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Chapter 5
Symbolic Increments
The main idea of di§ erential calculus is that small changes in smooth functions are approximately linear. In Chapter 3, we saw that ìmostî microscopic views of graphs appear to be linear, but we want a symbolic way to predict the view in a powerful microscope without actually having to use graphical magniÖcation. The computations in this chapter give us that prediction.
Direct computation of the microscopic ìgapî is hard work, but the formulas show when the gap gets small. This guarantees that a magniÖed graph will appear linear before we draw the graph. The direct computations of this chapter are proofs of speciÖc di§erentiation rules. The next chapter develops the rules systematically as a procedure or ìcalculusîof derivatives. You could just accept the speciÖc results of this chapter and go on to Chapter 6 to learn the ìrules,î but di§erentiation rules are actually theorems that say a local linear approximation is guaranteed by certain systematic computations. You should understand that this means the ìgapî tends to zero at all ìgoodî points. If the gap does not go to zero at a point, we will not see a straight line when we focus our microscope there. We know from Exercise 3.2.4 that the perfectly reasonable function
f [x] =
p x^2 + 2x + 1
does not have the local linearity property at x = 1. (Its graph has a kink no matter how much you magnify it.) Some less reasonable ones like Weierstrassífunction
W [x] = Cos[x] + Cos[3x] 2
Cos[3^2 x] 22
Cos[3nx] 2 n^
are continuous but not locally linear at any point (see Figure 3.2.8). Calculus gives a procedure to Önd out if a function given by a formula is locally linear. Once we have the basic rules of this chapter and some functional rules from the next, we will di§erentiate the kink function f [x] =
p x^2 + 2x + 1 ) f 0 [x] = x + 1 p x^2 + 2x + 1
but then notice that f 0 [ 1] is not deÖned, so that we cannot make any conclusion about local linearity at this point. (General di§erentiation rules cannot say there is a kink, only that no general conclusion is possible when the rules do not apply.) At every other x, the rules do apply and this function is locally linear.
5.1 The Gap for Power Functions
The gap " for y = f [x] = xp^ with f 0 [x] = p xp ^1
In Chapter 3, we formulated the deviation of y = f [x] from a straight line geometrically. The observed error or gap between the curve and line at magniÖcation 1 =x, denoted ", satisÖes
f [x + x] f [x] = m x + " x
The term " x is the actual unmagniÖed error that appears in the formula above. We observe " because of magniÖcation. The number m is the slope of the microscopic straight line dy = m dx in (dx; dy)-coordinates focused over x. (See Chapter 1 for equations of lines in local coordinates.) ìApproximate linearityîmeans that the error " is too small to measure, " 0 , when x 0 is ìsmall enough.î Notice that m may depend on x, but not x, because the slope of the curve depends on the point but not the magniÖcation. As we move the focus point of the microscope, the slope may change, but the graph should always appear straight under the microscope. Since m depends on x and not on x, it is customary to denote this slope by f 0 [x] rather than m. The function f 0 [x] is called the derivative of the function f [x] and the local linear equation dy = f 0 [x] dx is called the di§erential of y = f [x]. (It is the equation of the tangent line at a Öxed value of x in local (dx; dy)-coordinates.) We will compute the symbolic gap for all the examples in this section (and Exercise 5.1.1) by the steps
Procedure 5.1 Computing the "-Gap
1 ) Compute f [x + x] f [x] x 2 ) Simplify the expression from the Örst part and compute
f 0 [x] = lim x! 0
f [x + x] f [x] x
Give an intuitive justiÖcation of why your limit is correct.
3 ) Use your limit f 0 [x] to solve for " = f [x + x] f [x] x f 0 [x]
4 ) Show that "! 0 as x! 0 , or " 0 is small when x = x 0 is small.
The error " is the ìgapîor amount of deviation from straightness we see above x + x at power 1 =x. We want to let x get ìsmall enoughî so that " is below the resolution of our microscope. This chapter shows symbolically that " tends to zero for various functions at ìgoodî points.
Example 5.1 Increments of y = f [x] = x^3
Let y = f [x] = x^3 and calculate the increment corresponding to a change in x of x. First, we know from Example 1.4.1 that
f [x+x] f [x] x = 3x
(^2) + (3x + x)x
Figure 5.1: The gap near x = 2= 3 on y = x^3
Figure 5.2: y = 1=x
Example 5.2 Exceptional Numbers and the Derivative of y =
x
We follow the steps of Procedure 5.1 and Exercise 5.1.
- Compute f [x + x] f [x] x
f [x + x] f [x] =
x + x
x
x (x + x) x(x + x) =
x(x + x) x
f [x + x] f [x] x
x(x + x)
- Compute
f 0 [x] = lim x! 0
f [x + x] f [x] x
intuitively. It is ìclearî that as x! 0 ,
lim x! 0
x(x + x)
x (x + 0)
x^2
(This is true unless x! 0 at the same time. In particular, if x = x, then (^) x(x +^1 x) is not even deÖned.) At least with x Öxed, we should have
f 0 [x] =
x^2
- We make the ìgapî error explicit by the formula
f [x + x] f [x] x
f 0 [x]
and put the expression on a common denominator
" =
x(x + x)
x^2
x^2
x(x + x)
= x + x x^2 (x + x)
x x^2 (x + x)
= x x^2 (x + x)
= x
x^2 (x + x)
- Show that "! 0 as x! 0. It seems ìclearî that
lim x! 0 " = lim x! 0 x
x^2 (x + x)
x^2 (x + 0)
but plugging in zero is really not quite enough because it misses the point to the approximation we want. We want the (x; y)-graph of the function
Fx[x] =
f [x + x] f [x] x
to approximate the graph of f 0 [x] when x is small. Another way to say this is that we want the whole function "[x; x] to be small independent of x provided x is su¢ ciently small. Then we can move the microscope over these values of x and continue to see a straight line approximating y = f [x].
- Compute f [x + x] f [x] x
f [x + x] f [x] =
p x + x
p x
=
p x + x
p x)(
p x + x +
p x) p x + x + p x =
p x + x +
p x
x
f [x + x] f [x] x
p x + x +
p x
- Compute
f 0 [x] = lim x! 0 f [x + x] f [x] x
Although there certainly may be di¢ culties near x = 0 (see Figure 5.4), if x is Öxed and positive,
lim x! 0
f [x + x] f [x] x
= lim x! 0
p x + x +
p x
p x + 0 +
p x
p x
- Calculate the error gap
" = f^ [x+xx] f^ [x] f 0 [x]
(see Exercise 28.7.3):
f [x + x] f [x] x
f 0 [x]
=
p x + x + p x
p x
=
p x(
p x + x +
p x)^2
x
- Show that " ! 0 as x! 0. It seems ìclearî that
lim x! 0 " = lim x! 0
p x(
p x + x + p x)^2
x
p x(
p x + 0 +
p x)^2
8 x p x
and this shortcut computation (plugging in 0) is justiÖed as a function approximation for all x as long as the term 8 xp^1 x cannot get large. This is guaranteed by making x b for some Öxed positive b > 0.
Figure 5.4: y =
p x
Increment of f [x] =
p x:
f [x + x] f [x] = f 0 [x] x + " x p x + x
p x =
p x
x + " x
with f 0 [x] = 2 p^1 s and " = x=(
p x(
p x + x +
p x)^2 ).
We summarize the knowledge that " can be made small by making x small by writing
y =
p x ) dy = 2 p^1 x dx
This notation means that under su¢ cient magniÖcation the gap between the curve and its tangent will be appear small as shown in Figure 5.5 for x = 2= 3. These formulas are not valid if x 0.
Figure 5.5: The gap near x = 2= 3 on y =
p x
Exercise Set 5.
- y = xp^ ) dy = p xp ^1 dx, p = 1; 2 ; 3 ; : : : For each f [x] = xp^ below:
5.2 Moving the Microscope
This section uses interval notation to give a technical deÖnition of local linearity. This ìuniformî limit allows us to ìmoveî the microscope.
A summary of the "-gap computations so far is
y = f [x] = xp^ ) dy = f 0 [x] dx = pxp ^1 dx; p = 1; 2 ; 3 ; 4 ; 5 y = f [x] = (^) x^1 ) dy = f 0 [x] dx = x 21 dx y = f [x] = (^) x^12 ) dy = f 0 [x] dx = x 32 dx y = f [x] =
p x ) dy = f 0 [x] dx = 2 p^1 x dx y = f [x] = 3
p x ) dy = f 0 [x] dx = 3 p (^31) x 2 dx
More than the summary, we know that the size of the gap (given by ") viewed in a microscope of power 1 =x goes to zero even as x varies, provided we avoid ìbadî points. For the integer powers, we need to have x bounded, jxj b for a Öxed b as you saw in Exercise 5.1.1. (ì1î are the ìbadî points in this case.) The other ìbadî points are fairly obvious from the summary above. If x = 0, the function 1 =x and its derivative 1 =x^2 are undeÖned. We have to expect trouble there. If x < 0 , the function
p x is undeÖned, but even if x = 0, where p x is deÖned, the derived function 1 =( p x) is undeÖned, so we expect trouble. All we need is a way to say where ìgoodî approximations take place. To give a general approach, we want to phrase the exceptions in terms of intervals.
DeÖnition 5.1 Notation for Open and Compact Intervals
If a and b are numbers, we deÖne open intervals as follows
(a; b) = fx : a < x < bg ( 1; b) = fx : x < bg (a; 1 ) = fx : a < bg
We deÖne compact (or "closed and bounded") intervals by [a; b] = fx : a x bg
The condition of ìtangencyî is expressed by the microscopic error formula given next.
Informal Smoothness:
The function f 0 [x] is the derivative of the function f [x] if whenever we make a small change x 0 in input x in the interval of di§erentiability (a; b), then the change in output satisÖes
f [x + x] f [x] = f 0 [x] x + " x
with error " 0. This looks like Informally, this approximation remains valid if we move the microscope anywhere inside an interval of ìgoodî points. The approximation means that a microscopic view of a tiny piece of the graph y = f [x] looks the same as the linear graph dy = f 0 [x] dx. (The lower case (small) Greek delta , indicates intuitively that when the di§erence in x is a su¢ ciently small amount, x 0 , then the error " 0 is
Figure 5.6: A symbolic microscope
small.) When we say f 0 [x] is the derivative of f [x] we mean that this local approximation is valid. We have shown this approximation directly for the functions summarized at the beginning of the section on appropriate compact intervals described in detail at the end of this section. The rules of calculus are wonderful: They tell you where the trouble is going to occur.
Procedure 5.2 The Graph of the Linear Function Given by
dy = m dx
in local (dx; dy)-coordinates at (x; f [x]) is the tangent line to the explicit nonlinear graph
y = f [x]
provided m = f 0 [x] and f 0 [x] can be computed by the rules yielding a formula valid in an interval around x.
SpeciÖcally,
Theorem 5.1 Successful Rules Imply Linear Approximation Suppose the derivative dydx = f 0 [x] can be computed from an explicit formula y = f [x] using the rules of Chapter 6. Also, suppose that both f [x] and f 0 [x] are deÖned on the compact interval [ ; ]. Then the size of the gap, " in f [x + x] f [x] = f 0 [x] x + " x
can be made small for all x in [ ; ] by choosing a su¢ ciently small x. (For all x in [ ; ] if x = x 0 , then " 0 .)
The complete technical deÖnition of smoothness is given in the Mathematical Background materials on the CD accompanying this text. The background also gives a proof of Theorem 5.1. Here is the technical deÖnition.
Example 5.6 Domains of Approximation for y = xp, p = 1= 2 ; 1 = 3 ; : : :
The function y =
p x and its derivative dydx =
p x are deÖned for all positive real x in (0; 1 ). Both
are NOT deÖned at x = 0. Theorem 5.1 says they are di§erentiable on the open interval (0; 1 ), so for pairs of real numbers 0 < < , the gap " can be made small over the compact interval [ ; ] by choosing a single, small enough x. You cannot make " small for the whole interval (0; 1 ).
Exercise Set 5. You can see this function convergence for yourself on the computer.
- " on the Computer Run the program DfctLimit to show graphically that the gap errors of all the following functions tend to zero AS FUNCTIONS OF x away from the ìbadî points.
y = f [x] = xp^ ) dy = f 0 [x] dx = pxp ^1 dx; p = 1; 2 ; 3 ; 4 ; 5 y = f [x] = (^1) x ) dy = f 0 [x] dx = x 21 dx y = f [x] = (^) x^12 ) dy = f 0 [x] dx = x 32 dx y = f [x] = p x ) dy = f 0 [x] dx = 2 p^1 x dx y = f [x] = 3
p x ) dy = f 0 [x] dx = 3 p (^31) x 2 dx
- A View in the Microscope You are told that a certain function y = f [x] has a derivative for all values of x, f 0 [x]. At the point x = 1, we know that f 0 [1] = 2 = 3. Sketch what you would see in a very powerful microscope focused on the graph y = f [x] above the point x = 1.
Compare your work on the next problem about kinks with Exercise 3.2.4.
Problem 5.2 " on the Computer and Analytically for a Kink Run the program DfctLimit on the function
y = f [x] =
p x^2 + 2x + 1
which has derivative dy dx = f 0 [x] =
x + 1 p x^2 + 2x + 1 Show that the function and its derivative are NOT BOTH deÖned at x = 1. Verify analytically that the gap you see on the computer does NOT tend to zero.
Figure 5.7: y =
p x^2 + 2x + 1 near x = 1
5.3 Trigonometric Derivatives
The gaps " for y = f [x] = Sin[x], y = f [x] = Cos[x], and y = f [x] = Tan[x] are calculated in this section by comparing the length of a segment of the unit circle with the vertical and horizontal projections from the ends of the segment.
The derivative of sine in radians is cosine and the derivative of cosine in radians is -sine. These important facts can be seen by magnifying the unit circle. We assume that you know the deÖnition of radian measure of angles and the associated fact that (Cos[]; Sin[]) is the (x; y)-point on the unit circle at the angle , measured counterclockwise from the x-axis. (See Chapter 28, Section 5 and Figure 5.8.) In this section, we use the informal version of DeÖnition 5.2 and work from the relationship between the sine and cosine and the length along the unit circle shown on Figure 5.8. Consider what happens as we move from a point (Cos[]; Sin[]) to a nearby point (Cos[ + ]; Sin[ + ]). We magnify the unit circle, noting on Figure 5.9 that the more we magnify, the straighter the magniÖed portion of the circle appears. The Ögure with small 0 appears to be a triangle at magniÖcation 1 =. The length of the hypotenuse of the apparent triangle is because we use radian measure. (Degrees are not the distance along a unit circle.) The radii coming from the larger Ögure appear to meet it at right angles, so the apparent triangle is similar to the large triangle at the left with hypotenuse 1 and sides Sin[], Cos[]. (You may have to do some geometry to convince yourself of this, since the corresponding edges are at right angles to one another.) The sides of the apparent triangle are the di§erences in sine and cosine, with cosine decreasing - hence a negative sign. Figure 5.9 is the microscopic view of the circle that gives us the results Consider the apparent similarity, comparing the long sides of the two triangles, Cos[] 1
sin
Sin[ + ] Sin[] Because we only know the apparent triangle up to a small error, we write only approximate similarity. To be explicit, let the di§erence equal,
" =
Cos[] 1
Sin[ + ] Sin[]
Now do a little algebra to see,
Sin[ + ] Sin[] = Cos[] + "
Figure 5.9: Derivatives of sine and cosine
valid only in radian measure. We take = 6 and = 180 0 ,
f [x + x] f [x] = f 0 [x] x + " x f [ + ] f [] = f 0 [] + " Sin[ + ] Sin[] = Cos[] + "
Sin[
]
p 3 2
Sin[
]
p 3 2
Sin[
] 0 : 484885
The computerís approximation of sine of 29 degrees is 0.48481.
Example 5.8 Limits of Sine
The previous example can be cast in limit notation as: Find
lim ! 0
Sin[ 6 + ] Sin[ 6 ] The solution is to recognize this as a special case of the limit deÖning the derivative,
lim ! 0
Sin[ + ] Sin[]
= Cos[]
with = = 6 , or to use the increment approximation,
Sin[ + ] Sin[] = Cos[] + " Sin[ + ] Sin[] = Cos[] + " Sin[ 6 + ] Sin[ 6 ]
= Cos[
] + "
and recall that
lim ! 0
Sin[ 6 + ] Sin[ 6 ]
= L
is the number L the expression approximates when = 0 is small,
Sin[ 6 + ] Sin[ 6 ] Cos[
]
5.3.1 Di§erential Equations and Functional Equations
It is intuitively clear that magniÖed circles appear straighter and straighter, but complete justiÖcation of the local linearity of sine and cosine requires that we really show that the magniÖed increment of the circle is close to a triangle. We will not do this here except to make two speciÖc uses of identities that are important in their own right. More details are contained in the Mathematical Background chapter on Functional Identities. The formula (Sin[])^2 + (Cos[])^2 = 1
simply says that sine and cosine lie on the unit circle. If x = Cos[] and y = Sin[], x^2 + y^2 = 1 is the equation of the unit circle. Rather than using the increment approximation based on a greatly magniÖed circle, we could use the exact addition formulas to obtain increments of trig functions. In the case of the sine,
Sin[ + ] = Sin[] Cos[] + Cos[] Sin[] For example,
Sin[=6 + ] =
Cos[] +
p 3 2 Sin[]
These are exact formulas for the increments, but we need to obtain the di§erential approximations
Sin[] = + " 1 Cos[] = 1 + " 2
to complete the last step in proving the local linear approximation. The point we wish to illustrate is this: The di§erential d(Sin[]) = Cos[] d
is a sort of simpliÖed version of the functional identity
Sin[ + ] Sin[] = Sin[] Cos[] Sin[] + Cos[] Sin[] = Cos[] + Cos[](Sin[] ) + Sin[](Cos[] 1) = Cos[] + "
that discards the error term ". We know " 0 because magniÖed circles appear straighter and straighter as the magniÖcation increases. This observation gives us two interesting limits. Since
" = Cos[]
Sin[]
Cos[] 1
Figure 5.10: An increment of tangent and the secant Function
Problem 5.3 Derivative of Tangent (Optional ) Find the di§ erential of the tangent function by examining an increment in the Ögures below. The segment on the line x = 1 between two rays from the circle is the increment of the tangent, because SOH- CAH-TOA with adjacent side of length 1 gives Tan[] as the length of the segment on x = 1 between the x-axis and the ray. The area of the triangle in Figure 5.10 with y = (Tan[]) as its tiny vertical side is 12 Tan[], because the ìheightî is 1 for the ìbaseî of Tan[]. The length of the ray at out to x = 1 is 1 = Cos[]. Why? Show the lower estimate 1 2
Cos^2 []
Tan[]
by Önding the area of the circular sector at the left in Figure 5.11. (Note that the area of a circular sector of radius r and angle is 12 r^2 .)
Figure 5.11: A lower estimate and an upper estimate
Similarly, show 1 Cos^2 []
Tan[]
Cos^2 [ + ]
(What is the radius of the sector on the right in Figure 5.11?) The area of both of the sectors in Figure 5.11 is (^) 2 Cos (^2) [] + " , with " 0 when 0. Why? When
is (^) Cos[^1 ] (^) Cos[^1 +] for 0? Prove that y = Tan[] ) dy =
Cos^2 []
d
provided Cos[] 6 = 0.
5.4 Derivatives of Log and Exp
The gaps " for y = ex^ and y = Log[x] are discussed in this section.
The important functional identities of exponential functions are as follows:
Theorem 5.2 Laws of Exponents For a positive base a > 0 and any real numbers p and q
a p^ = (^) a^1 p ap^ aq^ = ap+q a^1 =p^ = p
p a (ap)q^ = ap q
We want to use these properties to show what we need to estimate in order to di§erentiate log and exponential functions. (Practice with the rules can be found in Chapter 28, Section 4, if your skills are rusty.)
Example 5.9 The Exact Increment of y = ax
We write an exact formula for the di§erence ax+x^ ax^ in terms of ax,
ax+x^ ax^ = ax^ ax^ ax = ax^ (ax^ 1) ax+x^ ax x = ax^
ax^ 1 x
Notice that the last formula says The rate of change of y = ax^ for a Öxed change x beginning at x is proportional to ax,
