Differentials: Linear and Quadratic Approximation

DIFFERENTIA
AND QUADRA
APPROXIMAT

By Kenny Abitbol, Sam Mote
Kirkland

ALS: LINEAR
ATIC
TION

e, and Evan

Introduction of Line
Approximation

= +

• : point whose tangent line as
• : point whose value is being a
• If () is concave up after =

underestimate, and if it is conca
overestimate

Linear approximation of =

at = 0.

ear

+ ’ −
an approximation of the function
approximated
, the approximation will be an
ave down, it will be an

= cos

Definition of Linea

“The equation of the tangent line
(, ()) is

= +
so…the tangent line at (, ()
curve = () when is near

= +
is called the linear approximatio
approximation of at .”

- Stewart, “Ca

ar Approximation

e to the curve = () at

’ – ,
)) [is] an approximation to the
.
+ ’ –
on or tangent line

alculus: Early Vectors”

Introduction of Qua
Approximation

= + ′ − + (12)

or
= + (12)′′()( − )2

Quadratic approximation of =
at = 0.

adratic

′′()( − )2

= ()

=

= ()

Definition of Quad
Approximation

The quadratic approximation a
approximate nearby values, bu
just a tangent line to do so.

This gives a closer
approximation because the
parabola stays closer to the
actual function.

dratic

also uses the point = to
ut uses a parabola instead of

for at
= 0.

History: Taylor’s Th

Linear and Quadratic approximations are
polynomials. The theorem is named afte
Taylor who designed a general formula f
functions after a small change of the x-v
1712. His theorem is:

≈ + ′ − + "()
2!

where a is a reference point, x is the nea
is the maximum amount of times the orig

heorem

e based off of Taylor’s theorem of
er 18th century mathematician Brook
for approximating the values of
value. The formula was first published in

) ( − )2+ ⋯ + − ,
!

arby value being approximated, and k
ginal function can be derived.

History: Differentials

• Began in 1680’s

• Bernouli brothers
• Gottfried Wilhelm Leibniz

• Applied to geometry and mec

ls

chanics

Producing Along St

A company requires that the b
have a volume in the range of
of the balls are produced to be
.01 , will the company be ab
standards?

tandards

bowling balls that it creates
4170 − 42003. If the radii

e 10 and have an error of
ble to produce along these

Error Calculation with
Approximation

= 4 3
3

= 42


= 4102


∆ =

h Linear

() = + ′ −

= + ′ −

− = ′ −

∆ = ′ ∆

∆


Error Calculation with
Approximation

Now using the formula ∆ =


= 400


∆ = 0.01

∆ = 400 .01 = 12.573

10 = 4 103 = 4188.793
3

h Linear

∆


3

Analysis

We found that the volume of th
would be 4188.793 with a d
creates a range of 4176.22 − 4
fit with the desired range of 41
will not be able to produce alo

he perfectly created ball
deviation of ±12.573. This
4201.263, which does not
170 − 42003. The company

ong these standards.

Surface Area Errors

In the design of a waterpark fo
below, the cost of paint is impo
$0.25/ℎ2, what would the
be if the radius of the circle ha
± .75 ℎ?What would be th

the radiu

s

ountain like the one shown
ortant. With the paint costing
e relative error in surface area
as an error of
he variation in price?

Take the inner radius to be 4,
us of the circle to be 37 ℎ,

and the height to be 2R.

Modifying the Linear A
Formula

= + ′

Adding up the surface areas:

Sphere: 22

Bottom of sphere: 2 − (4)2
(4)2+2
Cylinder: 4 2 (top

Surface Area = 42

Proposed Area = 4(37)2
= 17,203.36

Approximation

− → ∆ = ∆


2

of cylinder doesn’t matter)

ℎ2

Error in Surface Are

Surface Area Error : ∆ = .75

Now take the derivative of the

respect to the radius of the he

=


ea

ℎ

surface area formula, with
emisphere.(() = 42)

8,

Error in Surface Are

∆ = ∆ → 8 37 .75 =


= ∆ = 697.43
17203.36

∆ = 697.43 0.25 = $17

ea

697.43 ℎ2
= 0.0405

74.36

Analysis

The small error in the radius o
substantial change in cost.

The cost of painting the structu
0 = 17203.36 .25 = $43
∆ = $174.36

Cost = $ 4300.84 ± 174.36

Percent Error of R: .75 × 100
Percent Error of A: 37

697.43 ×
17203.36

of the sphere can cause a
ure: Cost = 0 ± ∆
300.84

= 2.03%
100 = 4.05%

Oscillation with Diffe

The water level in an artificial p
constant drainage, due to abso
a periodically active source, ov
given by the approximation

ℎ = 5sin

where ℎ is given in ft. This equ
measurement tests every 30 m
depth at many other times inde
that = 0 is midnight, = 1 is
would the depth have been at

ferentials

pond, which has both a
orption and evaporation, and
ver a given 24-hour period is

n + 25,
2

uation was the result of taking

minutes. That leaves the

efinite, but assumable. Given

s 1:00 A.M., and so on, what

11:03 A.M.?

Oscillation with Diffe

The first thing to do is identify
ℎ = 5sin 2 + 25,

= 11 + 3 11.05 .
60

And we know that the linear a

formulas are
= ℎ + ℎ′

= +

ferentials

given values.
= 11 ,

and quadratic approximation

−
ℎ"
+ 2 ( − )2

Oscillation with Diffe

Now we plug in values and so

ℎ = 5sin 2 + 25, ℎ ℎ′

11.05 = ℎ 11 + ℎ′ 11 11.

11.05 = 21.472 + 1.772 .0
11.05 = 21.5606

So according to the linear app
water’s depth was 21.5606 ft.

ferentials

olve where we can. So if

= 2.5cos 2 . Now :

.05 − 11 

05 

proximation, at 11:03, the

Oscillation with Diffe

Now to find the depth of the w
approximation, we need ℎ"()

ℎ’ = 2.5

then
ℎ"() = −1

ferentials

water according to a quadratic

). So if
5cos 2
,

1.25sin .
2

Oscillation with Diffe

Now:

11.05 = 11.05 + ℎ"(11) (1
2

11.05 = 21.5606 + .882 .0
2
11.05 = 21.5617

So now according to the quad
depth of the water was 21.561

ferentials

11.05 − 11)2 
05 2 

dratic approximation, the
17 ft.

Oscillation with Diffe

When = 11.05 is plugged int
depth is 21.5620 ft. To calculat
approximation it is

ℎ 11.05 −
ℎ(11.0

which is an error of 0.0065%. F
approximation error it is

ℎ 11.05 −
ℎ(11.0

which is an error of .0014%. So
have very low error, the Quadr
more accurate.

ferentials: Analysis

to the original equation, the
te error for the Linear

(11.05) ,
05)

For the Quadratic

(11.05) ,
05)

o while both approximations
ratic approximation is even

Visual Map

Differentials

Linear/Qu
Approxim

Method of Finite
Differences
(Create 1st order methods for
solving/approximating
solutions)

s Functions Relative
Errors
uadratic
mation

Determining
an Accurate
Inverse
Function

Common Mistakes

• Omission of 1 in equation fo
2

• ( − ) is used instead of (

approximation

• Chain rule forgotten when tak

• Plug ′() instead of ′() in

approximation

s

or quadratic approximation
− )2 in the quadratic

king derivatives
nto linear or quadratic

References

"Linear Approximation." Wikipedia.org. 6 Nov
<http://en.wikipedia.org/wiki/Linear_

Stewart, James. "Calculus: Early Vectors." 1s
Print.

"Taylor's Theorem." Wikipedia.org. 6 Novemb
<http://en.wikipedia.org/wiki/Taylor'

vember 2012. Web. 3 June 2012.
_approximation>.

st ed. Pacific Grove: Brooks/Cole, 1999.

ber 2012. Web. 5 November 2012.
's_theorem>.