Cubic function
cube function
cube coefficient
Chapter 13
Cubic Functions: Local
Analysis
Input-Output Pairs, 210 – Local I-O Rule Near ∞, 212 – Local Graph
Near ∞, 213 – Local Features Near ∞, 214 – Addition Formula For
Cubes, 216 – Local I-O Rule Near x0 , 219 – Local Coefficients Near x0 ,
221 – Local Graph Near x0 , 225 – Local Features Near x0 , 226.
Cubic functions are the result of adding-on a quadratic function to a
cube function, that is a power function with exponent +3, or, the other
way around, adding on a cube function to a quadratic function. Either way,
cubic functions are specified by a global input-output rule of the form
CU BICa,b,c,d
x −−−−−−−−−−−−−→ CU BICa,b,c,d (x) = ax3 + bx2 + cx + d
where CU BICa,b,c,d is the name of the cubic function and where a, b, c, d
stand for the four bounded numbers that are needed to specify the cubic
function CU BICa,b,c,d .
In order to facilitate our investigation of cubic functions, it is necessary
to begin by introducing some language special to cubic functions.
Given the cubic function CU BICa,b,c,d , that is the function specified by
the global input-output rule
CU BICa,b,c,d
x −−−−−−−−−−−−−→ CU BICa,b,c,d (x) = ax3 + b2 x + cx + d
• The given signed number a is called the cube coefficient of the cubic
function CU BICa,b,c,d and ax3 , that is the cube coefficient multiplied by
209
210
cube term
square coefficient
square term
linear coefficient
linear term
constant coefficient
constant term
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
three copies of the input, is called the cube term of the cubic function
CU BICa,b,c,d .
• The given signed number b is called the square coefficient of the cubic
function CU BICa,b,c,d and bx2 , that is the square coefficient multiplied by
two copies of the input, is called the square term of the cubic function
CU BICa,b,c,d
• The given signed number c is called the linear coefficient of the cubic
function CU BICa,b,c,d and cx1 , that is the linear coefficient multiplied
by one copy of the input, is called the linear term of the cubic function
CU BICa,b,c,d
• The given signed number d is called the constant coefficient of the
cubic function CU BICa,b,c,d and dx0 , the constant coefficient multiplied
by no copy of the input, is called the constant term of the cubic function
CU BICa,b,c,d .
In other words, the constant term is the same as the constant coefficient
and what words we will use will depend on what viewpoint we will be
taking.
EXAMPLE 1. Given the function cubic function T AT A−3,+5,−2,+4 , that is the
function specified by the global input-output rule
T AT A−3,+5−2,+4
x −−−−−−−−−−−−−−−−−→ T AT A−3,+5,−2,+4 (x) = −3x3 + 5x2 − 2x + 4
is the cubic function whose
• cube term is (−3)x3 , that is the cube coefficient −3 multiplied by three copies of
the input x
• square term is (+5)x2 , that is the square coefficient +5 multiplied by two copies
of the input x
• linear tern is (−2)x, that is the linear coefficient −2 multiplied by one copy of the
input x
• constant term is +4, that is the constant coefficient +4 multiplied by no copy of the
input x. In other words, the constant term is the same as the constant coefficient.
13.1
Input-Output Pairs
Given the cubic function CU BICa,b,c,d , that is the function specified by the
global input-output rule
CU BIC
x −−−−−−−−−−→ CU BIC(x) = ax3 + bx2 + cx + d
and given an input x0 , in order to get the output CU BIC(x0 ), we proceed
as follows.
211
13.1. INPUT-OUTPUT PAIRS
i. We read the global input-output rule:
The output of CU BICa,b,c,d is obtained by multiplying a by 3
copies of the input, adding b multiplied by 2 copies of the input, adding c multiplied by 1 copy of the input and adding d.
ii. We indicate that x is to be replaced by the given input x0
CU BIC
x
x←x0
−−−−−−−→ CU BIC(x) x←x0
= ax3 + bx2 + cx + d =
ax30
+
bx20
x←x0
+ cx + d
iii. When we are given specific numbers for a, b, c, d and for x0 , we can
then identify the specifying-phrase ax30 + bx20 + cx0 + d to get the specific
value of the output CU BICa,b,c,d (x0 ).
EXAMPLE 2. Given the function T HORA specified by the global input-output rule
T HORA
x −−−−−−−−→ T HORA(x) = −42.17x3 − 32.67x2 − 52.39x + 71.07
and given the input −3, in order to get the input-output pair, we proceed as follows.
i. We read the global input-output rule:
“The output of T HORA is obtained by multiplying −32.67 by 2 copies of the
input, adding −52.39 multiplied by 1 copy of the input, and adding +71.07”
ii. We indicate that x is to be replaced by the given input −3
x x←−3
T HORA
−−−−−−−−→ T HORA(x) x←−3
= −42.17x3 − 32.67x2 − 52.39x + 71.07 x←−3
which gives us the following specifying-phrase
= (−42.17)(−3)3 + (−32.67)(−3)2 + (−52.39)(−3) + (+71.07)
iii. We identify the specifying-phrase
= +1138.59 − 294.03 + 157.17 + 71.07
= +1072.80
iv. Altogether, depending on the circumstances, we can then write any of the
following:
T HORA
−3 −−−−−−−−→ +1072.80
T HORA(−3) = +1072.80
(−3, +1072.80) is an input-output pair for T HORA
.
212
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
LOCAL ANALYSIS NEAR INFINITY
Doing local analysis means working in a neighborhood of some given input
and thus counting inputs from the given input since it is the center of the
neighborhood. When the given input is ∞, counting from ∞ means to set
x ← large and to compute with powers of large in descending order of sizes.
NOTE. As it happens, the powers in global input-output rules are usually given in
descending order of exponents which, when x is large, is in descending order of sizes.
However, this is not an obligation and, when the powers in a global input-output rule
are not given in descending order of exponents, in order to work in a neighborhood of
∞, the powers must be rearranged in descending order of exponents so as to be in
descending order of sizes.
13.2
Local Input-Output Rule Near Infinity
Given a cubic function CU BIC, we look for a function whose input-output
rule will be simpler than the input-output rule of CU BIC but whose local
graph near ∞ will still be qualitatively the same as the local graph near ∞
of CU BIC.
1. Given a cubic function CU BICa,b,c that is a function specified by the
global input-ouput rule
CU BICa,b,c,d
x −−−−−−−−−−−−−→ CU BICa,b,c,d (x) = ax3 + bx2 + cx + d
where a, b, c, d are bounded, when the input is large, the cube term of CU BICa,b,c,d
is the principal term near ∞ of the output, because when the input is near
∞:
• The bounded number a multiplied by three copies of a large input gives
a result that is large,
while
• A bounded number multiplied by two copies of a large input will also be
large but not as large and thus, regardless of its sign, will not change
the size of the result which which will remain large
• A bounded number multiplied by one copy of a large input will also be
large but not as large and thus, regardless of its sign, will not change
the size of the result which which will remain large
and
• adding the constant which is also bounded will even less change the size
from being large.
13.3. LOCAL GRAPH NEAR ∞
213
2. However, even though, for large inputs, the quadratic part of CU BIC
is “too small to matter here”, it is not 0 and so we cannot write that
CU BICa,b,c,d (x) is equal to the cube term ax3 and, in mathematics, we
do want to write equalities, if only because they are easier to work with. So
we will use the symbol +[...] to mean “plus something too small to matter
here” and we will write
CU BICa,b,c,d
x|x near ∞ −−−−−−−−−−−→ CU BICa,b,c,d (x)|x near ∞ = ax2 + [...]
which we will of course call the approximate local input-output rule near ∞
of the cubic function CU BICa,b,c,d .
EXAMPLE 3. Given the cubic function M OAN+93.37−21.74,+31.59.−71.09 , that is the
function specified by the global input-ouput rule
M OAN
x −−−−−−−→ M OAN (x) = +93.37x3 − 21.74x2 + 31.59x − 71.09
we can write
M OAN
x|x near ∞ −−−−−−−−→ M OAN (x)|x near ∞ = +93.37x3 + [...]
In other words, we have:
THEOREM 1 (Approximation Near ∞). Near ∞ the output of a cubic
function is principally its cube term.
13.3
Local Graph Near Infinity
We get the qualitative local graph near ∞ as follows:
i. We use the Approximation Near ∞ Theorem to get the
approximate local input-output rule near ∞
ii. We normalize it so that, depending on the sign of the coefficient, we get the local input-output rule of either the cube
function or the opposite cube function,
iii. We draw the corresponding local graph near ∞..
EXAMPLE 4. Given the cubic function ALM A−38.38,−21.36,−45.78,+53.20 , find its local graph near ∞.
ALBA−21.36,−45.78,+53.20 is specified by the global input-output rule:
ALM A
x −−−−−−−→ ALM A(x) = −38.38x3 − 21.36x2 − 45.78x + 53.20
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CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
i. The approximate local input-output rule near ∞ is:
ALM A
x|x near ∞ −−−−−−−−−−→ ALM A(x)|x near ∞ = −38.38x+3 + [...]
ii. We normalize ALM A
ALM A∗
x|x near ∞ −−−−−−−−−−→ ALM A(x)|x near ∞ = −x+3 + [...]
Output Ruler
+∞
)
iii. The local graph near ∞ of ALM A is:
(
Screen
–∞
–∞
13.4
)
)
+∞
Input
Ruler
Local Features Near Infinity
Once we have the local graph of the function CU BIC near ∞, we can “read”
the local features off the local graph near ∞ as in Chapter 4. This is a
good way to proceed but not in the long run because it is too slow. And,
in fact, once we have a mental picture of the two possible qualitative local
graphs near ∞, in order to decide which of the two is the actual case in the
instance, we need only know what the place of the local graph is and this is
given by
THEOREM 2 (Height-sign Near ∞). Given the function CU BICa,b,c,d :
• W hen Sign a = +, Height-sign CU BIC |near ∞ = (+, −)
• W hen Sign a = −, Height-sign CU BIC |near ∞ = (−, +)
And of course we also have the other two local features near ∞:
THEOREM 3 (Slope-sign Near ∞). Given the function CU BICa,b,c,d :
• When Sign a = +, Slope-sign CU BIC |near x0 = (, )
• When Sign a = −, Slope-sign CU BIC |near x0 = (, )
(Keep in mind that the feature signs near ∞ are coded while facing ∞ and
not while facing 0.)
215
13.4. LOCAL FEATURES NEAR ∞
THEOREM 4 (Concavity-sign Near ∞). Given the function CU BICa,b,c,d :
• W hen Sign a = +, Concavity-sign CU BIC |near ∞ = (∪, ∩)
• W hen Sign a = −, Concavity-sign CU BIC |near ∞ = (∩, ∪)
EXAMPLE 5. Given the cubic function M AON−38.82,+34.54,−40.38,−94.21 , that is the
function specified by the global input-output rule
M AON
x −−−−−−−−−−→ M AON (x) = −38.82x3 + 34.54x2 − 40.38x − 94.21
we find its local features near ∞ as follows.
i. The local features near ∞ according to the Local Features Theorem are:
• Height-sign |x near ∞ = (−, +)
• Slope-sign |x near ∞ = (−, −)
• Concavity-sign |x near ∞ = (−, +)
which, translated back into our language, gives
• Height-sign |x near ∞ = (−, +)
• Slope-sign |x near ∞ = (, )
• Concavity-sign |x near ∞ = (∪, ∩)
ii. Should we want to check that the Local Features Theorem gave us the correct
information, we would get the normalized approximate local input-output rule near ∞
M AON ∗
x|x near ∞ −−−−−−−−−−−→ M AON ∗ (x)|x near ∞ = −x3 + [...]
Output Ruler
)
+∞
Screen
(
and then the local graph near ∞ from
which we get the following local features
–
+
–∞
–∞
)
)
+∞
Input
Ruler
which are the same as those that were given by the Local Features Theorem keeping in
mind that the local graph near ∞ is viewed here by Mercator rather than by Magellan.
LOCAL ANALYSIS NEAR A FINITE INPUT
Doing local analysis means working in a neighborhood of some given input
and thus counting inputs from the given input since it is the center of the
neighborhood. When the given input is x0 , we localize at x0 , that is we set
x = x0 + h where h is small and we compute with powers of h in descending
order of sizes. (See Section 2.7 Analysis Near Finite Inputs.
EXAMPLE 6. Given the input +2, then the location of the number +2.3 relative to
+2 is +0.3:
216
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
addition formula for cubes
+2
(
+0.3
0
Neighborhood of +2
Input
Ruler
)
+3
+2
+1
0
13.5
+2.3 = +2 +0.3
Addition Formula For Cubes
In order to get the local input-output rule of a given cubic function near
a given finite input x0 , we need first to get an addition formula to give
us (x0 + h)3 in terms of x0 and h and which we will call the addition
formula for cubes. We prefer to establish it here, ahead of time, rather
than in the midst of the developing the local input-output rule. There are
two approaches.
1. The computational approach. We “just” multiply three copies of (x0 +
h):
a. We begin by multiplying two copies of (x0 + h):
x0
+ h
x0
+ h
x20
x20
+
h2
+
x0 h
x0 h
+
2x0 h
+
h2
b. We multiply x20 + 2x0 h + h2 , the result of the multiplication of two
copies of (x0 + h), by a third copy of (x0 + h):
x20
+ 2x0 h + h2
x0
+ h
x20 h
2x20 h
+
+
2x0 h2
x 0 h2
+
h3
x30 + 3x20 h
2. The graphic approach.
+
3x0 h2
+
h3
+h
We go back to the real world and
to the definition of double multiplication in terms of the volume of
a rectangle so that (x0 + h)3 is the
volume of a x0 + h by x0 + h by
x0 + h cube:
x0+h
x0
+
x0+h
x30
(x0+h)3
217
13.5. ADDITION FORMULA FOR CUBES
What we then do is to get this volume by starting with an x0 by x0 by x0
cube, the initial cube, and then augment the sides of the initial cube by h
and see how the volume increases. For the sake of clarity, we will augment
the cube one step at a time:
x0
i. The sides of the initital cube
are equal to x0 and the volume of
the initial cube is therefore x30 :
ii. We now augment the sides of
the initial cube by h in each dimension which adds three x0 + h
by x0 + h by h slabs to the initial
cube:
x0
x0
x0
x0
h
x0
h
x0
x0
x0
h
iii. We glue the three slabs to the
initial cube which, however, leaves
three grooves:
h
iv. We fill the three grooves with
three x0 by h by h rods:
h
x0
h
h
x0
x0
h
h
218
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
v. We glue the three rods in the
three grooves which leaves a hole
in the corner:
vi. We complete the enlargement
of the initial cube by adding one h
by h by h cube to fill the indentation in the corner:
vii. Altogether, the volume of the
enlarged cube, whose side is x0 + h
is therefore:
x30 + 3x20 h + 3x0 h + h2 + h3
h
h
h
h
x0
h
x0
h
h
x0
3. Contrary to what one might think at first, the graphic approach is
greatly preferable because it has three major advantages over the computational approach:
i. The terms in the sum automatically come in order of diminishing sizes.
Indeed, since x0 is finite and h is small,
• all three dimensions of the “initial cube” are finite, say “in the ones”,
so x30 is also finite or “in the ones”,
• two dimensions of the “slabs” are finite, say “in the ones”, but the third
dimension is small, say “in the tenths”, so the volume of the slabs is
small: since x0 is “in the ones” and h is “in the tenths”, then 3x20 h will
also be “in the tenths”,
• one dimension of the “rods” is finite, say “in the ones”, but the other
two dimensions are small, say “in the tenths”, so the volume of the rods
is smaller: since x0 is “in the ones” and h is “in the tenths”, then 3x0 h2
will be “in the hunfredths”,
• all three dimensions of the “little cube” are small so that the volume of
the little cube is even smaller than the volume of the slabs: if h is “in
the tenths”, then h3 will be “in the thousandths”.
ii. If all we need is only a particular one of the terms, and this will very
13.6. LOCAL I-O RULE NEAR X0
219
often be the case, we can get it straight from the picture without having to
go through the whole multiplication.
iii. Later on, when we shall need formulas for (x0 + h)4 , (x0 + h)5 , etc, not
only will the length of the computational approach get very rapidly out of
hand but, as we shall see, since we will never need more than the first two
or three terms of the result, the computational approach will also become
more and more inefficient. On the other hand, even though we will not be
able to draw pictures as we have been able to do so far, one can extend
the patterns we have found so far in the graphic approach and the graphic
approach will thus survive.
In any case, we have
THEOREM 5 (Addition Formula For Cubes).
(x0 + h)3 = x30 + 3x20 h + 3x0 h2 + h3
13.6 Local Input-Output Rule Near A Finite Input
Since x0 is given for the duration of the local investigation and since it is
therefore h that will be the actual input, we will think in terms of a local
function, that is of a function which returns for h the same output that the
given global function would return for x = x0 + h.
In order to get the input-output rule of the local function function when
the given function is a cubic function, we proceed just as we did in the case
of quadratic functions:
i. We replace x by x0 + h (where h is small) in the global inputoutput rule,
ii. We use the Addition Formula For Squares to compute
(x0 + h)2 ,
iii. We use the Multiplication Theorem to compute with h.
More precisely, given the cubic function CU BICa,b,c,d , that is the function
specified by the global input-output rule
CU BIC
x −−−−−−−−−−→ CU BIC(x) = ax3 + bx2 + cx + d
and given a finite input x0 ,
220
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
i. We localize the function CU BIC at the given input x0 , that is we
count inputs from the given imput x0 instead of from the origin of the input
ruler. In other words, we use the location of x in relation to the given input
x0 , that is we replace in the global input-output rule x by x0 + h, (where h
is small).
CU BICa,b,c,d
x|x←x0 +h −−−−−−−−−→ CU BICa,b,c,d (x)|x←x0 +h = ax3 + bx2 + cx + d
x←x0 +h
= a[x0 + h]3 + b[x0 + h]2 + c[x0 + h] + d
ii. Using the Addition Formula Theorems, we get
h
i
h
i
h
i
= a x30 + 3x20 h + 3x0 h2 + h3 + b x20 + 2x0 h + h2 + c x0 + h + d
= ax30 + 3ax20 h + 3ax0 h2 + ah3 + bx20 + 2bx0 h + bh2 + cx0 + ch + d
iii. We collect terms of like size in order of diminishing sizes:
[
] [
] [
]
[]
= ax30 + bx20 + cx0 + d + 3ax20 + 2bx0 + c h + 3ax0 + b h2 + a h3
We then have:
THEOREM 6 (Local Input-Output Rule). Given the cubic function
CU BIC specified by the global input output rule
CU BIC
x −−−−−−−−−−→ CU BIC(x) = ax3 + bx2 + cx + d
the local function CU BIC(x0 ) is specified by the local input-output rule
CU BIC(x
)
0
h −−−−−−−−
→ CU BICx0 (h) =
[ax
3
0
] [
] [
]
[]
+ bx20 + cx0 + d + 3ax20 + 2bx0 + c h + 3ax0 + b h2 + a h3
BEN
EXAMPLE 7. Given the cubic function x −−−−−→ BEN (x) = −2x3 + 3x2 − 5x + 7,
and given an input, say x0 = −4, we get the local input-output rule near x0 = −4 by
localizing at −4 and then “just” do the computations as follows:
BEN
−4 + h −−−−−−−−−→ BEN (−4 + h) = −2x3 + 3x2 − 5x + 7x←−4+h
3
2
= −2 − 4 + h + 3 − 4 + h − 5 − 4 + h + 7
221
13.7. LOCAL COEFFICIENTS NEAR X0
and, using the Addition Formula Theorem with x0 = −4,
i
h
i
h
= −2 (−4)3 + 3(−4)2 h + 3(−4)h2 + h3 + 3 (−4)2 + 2(−4)h + h2
h
i
−5 −4+h +7
h
i
h
i
h
i
= −2 − 64 + 48h − 12h2 + h3 + 3 + 16 − 8h + h2 − 5 − 4 + h + 7
local
local
local
local
constant coefficient
linear coefficient
quadratic coefficient
cubic coefficient
= +128 − 96h + 24h2 − 2h3 + 48 − 24h + 3h2 + 20 − 5h + 7
We collect terms of like size in order of diminishing sizes:
=
[ + 203] + [ − 125]h + [ + 27]h2 + [ − 2]h3
The local function BEN(−4) is therefore specified by the local input-output rule
BEN(−4)
h −−−−−−−−−−−→ BEN−4 (h) = +203 − 125h + 27h2 − 2h3
13.7
Local Coefficients Near A Finite Input
When looking only for one of the local features, instead of computing the
whole local input-ouput rule, we will only compute the single term of the
local input-ouput rule that controls the local feature.
More precisely, given a function CU BIC specified by the global inputoutput rule:
CU BIC
x −−−−−−−−−−→ CU BIC(x) = ax3 + bx2 + cx + d
and whose local input-output rule near x0 is therefore
CU BIC(x0 )
h −−−−−−−−−−−−→ CU BIC(x0 )|x←x0 +h
[
] [
] [
]
[]
= ax30 + bx20 + cx0 + d + 3ax20 + 2bx0 + c h + 3ax0 + b h2 + a h3
we will say that
• ax30 + bx20 + cx0 + d is the local constant coefficient,
[
]
• [3ax + 2bx + c] is the local linear coefficient,
• [3ax + b] is the local quadratic coefficient.
• [a] is the local cubic coefficient.
2
0
0
0
and we will now investigate how to get just a single one of the local coefficients without getting the whole local input-output rule. At first, getting
222
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
just this one single term rather than the whold local input-output rule will
look more difficult than getting the whole local input-output rule but, with
a little bit of practice, writing less and less each time, this will soon get easy.
We begin by writing
CU BICa,b,c,d
x|x←x0 +h −−−−−−−−−→ CU BICa,b,c,d (x)|x←x0 +h = ax3 + bx2 + cx + d
x←x0 +h
= a[x0 + h]3 + b[x0 + h]2 + c[x0 + h] + d
a. To find the local constant coefficient, we proceed as follows:
i. Since the Addition Formula for Cubes is:
(x0 + h)3 = x30 + 3x20 h + 3x0 h2 + h3
the contribution of a[x0 + h]3 to the local constant term will be a · x30 h0
ii. Since the Addition Formula for Squares is:
(x0 + h)2 = x20 + 2x0 h + h2
the contribution of b[x0 + h]2 to the local constant term will be b · x20 h0
iii. The contribution of c[x0 + h] to the constant term will be c · x0 h0
iv. The contribution of d to the local constant term will be d · h0
Altogether, the contributions to the local constant term add up to:
a · x30 h0 + b · x20 h0 + c · x0 h0 + d · h0 = ax30 h0 + bx20 h0 + cx0 h0 + dh0
[
]
= ax30 + bx20 + cx0 + d h0
and therefore the local constant coefficient is:
ax30 + bx20 + cx0 + d
b. To find the local linear coefficient, we proceed as follows
i. Since the Addition Formula for Cubes is:
(x0 + h)3 = x30 + 3x20 h + 3x0 h2 + h3
the contribution of a[x0 + h]3 to the local linear term will be a · 3x20 h
ii. Since the Addition Formula for Squares is:
(x0 + h)2 = x20 + 2x0 h + h2
the contribution of b[x0 + h]2 to the local linear term will be b · 2x0 h
223
13.7. LOCAL COEFFICIENTS NEAR X0
iii. The contribution of c[x0 + h] to the local linear term will be c · h
iv. The contribution of d to the local linear term will be nothing.
Altogether, the linear term adds up to:
a · 3x20 h + b · 2x0 h + c · h = 3ax0 h2 + 2bx0 h + ch
[
]
= 3ax20 + 2bx0 + c h
and therefore the local linear coefficient is:
3ax20 + 2bx0 + c
c. To find the local square coefficient, we proceed as follows:
i. Since the Addition Formula for Cubes is:
(x0 + h)3 = x30 + 3x20 h + 3x0 h2 + h3
the contribution of a[x0 + h]3 to the local square term will be a · 3x0 h2
ii. Since the Addition Formula for Squares is:
(x0 + h)2 = x20 + 2x0 h + h2
the contribution of b[x0 + h]2 to the local square term will be b · h2
iii. The contribution of c[x0 + h] to the local square term will be nothing
iv. The contribution of d to the local square term will be nothing.
Altogether, the square term adds up to:
a · 3x0 h2 + b · h2 = 3ax0 h2 + bh2
[
]
= 3ax0 + b h2
and therefore the local square coefficient is:
3ax0 + b
d. To find the local cube coefficient we proceed as follows:
i. Since the Addition Formula for Cubes is:
(x0 + h)3 = x30 + 3x20 h + 3x0 h2 + h3
the contribution of a[x0 + h]3 to the local cube term will be a · h3
local cube coefficient
224
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
ii. Since the Addition Formula for Squares is:
(x0 + h)2 = x20 + 2x0 h + h2
the contribution of b[x0 + h]2 to the local cube term will be nothing
iii. The contribution of c[x0 + h] to the local cube term will be nothing
iv. The contribution of d to the local cube term will be nothing.
Altogether, the cube term “adds up” to:
a · h3 = ah3
[]
= a h3
and therefore the local cube coefficient is:
a
In fact, it is usually better to write only the terms that are relevant to
what we want.
BEN
EXAMPLE 8. Given the cubic function x −−−−−→ BEN (x) = −2x3 + 3x2 − 5x + 7,
and given the input −4, find the local square term.
We start as usual:
BEN
−4 + h −−−−−−−−−→ BEN (−4 + h) = −2x3 + 3x2 − 5x + 7x←−4+h
3
2
= −2 − 4 + h + 3 − 4 + h − 5 − 4 + h + 7
now, instead of using the full Addition Formula Theorems, we just pick the h2 terms
in each addition formula:
h
i
h
i
h i
= −2 (· · · + 3(−4)h2 + · · · + 3 · · · + h2 − 5 · · · + · · ·
where 3(−4)h2 are the three rods in the enlarged cube and +h2 is the little square in
the enlarged square. So,
h
i
h
i
h
i
= −2 · · · − 12h2 · · · + 3 · · · + h2 − 5 · · · + · · ·
= · · · + 24h2 · · · + · · · + 3h2 + · · · + · · ·
and, collecting “like” terms,
=
[ · · · ] + [ · · · ]h + [ + 27]h2 + [ · · · ]h3
So, the quadratic term in the local input-output rule of BEN when x is near −4 is:
+27h2
225
13.8. LOCAL GRAPH NEAR X0
13.8
Local Graph Near A Finite Input
The local graph of a given cubic function CU BICa,b,c,d near a given input
x0 is the graph of the local function CU BIC(x0 ) , that is the graph of the
function specified by the local input-output rule of CU BICa,b,c,d near x0 .
So, in order to get the local graph of the given cubic function CU BIC
near the given input x0 , we:
i. Get the local graph of the constant term of CU BIC(x0 ) ,
ii. Get the local graph of the linear term of CU BIC(x0 ) ,
iii. Add-on (as we did in Chapter 8) the local graph of the linear term
to the local graph of the constant term to get the local graph of the affine
part of CU BIC(x0 )
iv. Get the local graph of the square term of CU BIC(x0 )
v. Add-on (as we did in Chapter 8) the local graph of the the local
graph of the square term to the local graph of the affine part to get the local
graph of CU BIC(x0 )
EXAMPLE 9. Given a cubic function SHIP and given that the local input-output
rule of SHIP near the input −4 is
SHIP(−4)
h−
−−−−−−−−−−−
→ SHIP(−4) (h) = +75 − 29h + 3h2 − 4h3
construct the local graph of SHIP near −4, that is the graph of SHIP(−4) .
i. We get the local graph of the constant term of SHIP(−4) :
h −−−−−→ +75
Output
Ruler
Screen
+75
0
that is
(
ii. We get the local graph of the linear
term of SHIP(−4) :
–4
)
Input
Ruler
)
Input
Ruler
Output
Ruler
Screen
h −−−−−→ −29h
that is
0
(
–4
226
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
iii. We get the local graph of the affine
part of SHIP(−4) by adding-on the linear term to the constant term:
Output
Ruler
Screen
h −−−−−→ +75 − 29h
+75
0
that is
(
iv. We get the local graph of the square
term of SHIP(−4) :
Output
Ruler
–4
)
Input
Ruler
)
Input
Ruler
Screen
h −−−−−→ +3h
2
0
that is
(
v. We get the full local graph of
SHIP(−4) by adding-on the square
term to the affine part:
Output
Ruler
h −−−−−→ +75 − 29h + 3h2
–4
Screen
vi. And
+75
0
that is
(
since the local graph has concavity, we are done!
13.9
–4
)
Input
Ruler
Local Features Near A Finite Input
Once we have the local graph of the function CU BIC near x0 , we can “read”
the local features off the local graph near x0 as in Chapter 4. This is a
good way to proceed but not in the long run because it is too slow. Given
the cubic function CU BICa,b,c,d , that is the function specified by the global
input-output rule
CU BIC
x −−−−−−−−−−→ CU BIC(x) = ax3 + bx2 + cx + d
what controls the local features of CU BIC near x0 are the local coefficients,
that is the coefficients in the local input-output rule near x0
CU BIC(x0 )
h −−−−−−−−−−−−→ CU BIC(x0 )|x←x0 +h
[
] [
] [
]
= ax30 + bx20 + cx0 + d + 3ax20 + 2bx0 + c h + 3ax0 + b h2 + [...]
227
13.9. LOCAL FEATURES NEAR X0
1. The local constant term
[ax
3
0
]
+ bx20 + cx0 + d
gives the Height-sign near x0 because it contributes most of the local height
since, as long as h remains small, the other two terms of the local input-ouput
rule, namely the linear term and the square term are small and therefore
contribute very little to the total height and certainly not enough to change
the sign of the total local height.
The local constant coefficient is therefore what determines the height-sign
near any finite input x0
THEOREM 7 (Height-sign Near x0 ). Given the function CU BICa,b,c,d
• W hen Local constant coefficient of CU BIC(x0 ) = +,
Height-sign CU BIC |near x0 = (+, +)
• W hen Local constant coefficient of CU BIC(x0 ) = −,
Height-sign CU BIC |near x0 = (−, −)
• W hen Local constant coefficient of CU BIC(x0 ) = 0,
Height-sign CU BIC|near
, x0 depends on
the sign of the local linear coefficient of CU BIC(x0 )
2. The local linear term
[3ax
2
0
]
+ 2bx0 + c h
contibutes most of the local slope because the constant term has no slope
and the square term contributes very little to the total slope—as long as h
remains small—certainly not enough to change the sign of the total slope.
The local linear coefficient is therefore what controls the Slope-sign near any
finite input x0 :
THEOREM 8 (Slope-sign Near x0 ). Given the function CU BICa,b,c,d
• W hen Local linear coefficient of CU BIC(x0 ) = +,
Slope-sign CU BIC |near x0 = (, )
• W hen Local linear coefficient of CU BIC(x0 ) = −,
Slope-sign CU BIC |near x0 = (, )
• W hen Local linear coefficient of CU BIC(x0 ) = 0,
Slope-sign CU BIC|near, x0 depends on
the sign of the local square coefficient of CU BIC(x0 )
228
CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS
3. The local square term
[3ax
0
]
+ b h2
contributes all of the local concavity because neither the constant term nor
the linear term have any concavity.
The local square term is therefore what controls the Concavity-sign near any
finite input x0 :
THEOREM 9 (Concavity-sign Near x0 ). Given the function CU BICa, b, c, d
• W hen Local square coefficient of CU BIC(x0 ) = +,
Concavity-sign CU BIC |near x0 = (∪, ∪)
• W hen Local square coefficient of CU BIC(x0 ) = −,
Concavity-sign CU BIC |near x0 = (∩, ∩)
• W hen Local square coefficient of CU BIC(x0 ) = 0,
Concavity-sign CU BIC|
, near x0 depends on
the sign of the local cube coefficient of CU BIC(x0 )
4. We look at a couple of examples.
EXAMPLE 10. Given the function specified by the global input-ouput rule
JIM
x −−−−−→ JIM (x) = +2x3 + 3x2 − 36x + 1
and given the input +2, in order to find the slope-sign near +2 we must get the linear
term in the local input-ouput rule near +2. Whichever way we proceed, we get the
local input-ouput rule of JIM near +2
JIM(+2)
h −−−−−−−−→ JIM(+2) (h) =
[
] + [ 0 ]h + [ ]h
+ [...]
2
Since the linear term vanished, the slope-sign will be given by the square term.
After recomputing, we get
JIM(+2)
h −−−−−−−−→ JIM(+2) (h) =
[
] + [ 0 ]h + [ + 15]h
2
+ [...]
and therefore, Slope-sign near +2 = (, ).
EXAMPLE 11. Given the function specified by the global input-ouput rule
KAY
x −−−−−→ KAY (x) = +1x3 + 3x2 − 9x + 17
229
13.9. LOCAL FEATURES NEAR X0
and given the input −1, in order to find the concavity-sign near −1 we must get the
square term in the local input-ouput rule near −1. Whichever way we proceed, we get
the local input-ouput rule of KAY near −1
KAY(−1)
h −−−−−−−−→ KAY(−1) (h) =
[
] + [ ]h + [ 0 ]h
2
+ [...]
Since the square term vanished, the concavity-sign will be given by the cube term.
After recomputing, we get
KAY(−1)
h −−−−−−−−→ KAY(−1) (h) =
[
] + [ ]h + [ 0 ]h + [ + 1]h
and therefore, Concavity-sign near −1 = (∩, ∪).
2
3