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