3020 Differentials and Linear Approximation AB Calculus

3020 Differentials and Linear Approximation

AB Calculus

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 ‘s new value?

IF…

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 earth’s radius 1 inch, how much larger would the circumference be ?

2𝜋 h𝑖𝑛𝑐 𝑒𝑠

2𝜋 h𝑖𝑛𝑐 𝑒𝑠

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

∆ 𝑥=4

3




houry

x yx

3

4

y m x

y

y


∆ 𝑥=3

94




houry

x yx

3

4

y m x

y

y


∆ 𝑥=2

64




houry

x yx

3

4

y m x

y

y


∆ 𝑥=32

98

Differentials and Linear Approximation in the News

Algebra to CalculusThe DIFFERENTIAL: “How much has y changed?”

“the first difference in y for a fixed change in x ”

( )

( )

dyf x

dxdy f x dx

dy:

( )dy f x dx

Notation:

Also written: df


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

𝑑𝑦𝑑𝑥

=cos 𝑥𝑑𝑦=cos (𝑥 )𝑑𝑥

𝑑𝑦=cos (𝜋6

)𝜋36

𝑑𝑦=√32∗ 𝜋

36=𝜋 √3

72≈ .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

𝑑𝑉𝑑 𝑟

=4𝜋𝑟 2 𝑑𝑉=4𝜋𝑟 2(𝑑𝑟 )

𝑑𝑉=4𝜋 42( 116 )=4𝜋≈ 12.566

43𝜋𝑟 3= 4

3𝜋 ( 65

16 )3

−43𝜋 ( 4 )3280.8463 − 268.0826=12.764

B: Linearization

“Make It Linear!”

Linearization:

Linearization:

y – y1 = m (x – x1 )

y = y1 + m (x – 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 – a)

Linearization

Find the Linearization of sin ( ) at 3

y x x

𝑦=sin (𝑥) 𝑦=√32

𝑦 ′=cos (𝑥) 𝑦 ′=12

L (𝑥 )=𝑦+𝑦 ′(𝑥−𝑎)

L (𝑥 )=√32

+ 12 (𝑥−

𝜋3 )𝑥−

Linearization

Find the Linearization of 2 1 at 5y x x

𝑦=√2 𝑥−1 𝑦 (5 )=3

𝑦 ′=12

(2𝑥−1 )−12 ∗2

𝑦 ′ (5 )=13

𝑦 ′=1

√2𝑥− 1

𝐿 (𝑥 )=3+13

(𝑥− 5 )

C: Tangent Line Approximation

What is the new value?

y2 – y1 = m ( x2 – x1 )

y2 = y1 + m (Δx)

givenfrie

ndly

#1 find friendly #

#2

#3 y

#4 y’

#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

#

𝑎=12𝜋36

𝑜𝑟𝜋3

∆ 𝑥 𝑔𝑖𝑣𝑒𝑛− 𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦𝑦=cos (𝑥 )𝑦 (𝜋3 )=1

2

𝑦 ′=−sin (𝑥)𝑦 ′( 𝜋3 )=−√3

2

∆ 𝑥=−𝜋36

𝑓 ( 12𝜋36

+(− 𝜋36 ))≈ cos (𝑥 )+¿

≈12+ −√3

2 (− 𝜋36 )

≈12+ √3𝜋

72

𝑐𝑜𝑠( 11𝜋36

≈12+ √3𝜋

72 )

Linear Approximation - Tangent Line Approximation

.

( ) ( )f a x f a f a dx

EXAMPLE: Approximate 16.5Wants the VALUE!

II. Error

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

A. Error in Measurement Tools

0 1 2

Choose either 1 or 2

.00,000,1 .00,000,2

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:

• the volume of a cube• the surface area of a cube• 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%.

B. Error in Approximation Formulas

ERROR: Approximation Formulas

For Linear Approximation:

The Error Bound formula is

Error = (actual value – approximation) either Pos. or Neg.

Error Bound = | actual – 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 ” (x))

x

Example 5: Approximation

For Linear Approximation:

The Error Bound formula is

Error = (actual value – approximation) either Pos. or Neg.

Error Bound = | actual – approximation |

21( )

2LE f x x

EX: Find the Error in the linear approximation of 16.5

Example 6: Approximation

21( )

2LE f x x

EX: Find the Error in the linear approximation of 11

cos36

Last Update:

• 11/04/11

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 :

Algebra to Calculus

x

y

( )

.

y f x

yALG m

x

y m x

How much has y changed?

Algebra to Calculus

x

y

( ) for ( ) linear

. ( )

( )

y f x f x

y dyCal m f x

x dx

y f x x

How much has y changed?

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3020 Differentials and Linear Approximation AB Calculus