertical Tangents
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 126: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/126.jpg)
Differentiability FAILVertical Tangents
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 127: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/127.jpg)
Differentiability FAILVertical Tangents
Example
Let f have the graph on the le -hand side below. Sketch the graph ofthe deriva ve f′.
.. x.
f(x)
.. x.
f′(x)
![Page 128: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/128.jpg)
Differentiability FAILWeird, Wild, Stuff
Example
.. x.
f(x)
This func on is differen ableat 0.
.. x.
f′(x)
.
But the deriva ve is notcon nuous at 0!
![Page 129: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/129.jpg)
Differentiability FAILWeird, Wild, Stuff
Example
.. x.
f(x)
This func on is differen ableat 0.
.. x.
f′(x)
.
But the deriva ve is notcon nuous at 0!
![Page 130: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/130.jpg)
Differentiability FAILWeird, Wild, Stuff
Example
.. x.
f(x)
This func on is differen ableat 0.
.. x.
f′(x)
.
But the deriva ve is notcon nuous at 0!
![Page 131: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/131.jpg)
Differentiability FAILWeird, Wild, Stuff
Example
.. x.
f(x)
This func on is differen ableat 0.
.. x.
f′(x)
.
But the deriva ve is notcon nuous at 0!
![Page 132: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/132.jpg)
OutlineRates of Change
Tangent LinesVelocityPopula on growthMarginal costs
The deriva ve, definedDeriva ves of (some) power func onsWhat does f tell you about f′?
How can a func on fail to be differen able?Other nota onsThe second deriva ve
![Page 133: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/133.jpg)
Notation
I Newtonian nota on
f′(x) y′(x) y′
I Leibnizian nota on
dydx
ddx
f(x)dfdx
These all mean the same thing.
![Page 134: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/134.jpg)
Meet the MathematicianIsaac Newton
I English, 1643–1727I Professor at Cambridge(England)
I Philosophiae NaturalisPrincipia Mathema capublished 1687
![Page 135: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/135.jpg)
Meet the MathematicianGottfried Leibniz
I German, 1646–1716I Eminent philosopher aswell as mathema cian
I Contemporarily disgracedby the calculus prioritydispute
![Page 136: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/136.jpg)
OutlineRates of Change
Tangent LinesVelocityPopula on growthMarginal costs
The deriva ve, definedDeriva ves of (some) power func onsWhat does f tell you about f′?
How can a func on fail to be differen able?Other nota onsThe second deriva ve
![Page 137: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/137.jpg)
The second derivative
If f is a func on, so is f′, and we can seek its deriva ve.
f′′ = (f′)′
It measures the rate of change of the rate of change!
Leibniziannota on:
d2ydx2
d2
dx2f(x)
d2fdx2
![Page 138: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/138.jpg)
The second derivative
If f is a func on, so is f′, and we can seek its deriva ve.
f′′ = (f′)′
It measures the rate of change of the rate of change! Leibniziannota on:
d2ydx2
d2
dx2f(x)
d2fdx2
![Page 139: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/139.jpg)
Function, derivative, second derivative
.. x.
y
.
f(x) = x2
.
f′(x) = 2x
.f′′(x) = 2
![Page 140: Lesson 7: The Derivative (slides)](https://reader034.vdocuments.site/reader034/viewer/2022051514/5492601dac7959ff2d8b45d6/html5/thumbnails/140.jpg)
SummaryWhat have we learned today?
I The deriva ve measures instantaneous rate of changeI The deriva ve has many interpreta ons: slope of the tangentline, velocity, marginal quan es, etc.
I The deriva ve reflects the monotonicity (increasing-ness ordecreasing-ness) of the graph