Related Rates :How fast is y changing as x is changing?-
Differentials:How much does y change as x changes?
I. Approximation
A. Differentials
Goal: Answer Two Questions - How much has y changed?
and - What is y βs new value?
IFβ¦
My waist size is 36 inches
IF I increases my radius 1 inch, how much larger would my belt need to be ?
The earths circumference is 24,367.0070904 miles.
IF I increases the earthβs radius 1 inch, how much larger would the circumference be ?
2π hπππ ππ
2π hπππ ππ
The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM: mileshours miles
houry
x yx
3
4
y m x
y
y
NOTE: Finds the change in y -- NOT the value of y
β π₯=4
3
The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM: mileshours miles
houry
x yx
3
4
y m x
y
y
NOTE: Finds the change in y -- NOT the value of y
β π₯=3
94
The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM: mileshours miles
houry
x yx
3
4
y m x
y
y
NOTE: Finds the change in y -- NOT the value of y
β π₯=2
64
The Change in Value : The Differential
The Differential finds a QUANTITY OF CHANGE !
REM: mileshours miles
houry
x yx
3
4
y m x
y
y
NOTE: Finds the change in y -- NOT the value of y
β π₯=32
98
Algebra to CalculusThe DIFFERENTIAL: βHow much has y changed?β
βthe first difference in y for a fixed change in x β
( )
( )
dyf x
dxdy f x dx
dy:
( )dy f x dx
Notation:
Also written: df
The Differential finds a QUANTITY OF CHANGE !
In Calculus dy approximates the change in y using
the TANGENT LINE.
( )
34
4
dy f x dx
dy
dy
y
ydy
x dx
NOTE: APPROXIMATES the change in y
3
3
1
3434
The smaller the the better the approximation
The Differential Function: Example 1
Find the Differential Function and use it to approximate change.
( ) 1 sin( )f x x
A). Find the differential function.
B). Approximate the change in y
at with
6x
36x
ππ¦ππ₯
=cos π₯ππ¦=cos (π₯ )ππ₯
ππ¦=cos (π6
)π36
ππ¦=β32β π
36=π β3
72β .0756
The Differential Function: Example 2
Find the Differential Function and use it to approximate the volume of latex in a spherical balloon with inside radius
and thickness 34( )
3V r r
A). Find the differential function.
B). Approximate the change in V.
C). Find the actual Volume.
1.
16in
4 .r in
πππ π
=4ππ 2 ππ=4ππ 2(ππ )
ππ=4π 42( 116 )=4πβ 12.566
43ππ 3= 4
3π ( 65
16 )3
β43π ( 4 )3280.8463 β 268.0826=12.764
Linearization:
Linearization:
y β y1 = m (x β x1 )
y = y1 + m (x β x1 )
The standard linear approximation of f at a
The point x = a is the center of the approximation
L(x) = f(a) + f / (a) (x β a)
Linearization
Find the Linearization of sin ( ) at 3
y x x
π¦=sin (π₯) π¦=β32
π¦ β²=cos (π₯) π¦ β²=12
L (π₯ )=π¦+π¦ β²(π₯βπ)
L (π₯ )=β32
+ 12 (π₯β
π3 )π₯β
Linearization
Find the Linearization of 2 1 at 5y x x
π¦=β2 π₯β1 π¦ (5 )=3
π¦ β²=12
(2π₯β1 )β12 β2
π¦ β² (5 )=13
π¦ β²=1
β2π₯β 1
πΏ (π₯ )=3+13
(π₯β 5 )
C: Tangent Line Approximation
What is the new value?
y2 β y1 = m ( x2 β x1 )
y2 = y1 + m (Ξx)
givenfrie
ndly
#1 find friendly #
#2
#3 y
#4 yβ
#5 values
π6
=6π36
π4
=9π36
π3
=12π36
Linear Approximation - Tangent Line Approximation
.
( ) ( )f a x f a f a dx
EXAMPLE: 11
Approximate cos36
Wants the VALUE!
Find
frie
ndly
#
π=12π36
πππ3
β π₯ πππ£ππβ ππππππππ¦π¦=cos (π₯ )π¦ (π3 )=1
2
π¦ β²=βsin (π₯)π¦ β²( π3 )=ββ3
2
β π₯=βπ36
π ( 12π36
+(β π36 ))β cos (π₯ )+ΒΏ
β12+ ββ3
2 (β π36 )
β12+ β3π
72
πππ ( 11π36
β12+ β3π
72 )
Linear Approximation - Tangent Line Approximation
.
( ) ( )f a x f a f a dx
EXAMPLE: Approximate 16.5Wants the VALUE!
ERROR:
There are TWO types of error:
A. Error in measurement tools- quantity of error
- relative error
- percent error
B. Error in approximation formulas- over or under approximation
- Error Bound - formula
y dydy
y
(100)dy
y
EXAMPLE 1: Measurement (A)Volume and Surface Area: The measurement of the edge of a cube is found to be 12 inches, with a possible error of 0.03 inches. Use differentials to approximate the maximum possible error in computing:
β’ the volume of a cubeβ’ the surface area of a cubeβ’ find the range of possible measurements in parts (a) and (b).
EXAMPLE 2: Measurement (A)
Volume and Surface Area: the radius of a sphere is claimed to be 6 inches, with a possible error of .02 inch
Use differentials to approximate the maximum possible error in calculating the volume of the sphere.
Use differentials to approximate the maximum possible error in calculating the surface area.
Determine the relative error and percent error in each of the above.
EXAMPLE 3: Measurement (B) : Tolerance
Area: The measurement of a side of a square is found to be15 centimeters.
Estimate the maximum allowable percentage error in measuring the side if the error in computing the area cannot exceed 2.5%.
relative error da
a
EXAMPLE 4: Measurement (B) : Tolerance
Circumference The measurement of the circumference of a circle is found to be 56 centimeters.
Estimate the maximum allowable percentage error in measuring the circumference if the error in computing the area cannot exceed 3%.
ERROR: Approximation Formulas
For Linear Approximation:
The Error Bound formula is
Error = (actual value β approximation) either Pos. or Neg.
Error Bound = | actual β approximation |
21( )
2LE f x x
Since the approximation uses the TANGENT LINE
the over or under approximation is determined by the
CONCAVITY (2ndDerivative Test)
In Calculus dy approximates the change in y using the
TANGENT LINE.
y
dy
x dx
The ERROR depends on distance from center( ) and the bend in the curve ( f β (x))
x
Example 5: Approximation
For Linear Approximation:
The Error Bound formula is
Error = (actual value β approximation) either Pos. or Neg.
Error Bound = | actual β approximation |
21( )
2LE f x x
EX: Find the Error in the linear approximation of 16.5
New Value : Tangent line Approximation
y
2 1
( )
( ) ( )
y dy
dy f x dx
y y dy
f a x f a f a dx
In words: _____________________________________________
1y
2y
2 1 ( )
( )
y y m x
f a x f a y
x
With the differential :
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