math 112 lecture 9: the derivative function -- continued...the derivative is denoted in different...

Alternate Notations for the DerivativeEstimating the Derivative at a Point

Higher Derivatives

Lecture 9: The Derivative Function – Continued

Alternate Notations for the DerivativeLeibnitz NotationThe Derivative Operator

Estimating the Derivative at a PointExample 33 – Estimating a Rate of Change

Higher DerivativesThe Second DerivativeBeyond the Second DerivativeUses of the Second DerivativeExample 34 – Second Derivative of x3

Example 35 – Graphs of First & Second Derivative

Clint Lee Math 112 Lecture 9: The Derivative Function – Continued 1/15


Higher Derivatives

Leibnitz NotationThe Derivative Operator

Leibnitz Notation

The derivative is denoted in different ways, depending on the context, and,in some cases, on the taste of the person writing the expression. If y = f (x) isa function then some standard notations for the derivative are

f ′(x) =dydx

=ddx

(

f (x))

=dfdx

= y′(x) = y′



Higher Derivatives


Leibnitz Notation


f ′(x) =dydx

=ddx

(

f (x))

=dfdx

= y′(x) = y′

Thedydx

notation is sometimes called Leibnitz notation. One of the strengthsof the Leibnitz notation is that allows you to indicate both the independentand dependent variables directly. It can be thought of coming naturally fromthe formula for the derivative as a rate of change:

dydx

= lim∆x→0

∆y∆x



Higher Derivatives


Leibnitz Notation


f ′(x) =dydx

=ddx

(

f (x))

=dfdx

= y′(x) = y′

Thedydx

notation is sometimes called Leibnitz notation. One of the strengthsof the Leibnitz notation is that allows you to indicate both the independentand dependent variables directly. It can be thought of coming naturally fromthe formula for the derivative as a rate of change:

dydx

= lim∆x→0

∆y∆x

If you want to use Leibnitz notation to evaluate at derivative at a point, youdo it like this

f ′(a) =dydx

∣

∣

∣

∣

a=

dydx

∣

∣

∣

∣

x=a



Higher Derivatives


The Derivative Operator

The expressionddx

is an operator which instructs you to take the derivativeof, differentiate, whatever is to the right of it.



Higher Derivatives



The expressionddx

is an operator which instructs you to take the derivativeof, differentiate, whatever is to the right of it. Many of our derivativeformulas will be written using this operator.



Higher Derivatives



The expressionddx

is an operator which instructs you to take the derivativeof, differentiate, whatever is to the right of it. Many of our derivativeformulas will be written using this operator. So far we have found that

ddx

x2



Higher Derivatives



The expressionddx


ddx

x2 = 2x



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

x



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2

and we have guessed that

ddx

(

sin x)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2


ddx

(

sin x)

= cos x



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2


ddx

(

sin x)

= cos xddx

(ex)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2


ddx

(

sin x)

= cos xddx

(ex) = ex



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2


ddx

(

sin x)

= cos xddx

(ex) = exddx

(

ln x)



Higher Derivatives



The expressionddx


ddx

x2 = 2xddx

(

x3)

= 3x2

ddx

(√x)

=1

2√

xddx

(

3√

x)

=1

3 3√

x2


ddx

(

sin x)

= cos xddx

(ex) = exddx

(

ln x)

=1x



Higher DerivativesExample 33 – Estimating a Rate of Change

Estimating the Derivative at a Point

We have estimated the derivative of a function when we are given its graph.We did this in Example 25. When we did this we noted that we get the bestestimate of the derivative by straddling the point, which is equivalent toestimating the derivative by taking points on the left and the right andaveraging. This applies to data given in tabular form as well.




Example 33 – Estimating a Rate of Change

The table below shows the number of millilitres of blood pumped by theheart per beat as a function of heart rate. This is called the beat capacity ofthe heart, C = f (r), where r is the heart rate in beats per minute.

heart rate(beats/min)

50 60 70 80 90 100

beat capacity(mL/beat)

5.0 4.5 4.2 4.0 3.5 3.0

Estimate the value ofdCdr

∣

∣

∣

∣

70using the data in the table. Give the units of your

answer and explain in practical terms what it means.




Solution: Example 33

Straddling across the r = 70 means that we will use the values of C at r = 60and r = 80. This gives

dCdr

∣

∣

∣

∣

70






dCdr

∣

∣

∣

∣

70≈

4.0 − 4.580 − 60

= −0.0025






dCdr

∣

∣

∣

∣

70≈

4.0 − 4.580 − 60

= −0.0025

The units aremL/beatbeat/min






dCdr

∣

∣

∣

∣

70≈

4.0 − 4.580 − 60

= −0.0025

The units aremL/beatbeat/min

This means that if the heart rate increases by one beat/min from70 beats/min the beat capacity of the heart will go down by 0.0025 mL/beat.



Higher Derivatives

The Second DerivativeBeyond the Second DerivativeUses of the Second DerivativeExample 34 – Second Derivative of x3Example 35 – Graphs of First & Second Derivative

The Second Derivative

For a function f , the derivative f ′ is a function. So we can take its derivative.Let y = f (x). Then the derivative is

dydx

= f ′(x)



Higher Derivatives




dydx

= f ′(x)

Taking the derivative a second time gives

ddx

(

dydx

)



Higher Derivatives




dydx

= f ′(x)


ddx

(

dydx

)

=d2ydx2



Higher Derivatives




dydx

= f ′(x)


ddx

(

dydx

)

=d2ydx2

= f ′′(x)



Higher Derivatives




dydx

= f ′(x)


ddx

(

dydx

)

=d2ydx2

= f ′′(x)

This is the second derivative of y = f (x).



Higher Derivatives


Beyond the Second Derivative

We can continue to obtain higher (order) derivatives.



Higher Derivatives




ddx

(

d2ydx2

)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3

= f ′′′(x)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3

= f ′′′(x) third derivative



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4

= f ′′′′(x)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4

= f ′′′′(x) fourth derivative



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4


The nth derivative is

ddx

(

dn−1ydxn−1

)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4



ddx

(

dn−1ydxn−1

)

=dnydxn

= f (n)(x)



Higher Derivatives




ddx

(

d2ydx2

)

=d3ydx3


ddx

(

d3ydx3

)

=d4ydx4



ddx

(

dn−1ydxn−1

)

=dnydxn

= f (n)(x)

When you change from using primes to denote a higher derivative and usingthe number in parentheses is partly a matter of taste. Four primes is probablyas many as you can expect someone to count easily.



Higher Derivatives


Uses of the Second Derivative

If you have taken physics, you have already encountered the secondderivative.



Higher Derivatives



If you have taken physics, you have already encountered the secondderivative. If s = f (t) is the position of a moving object at time t, thenv = f ′(t) is the velocity of the body at time t.



Higher Derivatives



If you have taken physics, you have already encountered the secondderivative. If s = f (t) is the position of a moving object at time t, thenv = f ′(t) is the velocity of the body at time t. The derivative of the velocity isthe rate of change of the velocity with respect to the time t.



Higher Derivatives



If you have taken physics, you have already encountered the secondderivative. If s = f (t) is the position of a moving object at time t, thenv = f ′(t) is the velocity of the body at time t. The derivative of the velocity isthe rate of change of the velocity with respect to the time t. This is theacceleration of the body.



Higher Derivatives



If you have taken physics, you have already encountered the secondderivative. If s = f (t) is the position of a moving object at time t, thenv = f ′(t) is the velocity of the body at time t. The derivative of the velocity isthe rate of change of the velocity with respect to the time t. This is theacceleration of the body. Hence

d2sdt2

= f ′′(t)



Higher Derivatives




d2sdt2

= f ′′(t) = acceleration of the body



Higher Derivatives




d2sdt2


Another important use of the second derivative is in the description of the

graph of a function.



Higher Derivatives




d2sdt2



graph of a function. If y = f (x), thendydx

= f ′(x) gives the slope of the graph

of f .



Higher Derivatives




d2sdt2





of f . So thatd2ydx2

= f ′′(x) gives the rate of change of the slope of the graph.



Higher Derivatives




d2sdt2





of f . So thatd2ydx2

= f ′′(x) gives the rate of change of the slope of the graph.

This gives the curvature, or concavity, of the graph.



Higher Derivatives


Concavity and the Second Derivative

Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive. This means that the slope isincreasing, so its rate of change is positive.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive. This means that the slope isincreasing, so its rate of change is positive. Sothe second derivative is positive.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive. This means that the slope isincreasing, so its rate of change is positive. Sothe second derivative is positive. In this casethe graph is curving upward.



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive. This means that the slope isincreasing, so its rate of change is positive. Sothe second derivative is positive. In this casethe graph is curving upward. Another way tosee this is to note that the graph lies above anytangent line.

f ′(x) > 0f ′′(x) > 0



Higher Derivatives



Knowing the signs of the first and secondderivative of a function gives you a lot ofinformation about the graph. Consider thisgraph. The slope is positive, but it is gettingmore positive. This means that the slope isincreasing, so its rate of change is positive. Sothe second derivative is positive. In this casethe graph is curving upward. Another way tosee this is to note that the graph lies above anytangent line. We say the graph is concave up.

f ′(x) > 0f ′′(x) > 0



Higher Derivatives



For this graph



Higher Derivatives



For this graph the slope is positive butdecreasing since the graph is curvingdownward.



Higher Derivatives



For this graph the slope is positive butdecreasing since the graph is curvingdownward. So the second derivative isnegative and the graph lies below any tangentline.

f ′(x) > 0f ′′(x) < 0



Higher Derivatives



For this graph the slope is positive butdecreasing since the graph is curvingdownward. So the second derivative isnegative and the graph lies below any tangentline. The graph is concave down.

f ′(x) > 0f ′′(x) < 0



Higher Derivatives



Two more cases are:



Higher Derivatives



Two more cases are:

Negative slope getting more negative(decreasing).



Higher Derivatives



Two more cases are:

Negative slope getting more negative(decreasing). Concave down, graph belowany tangent line. f ′(x) < 0

f ′′(x) < 0



Higher Derivatives



Two more cases are:

Negative slope getting more negative(decreasing). Concave down, graph belowany tangent line.

Negative slope getting less negative(increasing).

f ′(x) < 0f ′′(x) < 0



Higher Derivatives



Two more cases are:

Negative slope getting more negative(decreasing). Concave down, graph belowany tangent line.

Negative slope getting less negative(increasing). Concave up, graph above anytangent line.

f ′(x) < 0f ′′(x) < 0

f ′(x) < 0f ′′(x) > 0



Higher Derivatives


Example 34 – Second Derivative of x3

In Example 28 we found the first derivative of f (x) = x3 to be f ′(x) = 3x2.

(a) Find f ′′(x).(b) Explain the relation between the graph of the function f and the signs

of the second derivative.



Higher Derivatives






Solution (a):



Higher Derivatives






Solution (a): Using Formula 3 gives



Higher Derivatives







f ′′(x) =



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h= 3 lim

h→0

x2 + 2xh + h2 − x2

h



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h= 3 lim

h→0

x2 + 2xh + h2 − x2

h

= 3 limh→0

2xh + h2

h



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h= 3 lim

h→0

x2 + 2xh + h2 − x2

h

= 3 limh→0

2xh + h2

h= 3 lim

h→0

h (2x + h)h



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h= 3 lim

h→0

x2 + 2xh + h2 − x2

h

= 3 limh→0

2xh + h2

h= 3 lim

h→0

h (2x + h)h

= 3 limh→0

(2x + h)



Higher Derivatives







f ′′(x) = limh→0

3(x + h)2 − 3x2

h= 3 lim

h→0

x2 + 2xh + h2 − x2

h

= 3 limh→0

2xh + h2

h= 3 lim

h→0

h (2x + h)h

= 3 limh→0

(2x + h) = 6x



Higher Derivatives


Solution: Example 34(b)

Recall that the graph of the function f (x) = x3 lookslike this.



Higher Derivatives



Recall that the graph of the function f (x) = x3 lookslike this. The second derivative f ′′(x) = 6x is positivefor x > 0 and negative for x < 0.



Higher Derivatives



Recall that the graph of the function f (x) = x3 lookslike this. The second derivative f ′′(x) = 6x is positivefor x > 0 and negative for x < 0. Thus, the graph of fis concave up for x > 0 and concave down for x < 0.



Higher Derivatives



The graphs of three functions are shown. Oneis the graph of a function f , and the other twoare the graphs of its first and secondderivatives. Determine which graph is which.

x

y(a)

(b)

(c)



Higher Derivatives




Solution:

x

y(a)

(b)

(c)



Higher Derivatives




Solution: Graph (a) is the derivative of graph(b),

x

y(a)

(b)

(c)



Higher Derivatives




Solution: Graph (a) is the derivative of graph(b), since where graph (a) crosses the x-axis,graph (b) has a high or low point,

(a)

(b)

(c)



Higher Derivatives




Solution: Graph (a) is the derivative of graph(b), since where graph (a) crosses the x-axis,graph (b) has a high or low point, and wheregraph (a) is above the x-axis graph (b) haspositive slope and negative slope elsewhere.

(a)

(b)

(c)



Higher Derivatives




Solution: Graph (a) is the derivative of graph(b), since where graph (a) crosses the x-axis,graph (b) has a high or low point, and wheregraph (a) is above the x-axis graph (b) haspositive slope and negative slope elsewhere.Graph (b) is the derivative of graph (c) forsimilar reasons.

(a)

(b)

(c)



Higher Derivatives




Solution: Graph (a) is the derivative of graph(b), since where graph (a) crosses the x-axis,graph (b) has a high or low point, and wheregraph (a) is above the x-axis graph (b) haspositive slope and negative slope elsewhere.Graph (b) is the derivative of graph (c) forsimilar reasons. So the final labeling looks likethis.

f ′′

f ′

f


Alternate Notations for the DerivativeLeibnitz NotationThe Derivative Operator

Estimating the Derivative at a PointExample 33 -- Estimating a Rate of Change

Higher DerivativesThe Second DerivativeBeyond the Second DerivativeUses of the Second DerivativeExample 34 -- Second Derivative of x3Example 35 -- Graphs of First & Second Derivative

math 112 lecture 9: the derivative function -- continued...the derivative is denoted in different...

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