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The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative
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Sections 2.1–2.2Derivatives and Rates of Changes
The Derivative as a Function
V63.0121, Calculus I
February 9–12, 2009
Announcements
I Quiz 2 is next week: Covers up through 1.6
I Midterm is March 4/5: Covers up to 2.4 (next T/W)
Outline
Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs
The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?
How can a function fail to be differentiable?
Other notations
The second derivative
The tangent problem
ProblemGiven a curve and a point on the curve, find the slope of the linetangent to the curve at that point.
Example
Find the slope of the line tangent to the curve y = x2 at the point(2, 4).
Upshot
If 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
The tangent problem
ProblemGiven a curve and a point on the curve, find the slope of the linetangent to the curve at that point.
Example
Find the slope of the line tangent to the curve y = x2 at the point(2, 4).
Upshot
If 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
Graphically and numerically
x
y
2
4
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
3
9
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
2.5
6.25
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
2.1
4.41
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
2.01
4.0401
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
1
1
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
1.5
2.25
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
1.9
3.61
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
Graphically and numerically
x
y
2
4
1.99
3.9601
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
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
3 5
2.5 4.25
2.1 4.1
2.01 4.01
limit 4
1.99 3.99
1.9 3.9
1.5 3.5
1 3
The tangent problem
ProblemGiven a curve and a point on the curve, find the slope of the linetangent to the curve at that point.
Example
Find the slope of the line tangent to the curve y = x2 at the point(2, 4).
Upshot
If 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
Velocity
ProblemGiven the position function of a moving object, find the velocity ofthe object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height canbe described by
h(t) = 50− 10t2
where t is seconds after dropping it and h is meters above theground. How fast is it falling one second after we drop it?
SolutionThe answer is
limt→1
(50− 10t2)− 40
t − 1= −20.
Velocity
ProblemGiven the position function of a moving object, find the velocity ofthe object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height canbe described by
h(t) = 50− 10t2
where t is seconds after dropping it and h is meters above theground. How fast is it falling one second after we drop it?
SolutionThe answer is
limt→1
(50− 10t2)− 40
t − 1= −20.
Numerical evidence
t vave =h(t)− h(1)
t − 12 −30
1.5 −25
1.1 −21
1.01 −20.01
1.001 −20.001
Velocity
ProblemGiven the position function of a moving object, find the velocity ofthe object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height canbe described by
h(t) = 50− 10t2
where t is seconds after dropping it and h is meters above theground. How fast is it falling one second after we drop it?
SolutionThe answer is
limt→1
(50− 10t2)− 40
t − 1= −20.
Upshot
If the height function is given by h(t), the instantaneous velocityat time t is given by
v = lim∆t→0
h(t + ∆t)− h(t)
∆t
Population growth
ProblemGiven the population function of a group of organisms, find therate of growth of the population at a particular instant.
Example
Suppose the population of fish in the East River is given by thefunction
P(t) =3et
1 + et
where t is in years since 2000 and P is in millions of fish. Is thefish population growing fastest in 1990, 2000, or 2010? (Estimatenumerically)?
SolutionThe estimated rates of growth are 0.000136, 0.75, and 0.000136.
Population growth
ProblemGiven the population function of a group of organisms, find therate of growth of the population at a particular instant.
Example
Suppose the population of fish in the East River is given by thefunction
P(t) =3et
1 + et
where t is in years since 2000 and P is in millions of fish. Is thefish population growing fastest in 1990, 2000, or 2010? (Estimatenumerically)?
SolutionThe estimated rates of growth are 0.000136, 0.75, and 0.000136.
Numerical evidence
r1990 ≈P(−10 + 0.1)− P(−10)
0.1≈ 0.000136
r2000 ≈P(0.1)− P(0)
0.1≈ 0.75
r2010 ≈P(10 + 0.1)− P(10)
0.1≈ 0.000136
Numerical evidence
r1990 ≈P(−10 + 0.1)− P(−10)
0.1≈ 0.000136
r2000 ≈P(0.1)− P(0)
0.1≈ 0.75
r2010 ≈P(10 + 0.1)− P(10)
0.1≈ 0.000136
Numerical evidence
r1990 ≈P(−10 + 0.1)− P(−10)
0.1≈ 0.000136
r2000 ≈P(0.1)− P(0)
0.1≈ 0.75
r2010 ≈P(10 + 0.1)− P(10)
0.1≈ 0.000136
Population growth
ProblemGiven the population function of a group of organisms, find therate of growth of the population at a particular instant.
Example
Suppose the population of fish in the East River is given by thefunction
P(t) =3et
1 + et
where t is in years since 2000 and P is in millions of fish. Is thefish population growing fastest in 1990, 2000, or 2010? (Estimatenumerically)?
SolutionThe estimated rates of growth are 0.000136, 0.75, and 0.000136.
Upshot
The instantaneous population growth is given by
lim∆t→0
P(t + ∆t)− P(t)
∆t
Marginal costs
ProblemGiven the production cost of a good, find the marginal cost ofproduction after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in ayear is
C (q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
Example
If q = 5, then C = 125, ∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
Marginal costs
ProblemGiven the production cost of a good, find the marginal cost ofproduction after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in ayear is
C (q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
Example
If q = 5, then C = 125, ∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
Comparisons
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
Marginal costs
ProblemGiven the production cost of a good, find the marginal cost ofproduction after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in ayear is
C (q) = q3 − 12q2 + 60q
We are currently producing 5 tons a year. Should we change that?
Example
If q = 5, then C = 125, ∆C = 19, while AC = 25. So we shouldproduce more to lower average costs.
Upshot
I The incremental cost
∆C = C (q + 1)− C (q)
is useful, but depends on units.
I The marginal cost after producing q given by
MC = lim∆q→0
C (q + ∆q)− C (q)
∆q
is more useful since it’s unit-independent.
Upshot
I The incremental cost
∆C = C (q + 1)− C (q)
is useful, but depends on units.
I The marginal cost after producing q given by
MC = lim∆q→0
C (q + ∆q)− C (q)
∆q
is more useful since it’s unit-independent.
Outline
Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs
The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?
How can a function fail to be differentiable?
Other notations
The second derivative
The definition
All of these rates of change are found the same way!
DefinitionLet f be a function and a a point in the domain of f . If the limit
f ′(a) = limh→0
f (a + h)− f (a)
h
exists, the function is said to be differentiable at a and f ′(a) isthe derivative of f at a.
The definition
All of these rates of change are found the same way!
DefinitionLet f be a function and a a point in the domain of f . If the limit
f ′(a) = limh→0
f (a + h)− f (a)
h
exists, the function is said to be differentiable at a and f ′(a) isthe derivative of f at a.
Derivative of the squaring function
Example
Suppose f (x) = x2. Use the definition of derivative to find f ′(a).
Solution
f ′(a) = limh→0
f (a + h)− f (a)
h= lim
h→0
(a + h)2 − a2
h
= limh→0
(a2 + 2ah + h2)− a2
h= lim
h→0
2ah + h2
h
= limh→0
(2a + h) = 2a.
Derivative of the squaring function
Example
Suppose f (x) = x2. Use the definition of derivative to find f ′(a).
Solution
f ′(a) = limh→0
f (a + h)− f (a)
h= lim
h→0
(a + h)2 − a2
h
= limh→0
(a2 + 2ah + h2)− a2
h= lim
h→0
2ah + h2
h
= limh→0
(2a + h) = 2a.
What does f tell you about f ′?
I If f is a function, we can compute the derivative f ′(x) at eachpoint x where f is differentiable, and come up with anotherfunction, the derivative function.
I What can we say about this function f ′?I If f is decreasing on an interval, f ′ is negative (well,
nonpositive) on that intervalI If f is increasing on an interval, f ′ is positive (well,
nonnegative) on that interval
Outline
Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs
The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?
How can a function fail to be differentiable?
Other notations
The second derivative
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous 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
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous 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
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
Differentiability is super-continuity
TheoremIf f is differentiable at a, then f is continuous 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
Note the proper use of the limit law: if the factors each have alimit at a, the limit of the product is the product of the limits.
How can a function fail to be differentiable?Kinks
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Kinks
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Kinks
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Cusps
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Cusps
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Cusps
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Vertical Tangents
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Vertical Tangents
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Vertical Tangents
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Weird, Wild, Stuff
x
f (x)
x
f ′(x)
How can a function fail to be differentiable?Weird, Wild, Stuff
x
f (x)
x
f ′(x)
Outline
Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs
The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?
How can a function fail to be differentiable?
Other notations
The second derivative
Notation
I Newtonian notation
f ′(x) y ′(x) y ′
I Leibnizian notation
dy
dx
d
dxf (x)
df
dx
These all mean the same thing.
Meet the Mathematician: Isaac Newton
I English, 1643–1727
I Professor at Cambridge(England)
I Philosophiae NaturalisPrincipia Mathematicapublished 1687
Meet the Mathematician: Gottfried Leibniz
I German, 1646–1716
I Eminent philosopher aswell as mathematician
I Contemporarily disgracedby the calculus prioritydispute
Outline
Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs
The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?
How can a function fail to be differentiable?
Other notations
The second derivative
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!
Leibniziannotation:
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! Leibniziannotation:
d2y
dx2
d2
dx2f (x)
d2f
dx2
function, derivative, second derivative
x
y
f (x) = x2
f ′(x) = 2x
f ′′(x) = 2