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Chapter 8: Functions of Several Variables
Section 8.4Differentials
Written by Richard GillAssociate Professor of Mathematics
Tidewater Community College, Norfolk Campus, Norfolk, VA
With Assistance from a VCCS LearningWare Grant
To begin this section on differentials in three variables, we will begin with a review of differentials in two variables. Consider the function y = f(x).Now consider a generic value of x with a tangent to the curve at (x, f(x)).Compare the initial value of x to a value of x that is slightly larger.
))(,( xxfxx
xx x
))(,( xfx
).(' xfdxdy
The slope of the tangent line is:
It was at this point that we first saw dx defined as .x
xdx
Since and since dx isbeing defined as “run” then dy becomes “rise” by definition.
dxdy
runrise
y
dy
For small values of dx, the rise of the tangent line, was used as an approximation for , the change in the function.y
dxxfdy )('
z
x y),( yx
)),(,,( yxfyx
Now consider the extension of the differential concept to functions of several variables. Consider an input (x,y) for this function. Its outputs z = f(x,y) create a set of points (x,y,z) that form the surface you see below.
z = f(x,y)
The differential dz will have two parts: one part generated by a change in x and the other part generated by a change in y.
Consider a new point in the domain generated by a small change in x.
x
),( yxx
Now consider the functional image of the new point.
)),(,,( yxxfyxx
We can use the function to calculate , the difference between the two z-coordinates.
z
By subtracting the z-coordinates: zyxfyxxf ),(),(
z
x y),( yx
)),(,,( yxfyx
Now consider the extension of the differential concept to functions of several variables. Consider an input (x,y) for this function and its output (x,y,z).
z = f(x,y)
The differential dz will have two parts: one part generated by a change in x and the other part generated by a change in y.
x
),( yxx
)),(,,( yxxfyxx
By subtracting the z-coordinates: zyxfyxxf ),(),(
Since the y-coordinate is constant, we can use that cross section to draw the tangent to )),(,,( yxfyx
Remember that, dz is the change in the height of the tangent line, and can be used to estimate the change in z.
dz
z
x y
)),(,,( yxfyx
z = f(x,y)
x
),( yxx
)),(,,( yxxfyxx
Since there has been no change in y we can express the differential so far in terms of the change in x:
dxxzdz
),( yx
Now track the influence on z when a new point is generated by a change in the y direction.
y
),( yyxx
)),(,,( yyxxfyyxx
Now that the change in z is generated by changes in x and y, we can define the total differential:
dyyxfdxyxfdz
ordyyzdx
xzdz
yx ),(),(
z
x y
)),(,,( yxfyx
z = f(x,y)
x
),( yxx
)),(,,( yxxfyxx
),( yx
y
),( yyxx
)),(,,( yyxxfyyxx dyydxxdz
dyyzdx
xzdz
46
z.for aldifferenti total theFind .23 :equation by the
generated is with ngbeen worki have graph we theSuppose 1. Example22 yxz
Try this on your own first.
Example 2.
hundredth.nearest the toRoundz. it to compare and dz aldifferenti total thecalculate
),08.1,1.2( and )1,2( evaluate ,cos),( If
ffxyyxf
Hint: first calculate dx and dy.Solution: dx = 2.1 – 2 = 0.1 and dy = 1.08 – 1 = 0.08
129.0))1.2(cos(08.1))1.2(cos(08.1
)1,2()08.1,1.2(416.0))1.2(cos(08.1)1,2(
545.0))1.2(cos(08.1)08.1,1.2(124.0.....0332.0....0909.0
)08.0(2cos)1.0(2sin18.0,1.0,)1,2(),(
cossin
cos
ffzff
dzdydxyxdyxdxxy
dyyzdx
xzdz
xyz
zfor ion approximat good a is dz
Definition of Differentiability
).0,0(),( as 0 and where
),(),( form in the expressedbecan if b)(a,at abledifferenti isfunction the,),(For
21
21
yx
yxybafxbafzzyxfz
yx
z. eapproximat todz use weerror when the
be to and consider can you ,),(),( Since 21
yxdyyxfdxyxfdz yx
The following theorem is presented without proof though you can usually find the proof in the appendix of a standard Calculus textbook.
b).(a,at abledifferenti is then b)(a,at continuous are and b)(a, containingregion a
inexist and sderivative partial theIf
f
ff yx
),(),(limb)(a,y)(x,
bafyxf
Theorem: If a function of x and y is differentiable at (a,b) then it is continuous at (a,b).
Solution: the objective is to show that
).,(),( as 0 and 0 and,, where
)),(()),(( definitionby Then,b).(a,at abledifferenti be ),(Let
21
21
bayxybyxax
ybafxbafzyxfz
yx
proof. thecompleteswhich ),(),(),(),(0),(),(
0. togoes expression above the,),(),( aslimit theTaking))(),(())(),((
)),(()),((),(),( above, From
.z that know also We
221
221
bayxasyxfbafyxfbafbayx
ybbafxabafybafxbafyxfbafz
,y)f(a,b)-f(x
x
x
Example 3. A right circular cylinder has a height of 5 ft. and a radius of 2 ft. These measurements have possible errors in accuracy as laid out in the table below. Complete the table below. Comment on the relationship between dV and for the indicated errors.V
0.1 ft. 0.1 ft.
0.01 ft. 0.01 ft.
0.001 ft. 0.001 ft.
r h dV VSolution:
dhrdrrhdV
dhhVdr
rVdV
hrhrfV
2
2
2
),(
3
2
3
2
3
2
0.0750.024
)001.0)(2()001.0)(5)(2(2001.0
754.24.004.2.0
)01.0)(2()01.0)(5)(2(201.0
540.74.24.02
)1.0)(2()1.0)(5)(2(21.0
ft
dVhr
ft
dVhr
ft
dVhr
7.54 cu ft.754 cu ft.075 cu ft
Example 3. A right circular cylinder has a height of 5 ft. and a radius of 2 ft. These measurements have possible errors in accuracy as laid out in the table below. Complete the table below. Comment on the relationship between dV and for the indicated errors.V
0.1 ft. 0.1 ft.
0.01 ft. 0.01 ft.
0.001 ft. 0.001 ft.
r h dV VSolution:
dhrdrrhdV
dhhVdr
rVdV
hrhrfV
2
2
2
),(
7.54 cu ft.754 cu ft.075 cu ft
075.0)5()2()001.5()001.2()5,2()001.5,001.2(
757.0)5()2()01.5()01.2()5,2()01.5,01.2(
826.7)5()2()1.5()1.2()5,2()1.5,1.2(
),(),(
22
22
22
ffV
ffV
ff
hrfhhrrfV
7.826 cu ft
0.757 cu ft
0.075 cu ft
smaller.get and asbetter gets Vfor ofion approximat The hrdV
Example 4. A right circular cylinder is constructed with a height of 40 cm. and a radius of 25 cm. What is the relative error and the percent error in the surface area if the possible error in the measurement of each dimension is ½ cm.Solution: If the measurements are correct the surface area will be
20508001250)40)(25(2)25(2
222
2
rhrA The total differential will generate an estimate for the possible error.
115)5(.50)5)(.80100()5)(.25(2)5))(.40(2)25(4(
2)24(
22 2
dhrdrhrdhhAdr
rAdA
rhrA
5.6% iserror percent The
056.02050115
AdA iserror relative The
This concludes the material for Lesson 8.4. This also concludes the lessons on Blackboard and the material for the semester.
As was the case in previous Bb lessons, we have posted three sets of exercises on Bb. Any exercises worked correctly will add to your thinkwell exercise totals.
The practice exam should be up and running on Bb. Good luck preparing for the exam and congratulations on getting to the end of one of the toughest courses in the curriculum.