Difference Equations to Differential Equations: Section 3.1 Best Affine Approximations
Difference Equations
to
Differential Equations
Section 3.1
Best Affine Approximations
We are now in a position to discuss the two central problems of calculus as mentioned in
Section 1.1. In this chapter we will take up the problem of finding tangent lines; in Chapter
4 we will consider the problem of finding areas. We choose this order only because the
work we do in solving the tangent line problem in this chapter will be of use, through the
Fundamental Theorem of Calculus, in solving area problems in the next.
We begin with some preliminary notation and terminology. If f is a function with
domain contained in the set A and range contained √
in the set B, then we may denote this
fact by writing f : A → B. For example, if g(t) = 1 − t2 and R denotes the set of real
numbers, then the statements g : R → R, g : [−1, 1] → R, and g : [−1, 1] → [0, 1] are all
correct. We will work exclusively with functions of the form f : R → R until Chapter 7,
where we will introduce functions of the form f : R → C and f : C → C, where C denotes
the set of complex numbers.
We call a function f : R → R linear if there is a constant m such that f (x) = mx for
all values of x. Graphically, linear functions are functions whose graphs are straight lines
passing through the origin. We call a function f : R → R affine if there are constants
m and b such that f (x) = mx + b for all values of x. Graphically, affine functions are
functions whose graphs are straight lines, not necessarily passing through the origin. Put
another way, an affine function is a first degree polynomial. Thus f (x) = 3x is both linear
and affine, whereas g(t) = 4t − 6 is affine but not linear.
The problem of finding the tangent line for the graph of a given function f at a point
(x0 , y0 ) is really the problem of finding the affine function T which best approximates
f for points close to x0 . In this section we will discuss how to solve this problem. In
the remaining sections of this chapter we will consider techniques for finding best affine
approximations and discuss some applications. In Chapter 5 we will see how to improve
upon affine approximations by using higher degree polynomials.
The following example should help to make these ideas more concrete.
√
Example Consider the problem of approximating the function f (x) = x for values of
x close to 1. For a first approximation, we might say that
x≈1
for x close to 1. In other words, if we let
T (x) = 1
for all x, then we are saying that the affine function T is a good approximation for f when
x is close to 1. Two facts characterize this statement. First, T and f agree at 1; that is,
T (1) = 1 = f (1).
1
(3.1.1)
Copyright c by Dan Sloughter 2000
2
Best Affine Approximations
Section 3.1
2
1.5
r ( x)
1
0.5
1
Figure 3.1.1 Graph of f (x) =
√
2
3
4
x and an approximating affine function
Second, the error committed by approximating f by T goes to 0 as x approaches 1. That
is, if we let
r(x) = f (x) − T (x),
then r(x) is the error made in approximating f by T at the point x, and
√
lim r(x) = lim (f (x) − T (x)) = lim ( x − 1) = 1 − 1 = 0.
x→1
x→1
x→1
(3.1.2)
Hence we have found an affine function which approximates our function f about x = 1
according to some reasonable criterion.
However, it is easy to see that any affine function T whose graph passes through (1, 1)
will satisfy (3.1.1) and (3.1.2). First note that if the graph of T is a straight line passing
through (1, 1) with slope m, then, using the point-slope form for the equation of a line,
T (x) = m(x − 1) + 1.
It then follows that
T (1) = 1 = f (1)
and, if we again let r(x) = f (x) − T (x),
√
lim r(x) = lim (f (x) − T (x)) = lim ( x − (m(x − 1) + 1)) = 1 − 1 = 0.
x→1
x→1
x→1
See Figure 3.1.1 for the geometrical interpretation. So now we must ask if there is a value
of m which makes T , in some sense, better than any other affine function for approximating
f for x near 1. In answering this question, it is convenient to let h = x − 1 and to define
R(h) = r(1 + h) = f (1 + h) − T (1 + h),
the amount of error committed when f is approximated by T at a point a distance h from
1. Since h approaches 0 as x approaches 1, (3.1.1) and (3.1.2) become, in terms of R,
R(0) = 0
(3.1.3)
Section 3.1
Best Affine Approximations
3
0.01
0.008
0.006
0.004
0.002
m = 0.5
-0.2
-0.1
0
0.1
0.2
Figure 3.1.2 Graphs of |R(h)| for different values of m
and
lim R(h) = 0.
(3.1.4)
h→0
In this case, we have
R(h) =
√
1 + h − (m((1 + h) − 1) + 1) =
√
1 + h − (mh + 1).
Figure 3.1.2 shows the graphs of |R(h)| on the interval [−0.2, 0.2] for m = 0.1, 0.3, 0.4,
0.5, 0.6, 0.7, and 0.9. Note that although all these functions approach 0 as h approaches 0,
one of the graphs clearly stands out from the others. Namely, when m = 0.5, the absolute
value of the approximation error appears to approach 0 at a significantly faster rate than
does the error for other values of m. To see why this is so, consider that
√
1 + h − (mh + 1)
√
√
1 + h + (mh + 1)
= ( 1 + h − (mh + 1)) √
1 + h + (mh + 1)
2
1 + h − (mh + 1)
= √
1 + h + mh + 1
1 + h − (m2 h2 + 2mh + 1)
√
=
1 + h + mh + 1
h(1 − 2m) − m2 h2
= √
.
1 + h + mh + 1
R(h) =
Note that when m = 0.5, the numerator reduces to −m2 h2 , whereas for other values of m
there is also the term h(1 − 2m). This explains why in Figure 3.1.2 the graph for m = 0.5
looks parabolic while the other graphs appear more as straight lines. Moreover, since, for
small values of h, h2 is significantly smaller than h (for example, (0.001)2 = 0.000001 is
much smaller than 0.001), we see why the approximation errors when m = 0.5 are so much
smaller than they are for other values of m.
4
Best Affine Approximations
Section 3.1
Intuitively, we should think that for small values of h, R(h) behaves like a multiple of
h when m 6= 0.5 and like a multiple of h2 when m = 0.5. To see this algebraically, it is
useful to consider the quotient
1 − 2m − m2 h
R(h)
=√
.
h
1 + h + mh + 1
Notice that
1 − 2m
R(h)
=
,
h→0
h
2
lim
which is 0 only when m = 0.5. We interpret this as an indication that R(h) behaves like
a multiple of h for small values of h when m 6= 0.5, but approaches 0 more rapidly than h
when m = 0.5.
In our example, we saw that
R(h)
=0
h→0
h
lim
when m = 0.5, but
R(h)
6= 0
h→0
h
lim
for all other values of m. We distinguish the two cases by saying that in the first case R(h)
is o(h), whereas in the second case R(h) is only O(h).
Definition
A function f is said to be o(h) if
f (h)
= 0.
h→0 h
lim
Definition
that
(3.1.5)
A function f is said to be O(h) if there exist constants M and > 0 such
f (h)
≤M
h
whenever − < h < .
Note that if
f (h)
= L,
h→0 h
lim
then we may find an > 0 such that
f (h)
−L ≤1
h
whenever |h| < . Hence
L−1≤
f (h)
≤L+1
h
(3.1.6)
Section 3.1
Best Affine Approximations
5
0.4
0.2
-0.4
0.2
-0.2
0.4
-0.2
-0.4
1
Figure 3.1.3 Rates of convergence to 0 of f (x) = x2 , g(x) = x, and k(x) = x 3
whenever |h| < . If we let M be the larger of |L − 1| and |L + 1|, then this shows that
f (h)
≤M
h
whenever − < h < . Hence we have the following proposition.
f (h)
exists, then f is O(h).
Proposition If lim
h→0 h
Note that a function which is o(h) is also O(h). Intuitively, we think of a function
which is o(h) as approaching 0 faster than h as h goes to 0, and a function which is O(h)
as approaching 0 at a rate which is at least as fast as that of h.
Example
1
Let f (x) = x2 , g(x) = x, and k(x) = x 3 . Then
lim f (h) = lim h2 = 0,
h→0
h→0
lim g(h) = lim h = 0,
h→0
h→0
and
1
lim k(h) = lim h 3 = 0.
h→0
h→0
However,
f (h)
h2
= lim
= lim h = 0,
h→0 h
h→0 h
h→0
lim
so f is o(h);
g(h)
h
= lim = lim 1 = 1,
h→0 h
h→0 h
h→0
so g is O(h), but not o(h); and
lim
1
k(h)
h3
1
lim
= lim
= lim 2 = ∞,
h→0 h
h→0 h
h→0 h 3
so k is neither o(h) nor O(h). Note in Figure 3.1.3 the difference in the way in which these
functions approach 0.
6
Example
Best Affine Approximations
Section 3.1
Let f (x) = x − x2 . Then
h − h2
f (h)
= lim
= lim (1 − h) = 1,
h→0 h
h→0
h→0
h
lim
so f is O(h), but not o(h).
√
Example Let g(x) = 2 1 + x − x − 2. Then
√
g(h)
2 1+h−h−2
lim
= lim
h→0 h
h→0
h
√
√
2 1 + h − (h + 2)
2 1 + h + (h + 2)
√
= lim
h→0
h
2 1 + h + (h + 2)
4(1 + h) − (h + 2)2
√
h→0 h(2 1 + h + (h + 2))
4 + 4h − (h2 + 4h + 4)
√
= lim
h→0 h(2 1 + h + h + 2)
−h2
√
= lim
h→0 h(2 1 + h + h + 2)
−h
= lim √
h→0 2 1 + h + h + 2
0
= = 0.
4
= lim
Thus g is o(h).
Example
Returning to the problem of approximating f (x) =
√
x for x close to 1, let
T (x) = m(x − 1) + 1
and
R(h) = f (1 + h) − T (1 + h).
We saw above that
R(h)
1 − 2m
=
.
h→0
h
2
lim
Thus R(h)
√ is o(h) if and only if m = 0.5. In other words, the error in approximating
f (x) = x by the affine function
T (x) =
1
(x − 1) + 1
2
(3.1.7)
goes to 0 faster as x approaches 1 than the error for any other affine function approximation.
Because of this, we will call (3.1.7) the best affine approximation to f at 1. Moreover, we
Section 3.1
Best Affine Approximations
7
2.5
2
1.5
1
0.5
1
2
Figure 3.1.4 Graph of f (x) =
√
3
4
x and its tangent line at (1, 1)
will call the graph of T , which is a straight line through (1, 1) with slope 0.5, the tangent
line to the graph of f at (1, 1). See Figure 3.1.4.
As an example of using T to approximate f , note that, to 4 decimal places,
√
1.1 = 1.0488,
while
T (1.1) =
1
(1.1 − 1) + 1 = 1.05,
2
giving a remainder of only
R(0.1) = 1.0488 − 1.0500 = −0.0012.
This approximation is remarkably accurate considering the simplicity of the calculations
used to obtain it. Of course, we expect the accuracy of the approximation to increase as
h decreases. For example, to 4 decimal places,
√
1.05 = 1.0247,
while
T (1.05) =
1
(1.05 − 1) + 1 = 1.025,
2
giving a remainder of only
R(0.05) = 1.0247 − 1.0250 = −0.0003.
Note that when we decreased h from 0.1 to 0.05, a factor of 21 , the error went from −0.0012
to −0.003, a factor of 14 . This is evidence of the quadratic nature of the error, the fact
that R(h) is approaching 0 like h2 , not like h.
Using the ideas of this example, we may now make the following definition.
8
Best Affine Approximations
Section 3.1
Definition Let f be a function defined in an open interval about a point c. If T is an
affine function such that T (c) = f (c) and
R(h) = f (c + h) − T (c + h)
is o(h), then we call T the best affine approximation to f at c. Moreover, the graph of T
is called the tangent line to the graph of f at (c, f (c)).
Using the point-slope form for the equation of a line, the equation of the tangent line
at (c, f (c)) may be written in the form
y − f (c) = m(x − c)
for some slope m. That is,
y = m(x − c) + f (c),
or, in other words, the best affine approximation has the form
T (x) = m(x − c) + f (c).
Thus, to determine T , we need only find the value of m. Since this number m is of such
importance, we give it a formal definition.
Definition
If
T (x) = m(x − c) + f (c)
is the best affine approximation to f at c, then we call m, the slope of the graph of T , the
derivative of f at c. This value is denoted by f 0 (c).
With this notation, the best affine approximation has the form
T (x) = f 0 (c)(x − c) + f (c).
Example
As a consequence of our previous example, if f (x) =
f 0 (1) =
(3.1.8)
√
x, then
1
.
2
Example Let f (x) = x2 and suppose we wish to find the best affine approximation to
f at 3. Then f (3) = 9, so we will let
T (x) = m(x − 3) + 9
and
R(h) = f (3 + h) − T (3 + h) = (3 + h)2 − (mh + 9).
Hence
R(h) = 9 + 6h + h2 − mh − 9 = h(6 + h − m),
Section 3.1
Best Affine Approximations
9
20
15
10
5
2
4
6
-5
Figure 3.1.5 Graph of f (x) = x2 and its tangent line at (3, 9)
and so
R(h)
h(6 + h − m)
= lim
= lim (6 − m + h) = 6 − m.
h→0
h→0
h→0
h
h
lim
Thus R(h) is o(h) if and only if m = 6. It follows then that f 0 (3) = 6 and the best affine
approximation to f at 3 is
T (x) = 6(x − 3) + 9.
The equation of the tangent line at (3, 9) is
y = 6(x − 3) + 9,
or, equivalently,
y = 6x − 9.
See Figure 3.1.5.
In Sections 3.2 through 3.5 we will explore techniques which will simplify greatly the
process of finding derivatives.
Problems
1. Consider the problem of finding an affine approximation for f (x) = sin(x) near 0.
Since f (0) = 0, we let T (x) = mx and
R(h) = f (h) − T (h) = sin(h) − mh.
(a) Plot |R(h)| on the interval [−0.2, 0.2] for m = 0.0, 0.2, 0.4, . . . , 2.0.
(b) Which value of m gives the smallest errors?
2. For each of the following, decide if the given function is O(h), o(h), or neither.
(a) f (x) = x3
(b) f (x) = x2 + 3x
10
Best Affine Approximations
(c) g(t) = 4t3 − 3t2
4
√
4+x−
x
−2
4
3
(f) g(t) = t − t 5
(e) f (t) = t 3
√
(d) g(x) =
Section 3.1
1
1
(x − 9) + 3, and S(x) = (x − 9) + 3.
2
6
(a) Graph f , T , and S together. Note that the graphs of T and S are straight lines
passing through the point (9, 3) on the graph of f .
(b) Let RT (h) = f (9 + h) − T (9 + h). Is RT (h) o(h)? Is it O(h)?
(c) Let RS (h) = f (9 + h) − S(9 + h). Is RS (h) o(h)? Is it O(h)?
(d) Which of T or S is the best affine approximation to f at 9?
√
√
√
(e) Use the best affine approximation to f at 9 to approximate 10, 8.9, and 9.3.
Compare these approximations with values from your calculator.
3. Let f (x) =
x, T (x) =
4. Let g(z) = z 2 , T (z) = 2(z − 1) + 1, and S(z) = 3(z − 1) + 1.
(a) Graph g, T , and S together. Note that the graphs of T and S are straight lines
passing through the point (1, 1) on the graph of g.
(b) Let RT (h) = g(1 + h) − T (1 + h). Is RT (h) o(h)? Is it O(h)?
(c) Let RS (h) = g(1 + h) − S(1 + h). Is RS (h) o(h)? Is it O(h)?
(d) Which of T or S is the best affine approximation to g at 1?
(e) Use the best affine approximation to g at 1 to approximate (1.1)2 and (0.999)2 .
Compare these approximations with values from your calculator.
5. Find the best affine approximation to f (x) = 2x2 at 1. What is f 0 (1)?
6. Find the best affine approximation to g(x) =
1
at 1. What is g 0 (1)?
x
7. Find the best affine approximation to f (t) = t2 + t − 1 at 0. What is f 0 (0)?