3.8 Local Linear 3.8 Local Linear Approximations;Approximations;

DifferentialsDifferentials(page 226)(page 226)

We have been interpreting We have been interpreting dy/dxdy/dx as a as a single entity representing the derivative of single entity representing the derivative of yy with respect to with respect to x.x.

We will now give the quantities We will now give the quantities dydy and and dx dx separate meanings that will allow us to separate meanings that will allow us to treat treat dy/dx dy/dx as a ratio. as a ratio.

dy dy terminology and the concept of terminology and the concept of differentials will also be used to differentials will also be used to approximate functions by simpler linear approximate functions by simpler linear functions.functions.

/ as a Ratiody dx

The ratio / can be interpreted as the slope of the secant line

joining the points , and , .

y x

x y x x y y

and Notationy dy

y f x x f x

'dyf x

dx 'dy f x dx

is the vertical change of the secant line. (rise)y

is the vertical change of the tangent line.dy

DifferentialsDifferentials(page 228)(page 228)

The symbols and are calleddifferentials.

dy dx

' is in differential form.dy f x dx

Historical NoteHistorical Note(See page 4)(See page 4)

Woolsthorpe, EnglandWoolsthorpe, England Not gifted as a youthNot gifted as a youth Entered Trinity College Entered Trinity College

with a deficiency in with a deficiency in geometry.geometry.

1665 to 1666 - 1665 to 1666 - discovered Calculus discovered Calculus

Calculus work not Calculus work not published until 1687published until 1687

Viewed value of work to Viewed value of work to be its support of the be its support of the existence of God.existence of God.

Isaac Newton (1642-1727)

Historical NoteHistorical Note(See page 5)(See page 5)

Leipzig, GermanyLeipzig, Germany Gifted GeniusGifted Genius Entered University of Entered University of

Altdorf at age 15Altdorf at age 15 Doctorate by age 20Doctorate by age 20 Developed Calculus Developed Calculus

in 1676in 1676 Developed the Developed the dy/dxdy/dx

notation we use notation we use todaytoday

Gottfried Wilhelm Leibniz(1646-1716

DifferentialsDifferentials(page 229)(page 229)

Differential NotationDifferential Notation(page 212)(page 212)

is fixed and is an

independent variable

that will be assigned

an arbitray value.

x dx

x dx

'dyf x

dx

Differential Form

Example Example

2This tells us that if we travel along the tangent to the curve at 3,

then a change of units in produces a change of 6 units in . If change

in is 4, then the change in along the

y x x

dx x dx y

x dx y

tangent is:

/ Not in this edition

Example 4Example 4(page 229)(page 229)

Example 2Example 2(page 229)(page 229)

Error CalculationError Calculation

Errory dy If error is positive then estimate is less than actual value.

If error is negative then estimate is greater than actual value.

Homework Example Homework Example #30, #36 page 233#30, #36 page 233

330. a) Let . Find and at 1 with 1

y x dy y xdx x

330. b) Sketch the graph of , showing and in the picture.

y xdy y

36. Find formulas for and at a general

point for sin .

dy dx

x y x

Local Linear Local Linear ApproximationApproximation

(page 226)(page 226)

If the graph of a function is magnified If the graph of a function is magnified at a point “P” that is differentiable, the at a point “P” that is differentiable, the function is said to be locally linear at function is said to be locally linear at “P”.“P”.

The tangent line through “P” closely The tangent line through “P” closely approximates the graph.approximates the graph.

A technique called “ local linear A technique called “ local linear approximation” is used to evaluate approximation” is used to evaluate function at a particular value. function at a particular value.

Locally Linear Function at Locally Linear Function at a Differentiable Point “P”a Differentiable Point “P”

(page 226)(page 226)

Function Not Locally Function Not Locally Linear at Point “P”Linear at Point “P”

Because Function Not Differentiable at Point “P”

Local Linear Local Linear Approximation FormulasApproximation Formulas

(page 227)(page 227)

Example 1Example 1(page 227)(page 227)

/ 1

Example 1Example 1(page 227)(page 227)

Error Propagation in Error Propagation in ApplicationsApplications

(page 230)(page 230)

In applications small errors occur in measured In applications small errors occur in measured quantities.quantities.

When these quantities are used in computations, When these quantities are used in computations, those errors are “propagated” throughout the those errors are “propagated” throughout the calculation process.calculation process.

To estimate the propagated error use the formula To estimate the propagated error use the formula below:below:

Example 6Example 6(page 231)(page 231)

/ 6

Homework Example - 20Homework Example - 20(page 233)(page 233)

320. Use an appropriate local linear approximation to estimate the value of 1.97 .