Short-Cuts to Differentiation

3.9	Linear Approximation and the Derivative

The Tangent Line Approximation

When we zoom in on the graph of a differentiable function, it looks like a straight line. In fact, the graph is not exactly a straight line when we zoom in; however, its deviation from straightness is so small that it can't be detected by the naked eye. Let's examine what this means. The straight line that we think we see when we zoom in on the graph of

has slope equal to the derivative,

, so the equation is

The fact that the graph looks like a line means that

is a good approximation to

. (See Figure 3.40.) This suggests the following definition:

The Tangent Line Approximation

Suppose

is differentiable at

. Then, for values of

near

, the tangent line approximation to

The expression

is called the local linearization of

near

. We are thinking of

as fixed, so that

and

are constant.

The error,

, in the approximation is defined by

It can be shown that the tangent line approximation is the best linear approximation to

near

. See Problem 43.

Figure 3.40:

The tangent line approximation and its error

Calculus Applet: Linear Approximation

Example 1

What is the tangent line approximation for

near

Solution

The tangent line approximation of

near

, then

, so

and

, and the approximation is

This means that, near

, the function

is well approximated by the function

. If we zoom in on the graphs of the functions

and

near the origin, we won't be able to tell them apart. (See Figure 3.41.)

Figure 3.41:

Tangent line approximation to

Example 2

What is the local linearization of

near

Solution

, then

and, by the chain rule,

, so

. Thus

becomes

This is the tangent line approximation to

near

. In other words, if we zoom in on the functions

and

near the origin, we won't be able to tell them apart.

Estimating the Error in the Approximation

Let us look at the error,

, which is the difference between

and the local linearization. (Look back at Figure 3.40.) The fact that the graph of

looks like a line as we zoom in means that not only is

small for

near

, but also that

is small relative to

. To demonstrate this, we prove the following theorem about the ratio

Theorem 3.6:

Differentiability and Local Linearity

Suppose

is differentiable at

and

is the error in the tangent line approximation, that is:

Then

Proof

Using the definition of

, we have

Taking the limit as

and using the definition of the derivative, we see that

Theorem 3.6 says that

approaches 0 faster than

. For the function in Example 3, we see that

for constant

is near

Example 3

Let

be the error in the tangent line approximation to

for

near 2.

(a)	What does a table of values for suggest about ?

(b)	Make another table to see that . Estimate the value of . Check that a possible value is .

Solution

(a)

Since

, we have

, and

. Thus,

and

, so the tangent line approximation for

near 2 is

Thus,

The values of

in Table 3.7 suggest that

approaches 0 as

Table 3.7


2.1	0.61
2.01	0.0601
2.001	0.006001
2.0001	0.00060001

(b)

Notice that if

, then

. Thus we make Table 3.8 showing values of

. Since the values are all approximately 6, we guess that

and

Since

, our value of

satisfies

Table 3.8


2.1	6.1
2.01	6.01
2.001	6.001
2.0001	6.0001

The relationship between

and

that appears in Example 1 holds more generally. If

satisfies certain conditions, it can be shown that the error in the tangent line approximation behaves near

This is part of a general pattern for obtaining higher-order approximations called Taylor polynomials, which are studied in Chapter 10.

Why Differentiability Makes a Graph Look Straight

We use the properties of the error

to understand why differentiability makes a graph look straight when we zoom in.

Example 4

Consider the graph of

near

, and its linear approximation computed in Example 1. Show that there is an interval around 0 with the property that the distance from

to the linear approximation is less than

for all

in the interval.

Solution

The linear approximation of

near 0 is

, so we write

Since sin

is differentiable at

, Theorem 3.6 tells us that

If we take

, then the definition of limit guarantees that there is a

such that

In other words, for

in the interval

, we have

, so

(See Figure 3.42.)

Figure 3.42:

Graph of

and its linear approximation

, showing a window in which the magnitude of the error,

, is less than

for all

in the window

We can generalize from this example to explain why differentiability makes the graph of

look straight when viewed over a small graphing window. Suppose

is differentiable at

. Then we know

. So, for any

, we can find a δ small enough so that

So, for any

in the interval

, we have

Thus, the error,

, is less than ε times

, the distance between

and

. So, as we zoom in on the graph by choosing smaller ε, the deviation,

, of

from its tangent line shrinks, even relative to the scale on the

. So, zooming makes a differentiable function look straight.

Exercises and Problems for Section 3.9

Exercises

Find the tangent line approximation for

near

With

, the chain rule gives

, so

and

. Therefore the tangent line approximation of

near

becomes

This means that, near

, the function

can be approximated by its tangent line

. (See figure 3.5.)

Figure 3.5

2.	What is the tangent line approximation to near ?

3.	Find the tangent line approximation to near .

4.	Find the local linearization of near .

5.	What is the local linearization of near ? With , we get a tangent line approximation of which becomes . Thus, our local linearization of near is .

6.	Show that is the tangent line approximation to near .

7.	Show that near .

8.	Local linearization gives values too small for the function and too large for the function . Draw pictures to explain why.

Using a graph like Figure 3.41, estimate to one decimal place the magnitude of the error in approximating sin

for

. Is the approximation an over- or an underestimate?

From Figure 3.6, we see that the error has its maximum magnitude at the end points of the interval,

. The magnitude of the error can be read off the graph as less than 0.2 or estimated as

The approximation is an overestimate for

and an underestimate for

Figure 3.6

10.

For

near 0, local linearization gives

Using a graph, decide if the approximation is an over- or underestimate, and estimate to one decimal place the magnitude of the error for

Problems

11.

(a)	Find the best linear approximation, , to near .

(b)	What is the sign of the error, for near 0? Positive for

(c)

Find the true value of the function at

. What is the error? (Give decimal answers.) Illustrate with a graph.

(d)	Before doing any calculations, explain which you expect to be larger, or , and why. larger

(e)	Find . 0.005

12.

(a)	Find the tangent line approximation to at .

(b)	Use a graph to explain how you know whether the tangent line approximation is an under- or overestimate for .

(c)	To one decimal place, estimate the error in the approximation for .

13.

(a)

Graph

(b)	Find and add to your sketch the local linearization to at . Since , we have , so , and the local linearization is .

(c)

Mark on your sketch the true value of

, the tangent line approximation to

and the error in the approximation.

14.

(a)	Show that is the local linearization of near .

(b)	Someone claims that the square root of 1.1 is about 1.05. Without using a calculator, do you think that this estimate is about right?

(c)	Is the actual number above or below 1.05?

15.

Figure 3.43 shows

and its local linearization at

. What is the value of

? Of

? Is the approximation an under- or overestimate? Use the linearization to approximate the value of

Figure 3.43

In Problems 16-17, the equation has a solution near

. By replacing the left side of the equation by its linearization, find an approximate value for the solution.

16.

17.

18.

(a)	Given that and , estimate .

(b)	Suppose also for all . Does this make your answer to part (a) an under- or overestimate?

19.

(a)

Explain why the following equation has a solution near 0:

(b)	Replace by its linearization near 0. Solve the new equation to get an approximate solution to the original equation. 0.1

20.

The speed of sound in dry air is

where

is the temperature in

. Find a linear function that approximates the speed of sound for temperatures near

21.

Air pressure at sea level is 30 inches of mercury. At an altitude of

feet above sea level, the air pressure,

, in inches of mercury, is given by

(a)

Sketch a graph of

against

(b)

Find the equation of the tangent line at

Using the chain rule, we have

Hence, at

, the slope of the tangent line is

, so the equation of the tangent line is

(c)

A rule of thumb used by travelers is that air pressure drops about 1 inch for every 1000-foot increase in height above sea level. Write a formula for the air pressure given by this rule of thumb.

The rule of thumb says

But since the pressure at sea level is 30 inches of mercury, this drop in pressure is also

, so

giving

(d)

What is the relation between your answers to parts (b) and (c)? Explain why the rule of thumb works.

(e)

Are the predictions made by the rule of thumb too large or too small? Why?

22.

On October 7, 2010, the Wall Street Journal⁸ reported that Android cell phone users had increased to 10.9 million by the end of August 2010 from 866,000 a year earlier. During the same period, iPhone users increased to 13.5 million, up from 7.8 million a year earlier. Let

be the number of Android users, in millions, at time t in years since the end of August 2009. Let

be the number of iPhone users in millions.

(a)	Estimate . Give units.

(b)	Estimate . Give units.

(c)	Using the tangent line approximation, when are the numbers of Android and iPhone users predicted to be the same?

(d)	What assumptions did you make in part (c)?

23.

Writing

for the acceleration due to gravity, the period,

, of a pendulum of length

is given by

(a)

Show that if the length of the pendulum changes by

, the change in the period,

, is given by

(b)	If the length of the pendulum increases by 2%, by what percent does the period change? 1% increase

24.

Suppose now the length of the pendulum in Problem 23 remains constant, but that the acceleration due to gravity changes.

(a)	Use the method of the preceding problem to relate approximately to , the change in .

(b)	If increases by 1%, find the percent change in .

25.

Suppose

has a continuous positive second derivative for all

. Which is larger,

? Explain.

Since

has a positive second derivative, its graph is concave up, as in Figure 3.9 or 3.10. This means that the graph of

is above its tangent line. We see that in both cases

(The diagrams show

positive, but the result is also true if

is negative.)

Figure 3.9

Figure 3.10

26.

Suppose

is a differentiable decreasing function for all

. In each of the following pairs, which number is the larger? Give a reason for your answer.

(a)

and

(b)

and 0

(c)

and

Problems 27-29 investigate the motion of a projectile shot from a cannon. The fixed parameters are the acceleration of gravity,

, and the muzzle velocity,

, at which the projectile leaves the cannon. The angle

, in degrees, between the muzzle of the cannon and the ground can vary.

27.

The range of the projectile is

(a)	Find the range with .

(b)	Find a linear function of that approximates the range for angles near .

(c)	Find the range and its approximation from part (b) for . True: Approx:

28.

The time that the projectile stays in the air is

(a)	Find the time in the air for .

(b)	Find a linear function of that approximates the time in the air for angles near .

(c)	Find the time in air and its approximation from part (b) for .

29.

At its highest point the projectile reaches a peak altitude given by

(a)	Find the peak altitude for . The peak altitude is meters

(b)

Find a linear function of

that approximates the peak altitude for angles near

We have

Thus, for angles,

, near

, we have

(c)

Find the peak altitude and its approximation from part (b) for

In Problems 30-32, find the local linearization of

near 0 and use this to approximate the value of

30.

31.

32.

In Problems 33-37, find a formula for the error

in the tangent line approximation to the function near

. Using a table of values for

near

, find a value of

such that

. Check that, approximately,

and that

33.

We have

and

. Thus

Values of

near

are in Table 3.2.

Table 3.2

	1.1	1.01	1.001
	0.641	0.060401	0.006004

From the table, we can see that

and

In addition,

, so

The same result can be obtained by rewriting the function

using

and expanding:

Thus,

For

near 1, the value of

is small, so we ignore powers of

higher than the first, giving

34.

35.

36.

37.

We have

and

. Thus

Values of

near

are in Table 3.3.

Table 3.3

	1.1	1.01	1.001

From the table, we see that

and

In addition,

, so

38.	Multiply the local linearization of near by itself to obtain an approximation for . Compare this with the actual local linearization of . Explain why these two approximations are consistent, and discuss which one is more accurate.

39.

(a)	Show that is the local linearization of near .

(b)

From your answer to part (a), show that near

(c)	Without differentiating, what do you think the derivative of is at ? 0

40.	From the local linearizations of and near , write down the local linearization of the function . From this result, write down the derivative of at . Using this technique, write down the derivative of at .

41.

Use local linearization to derive the product rule,

Note that

We use the hint: For small

, and

. Therefore

Therefore

A more complete derivation can be given using the error term discussed in the section on Differentiability and Linear Approximation in Chapter 2. Adapting the notation of that section to this problem, we write

where

. (This implies that

We have

The terms

and

cancel out. All the remaining terms on the right, with the exception of the second and third terms, go to zero as

. Thus, we have

42.	Derive the chain rule using local linearization. [Hint: In other words, differentiate , using .]

43.

Consider a function

and a point

. Suppose there is a number

such that the linear function

is a good approximation to

. By good approximation, we mean that

where

is the approximation error defined by

Show that

is differentiable at

and that

. Thus the tangent line approximation is the only good linear approximation.

44.	Consider the graph of near . Find an interval around with the property that throughout any smaller interval, the graph of never differs from its local linearization at by more than .

Strengthen Your Understanding

In Problems 45-46, explain what is wrong with the statement.

45.

To approximate

, we can always use the linear approximation

46.	The linear approximation for near is an underestimate for the function for all , .

In Problems 47-49, give an example of:

47.	Two different functions that have the same linear approximation near .

48.	A non-polynomial function that has the tangent line approximation near .

49.	A function that does not have a linear approximation at . The linear approximation of a function , for values of near , is given by . Since does not have a derivative at this function does not have a linear approximation for near . Other answers are possible.

50.

Let

be a differentiable function and let

be the linear function

for some constant

. Decide whether the following statements are true or false for all constants

. Explain your answer.

(a)	is the local linearization for near ,

(b)	If , then is the local linearization for near .