2

1 0.20.2 .00

1 02

0 10f

Use the Linear Approximation to estimate How accurate is your estimate?

1

10 0.2&

f xx

a x

0.00196 0.002

0.00004To 5 decimal places!

1 1

10.2.

10

Differential Notation The Linear Approximation to y = f (x) is often written using the “differentials” dx and dy. In this notation, dx is used instead of Δx to represent the change in x, and dy is the corresponding vertical change in the tangent line:

'y dy f a dx

This is simply another way of writing Δf ≈ f (a)Δx. '

2/31 0.1

8 0.13 3 4

0.00833f

Differential Notation How much larger is 3 38.1 than 8 2.

1/3

8 0 1& .

f x x

a x

0.008

'dy f a dx

5 4

12

4 4

1.7 10 12 2.04 10 cm/ C

2.04 10 3 6.12 10

L

dLKL

dT

dLL dT

dT

Thermal Expansion A thin metal cable has length L = 12 cm when the temperature is T = 21°C. Estimate the change in length when T rises to 24°C, assuming that

where k = 1.7 × 10−5°C (k is called the coefficient of thermal expansion).

, 3dL

L dL dT dTdT

dLKL

dT

cm

Differentials

Suppose that we measure the diameter D of a circle and use this result to compute the area of the circle. If our measurement of D is inexact, the area computation will also be inexact. What is the effect of the measurement error on the resulting area computation? This can be estimated using the Linear Approximation, as in the next example.

2

' 50 50 1.

3

22

0 94.248 cm

A dx

2

2

2 4

DA D D

Effect of an Inexact Measurement The Bonzo Pizza Company claims that its pizzas are circular with diameter 50 cm

(a) What is the area of the pizza?(b) Estimate the quantity of pizza lost or gained if the diameter is off by at most 1.2 cm.

262550 cmA

'y dy f a dx

Area in terms of diameter

Approximating f(x) by Linearization

If f is differentiable at x = a and x is closed to a, then

'f x L x f a x a f a

Compute the linearization of at a = 1.

1'

2

1 1

1' 1

2

f xx

f

f

'f x L x f a x a f a

11 1

2L x x

f x x

2 14

0.02 2 0.02 14

Calculator tan 0.02

'

1.04

0

1.04084

0.0008Percentage Error 100

1..0

4%

877

0 0

L x x

L

f x L x f a x a f a

2' sec

14

' 24

f x x

f

f

Estimate and compute the percentage error.

errorPercentage error 100 percent

actual value

Linear Approximation

tan 0.024

tanf x x

The Size of the Error

The examples in this section may have convinced you that the Linear Approximation yields a good approximation to Δf when Δx is small, but if we want to rely on the Linear Approximation, we need to know more about the size of the error:

Remember that the error E is simply the vertical gap between the graph and the tangent line.

where K is the maximum valueof |f (x)| on the interval froma to a + Δx.

"

'E Error f f a x

The Size of the ErrorThe Error Bound tells us two important things. First, it says that the error is small when the second derivative (and hence K) is small. This makes sense, because f (x) measures how quickly the tangent lines change direction. When |f (x)| is smaller, the graph is flatter and the Linear Approximation is more accurate over a larger interval around x = a (compare the graphs).

""

The Size of the ErrorSecond, the Error Bound tells us that the error is of order two in Δx, meaning that E is no larger than a constant times (Δx)2. So if Δx is small, say Δx = 10−n, then E has substantially smaller order of magnitude (Δx)2 = 10−2n. In particular, E/Δx tends to zero (because E/Δx < KΔx), so the Error Bound tells us that the graph becomes nearly indistinguishable from its tangent line as we zoom in on the graph around x = a. This is a precise version of the “local linearity” property discussed in Section 3.2

4.1 SUMMARY Let Δf = f (a + Δx) − f (a). The Linear Approximation is the estimate

Differential notation: dx is the change in x, dy = f (a)dx, Δy = f (a + dx) − f (a). In this notation, the Linear Approximation reads

'

The linearization of f (x) at x = a is the function

The Linear Approximation is equivalent to the approximation

The error in the Linear Approximation is the quantity

In many cases, the percentage error is more important than the error itself:

'E Error f f a x

errorPercentage error 100 percent

actual value