Capítulo III: Evaluación Externa
3.5. El Sector Industrial de la Caña de Azúcar en el Perú y sus Competidores
The hyperbolic trig functions are defined as follows. sinh x = 1 2 e x− e−x cosh x = 1 2 e x+ e−x and tanh x = sinh x cosh x .
Their inverses can be calculated using the following relations: sinh−1x = lnx +px2+ 1 cosh−1x = lnx +px2− 1 tanh−1x =1 2ln 1 + x 1 − x
The derivatives are as follows:
(sinh x)0= cosh x (sinh−1x)0 = (x2+ 1)−1/2 (cosh x)0= sinh x (cosh−1x)0 = (x2− 1)−1/2
(tanh x)0 = (cosh x)−2 (tanh−1x)0= (1 − x2)−1
Review problems
a1 For what set of angles θ do we have both sin θ < 0 and
cos θ < 0? . Solution, p. 227
a2 Let the function f be defined by f (x) = x3 + 1. Find an
expression for the function f−1. √
Problems
c1 Differentiate ln(2t + 1) with respect to t.
. Solution, p. 227
c2 Differentiate a sin(bx + c) with respect to x.
. Solution, p. 227
c3 Differentiate the following with respect to x: e7x, eex. (In the latter expression, as in all exponentials nested inside exponentials, the evaluation proceeds from the top down, i.e., e(ex), not (ee)x.)
. Solution, p. 227
c4 The range of a gun, when elevated to an angle θ, is given by R = 2v
2
g sin θ cos θ .
Find the angle that will produce the maximum range.
. Solution, p. 228
c5 Prove, as claimed on p. 129, that the derivative of tan θ with respect to θ is (cos θ)−2. Assume that the derivatives of the sine and
cosine are already known. . Solution, p. 228
c6 Show that the function sin(sin(sin x)) has maxima and min- ima at all the same places where sin x does, and at no other places.
. Solution, p. 228
c7 Find any extrema of the hyperbolic cosine function defined
on p. 129. . Solution, p. 229
In problems e1-e4, differentiate the given functions.
e1 sin cos tan x √
e2 ln cos ex √
e3 exp sin ln x √
e4 tan−1√ln x √
e5 Differentiate the function xx. √
A Mercator projection, prob- lem e6. Note the extremely exaggerated scale at the poles.
e6 On a map drawn using a Mercator projection, the y coor- dinate on the paper is given by y = a tanh−1sin φ, where φ is the latitude, a is a constant, and the inverse hyperbolic tangent function is defined on p. 129. (a) Find the derivative dy/ dφ, which indicates the latitude-dependent scale of the map in the north-south direc- tion. (b) The approximations tanh x ≈ x and sin x ≈ x are valid for small x. Use these approximations to approximate the behavior of y(φ) for small φ, and use this to check your answer to part a.√
f1 A cold bottle of beer is left outside under a shady tree at a picnic. Its temperature as a function of time is given by
T = a − be−ct , where a, b, and c are constants.
(a) Infer the units of a, b, and c. (For examples of how to do this, see example 9 on p. 27 and example 1 on p. 119.)
(b) Find the derivative dT / dt, which measures how fast the beer is warming up. Check that its units make sense.
(c) Interpret both the original equation and your answer to part b in the limit where t → ∞.
(d) Interpret the constants a, b, and c physically.
. Solution, p. 229
f2 A person is parachute jumping. During the time between when she leaps out of the plane and when she opens her chute, her altitude is given by an equation of the form
y = b − c
t + ke−t/k
.
where b, c, and k are constants. Because of air resistance, her ve- locity does not increase at a steady rate as it would for an object falling in vacuum.
(a) What units would b, c, and k have to have for the equation to make sense? (For examples of how to do this, see example 9 on p. 27, example 1 on p. 119, and problem f1 above.)
(b) Find the person’s velocity, v, as a function of time. √ (c) Use your answer from part b to get an interpretation of the con- stant c.
(d) Find the person’s acceleration, a, as a function of time. √ (e) Use your answer from part d to show that if she waits long enough to open her chute, her acceleration will become very small.
f3 If an object is vibrating, and the vibration is gradually dying out, its motion is typically of the form
x(t) = A cos(ωt + δ)e−bt , where A, ω, δ, and b are constants.
(a) Infer the units of each of the four constants, and give a physical interpretation. (For examples of how to infer the units, see example 9 on p. 27, example 1 on p. 119, and problem f1 above.)
(b) Find the velocity.
(c) Check that the units of your answer to part b make sense. √
f4 Sometimes doors are built with mechanisms that automati- cally close them after they have been opened. The designer can set both the strength of the spring and the amount of friction. If there is too much friction in relation to the strength of the spring, the door takes too long to close, but if there is too little, the door will oscillate. For an optimal design, we get motion of the form
x = cte−bt ,
where x is the position of some point on the door, and c and b are positive constants. (Similar systems are used for other mechanical devices, such as stereo speakers and the recoil mechanisms of guns.) In this example, the door moves in the positive direction up until a certain time, then stops and settles back in the negative direction, eventually approaching x = 0. This would be the type of motion we would get if someone flung a door open and the door closer then brought it back closed again. (a) Infer the units of the constants b and c. (For examples of how to do this, see example 9 on p. 27, example 1 on p. 119, and problem f1 above.)
(b) Find the door’s maximum speed (i.e., the greatest absolute value of its velocity) as it comes back to the closed position. √ (c) Show that your answer has units that make sense.
Problem g2. Probability of death in the U.S. in the year 2003. Note the logarithmic scale on the vertical axis. Between the ages of about 30 and 95, the death rate rises exponentially, as shown by the linearity of the data on the logarithmic graph.
g1 Credit card fraud creates costs (including both economic costs and inconvenience) for businesses, credit card holders, and the credit card companies. If the company institutes a particular measure to prevent fraud, it may be able to eliminate some fraction of the fraud that would otherwise have occurred. Putting some additional measure in place may then eliminate some fraction of the remaining fraud, further reducing the total amount. Let the amount the company spends on prevention be p. For the reasons described above, it’s reasonable to imagine that fraud falls off exponentially as a function of p, so that the total cost to the company is
C(p) = p + ae−bp .
Here a and b are constants, the first term represents the cost of carrying out the fraud prevention, and the second term represents the cost of the fraud that was not prevented.
(a) Find the value of p that minimizes the cost. √ (b) Check that the units of your answer make sense.
(c) For what values of the parameters a and b does your answer not produce a meaningful result? Check that this makes sense.
(d) Suppose that legislation forces the credit card company to suffer more of the consequences of the fraud, rather than making their customers bear the brunt. What change does this imply in the parameters of the model? Check that your answer to part a shows the right trend when this change is applied.
g2 Benjamin Gompertz (1779-1865) was a British mathemati- cian and pioneering actuarial scientist, who overcame significant so- cial barriers due to antisemitism. We would all like to live forever, and actuaries are in the business of telling us that we probably can’t. Based on mortality data, Gompertz constructed a model in which an initial population No of babies born at t = 0 becomes at a later
time t a surviving population
N = Noe1−e
t
,
where I’ve simplified the expression by leaving out some constants. If you’ve survived to age t, then your probability of dying in the coming year is
−∆N
N ,
where −∆N is the number of deaths per year. Therefore the death rate is
−1 N
dN
dt .
Show that in the Gompertz model, this death rate is proportional to et. This exponential rate of increase is demonstrated in the figure.
g3 In problem g1 on p. 134, we minimized a function that looked like
y = x + ae−bx ,
where x, a, and b were all positive. Suppose instead that the function had been
y = x2+ ae−bx ,
with the corresponding quantities still being positive. Using the same technique to find its minimum, we obtain an equation of a type called a transcendental equation, which cannot be solved exactly for x in terms of elementary functions. Use the intermediate value theorem to prove that such a minimum nevertheless exists, as long as a and b are both greater than zero.
k1 Proof by induction was introduced in section 2.6.1, p. 54. Use induction to prove that
dn dxnb
x= (ln b)nbx .
To understand what’s going on, you may wish to calculate the first few derivatives; however, doing this and observing the pattern does not constitute a proof.
k2 The function
f (x) = e−12x 2
defines the standard “bell curve” of statistics. (Note that exponenti- ation is not associative, and that in exponentiation, xyz means x(yz), not (xy)z; an expression of the latter form is not very interesting, since it simply equals x(yz).)
Proof by induction was introduced in section 2.6.1, p. 54. Use in- duction to prove that the nth derivative of f is of the form
f(n)(x) = Pn(x)e−
1 2x
2
,
where Pn is an nth order polynomial. To understand what’s going
on, you may wish to calculate the first few derivatives; however, doing this and observing the pattern does not constitute a proof.
.Box 6.1 More indetermi- nate forms
We will mainly be con- cerned with the indeterminate form 0/0, but there are other ones as well. Suppose we try to evaluate the limit
lim θ%π/2 π 2 − θ tan θ by plugging in θ = π/2. This fails because the first factor goes to zero, but the tangent factor blows up to infinity. This is an example of the indetermi- nate form 0 · ∞. The limit is defined and equals 1, but plug- ging in won’t tell us that.
The limit lim
x→∞
√
x + 1 −√x − 1 is an example of the indetermi- nate form ∞ − ∞. It equals zero.
Chapter 6
Indeterminate forms and
L’H ˆopital’s rule
6.1
Indeterminate forms
6.1.1 Why 1/0 and 0/0 are not morally equivalent
If you enter 1/0 and 0/0 into your calculator, it probably flashes the same error message in both cases. You learned in grade school that division by zero is “undefined.” But there are completely dif- ferent reasons why these two types of division by zero are undefined. Briefly:
• 1/0 is undefined as a real number because it would have to be infinite, and the real number system doesn’t include infinite numbers.1
• 0/0 is undefined because writing this expression doesn’t give enough information to say what it equals.
Suppose that for some real number x, we had 0
0 = x .
Multiplying by 0 on both sides gives a condition 0 = 0x
that x should satisfy. But every real number has this property, so writing 0/0 doesn’t give enough information to say whether x is defined and, if so, what its value is. Expressions of this “not- enough-information” type are called indeterminate forms.
6.1.2 Indeterminate forms from brute force on a limit
When we try to evaluate a limit, usually our first attempt is simply to plug in and see if a number comes out. For example, if we want to evaluate
lim
x→0
1 + x
3 + x ,
we will naturally try plugging in x = 0, get the result 1/3, and we’re done. This is not an indeterminate form. But, for example, suppose
1
See section 2.9, p. 60, and example 11, p. 105.
that f (x) = x2 and we want to evaluate f0(1). The definition of the derivative in terms of a limit gives
lim
h→0
(1 + h)2− 1
h ,
and attempting to plug in h = 0 results in the indeterminate form 0/0. This limit is well defined; it equals 2. But the indeterminate form tells us that the brute-force technique was too crude, and we needed to handle the calculation a little more delicately.
The indeterminate form 0/0 can also be undefined. For example, limx&0
√ x x2 = ∞.
6.2
L’H ˆopital’s rule in its simplest form
Every derivative, if defined, can be seen as a case of the indetermi- nate form 0/0. Conversely, we can often convert a 0/0-type limit into a problem in evaluating derivatives. Suppose that we want to calculate a limit of the form
lim
x→a
u(x)
v(x) ,
where u(a) = 0 and v(a) = 0. Then ∆u = u(x) − u(a) means the same thing as u, and similarly, ∆v equals v. So we can rewrite our limit as lim x→a ∆u ∆v , or lim x→a ∆u/∆x ∆v/∆x .
If v0(a) 6= 0, then by property P6 of the limit, p. 87, our limit
becomes
limx→a∆u/∆x
limx→a∆v/∆x
,
which equals
u0(a) v0(a) . We have proved the following.
a/Guillaume de L’H ˆopital (1661-1704) was a French marquis. Born into a military family, he eventually became a mathematician because of bad eyesight. He wrote the first calculus textbook. As acknowledged in the preface, the results given in the book originated with Leibniz and the Bernoulli brothers, but L’H ˆopital’s own name has become attached to the theorem known as L’H ˆopital’s rule. When students meet the Marquis, they always wonder about his name, which looks like the English word “hospi- tal.” Actually, he spelled it with an “s,” and it is the same word in French. The “H” is silent, and the accent is on the “a.” As French people gradually stopped pronouncing the “s,” they stopped writing it, but put the housetop accent on the “ ˆo” to show what they were leaving out. The family name probably comes from an early association with a “hospital,” a word that in medieval times had a broader meaning, encompassing institutions such as guest-houses for pilgrims and what we would today call subsidized public housing.
Theorem: L’Hˆopital’s rule (simplest form)
If u and v are functions with u(a) = 0 and v(a) = 0, the derivatives u0(a) and v0(a) are defined, and the derivative v0(a) 6= 0, then lim x→a u v = u0(a) v0(a) .
We will generalize L’Hˆopital’s rule in section 6.3, p. 140.
Example 1 .Evaluate lim x →0 sin x x + x3
. Attempting to plug in x = 0 gives the indeterminate form 0/0, and this suggests applying L’H ˆopital’s rule. The derivative of the top is cos x , and the derivative of the bottom is 1+3x2. Evaluating these at x = 0 gives 1 and 1, so the answer is 1/1 = 1.
Example 2 The limit lim x →1 3x2− x − 2 x2− 1
is of the form00, so we can try to apply l’H ˆopital’s rule. We get
lim x →1 3x2− x − 2 x2− 1 = 6x − 1 2x = 5 2
6.3
Fancier versions of L’H ˆopital’s rule
Mathematical theorems are sometimes like cars. I own a Honda Fit that is about as bare-bones as you can get these days, but persuading a dealer to sell me that car was like pulling teeth. The salesman was absolutely certain that any sane customer would want to pay an extra $1,800 for such crucial amenities as upgraded floor mats and a chrome tailpipe. L’Hˆopital’s rule in its most general form is a much fancier piece of machinery than the stripped-down model described in section 6.2. The price you pay for the deluxe model is that the proof becomes much more complicated. I’ll state the fancier versions of L’Hˆopital’s rule without proof.
6.3.1 Multiple applications of the rule
In the following example, we have to use l’Hˆopital’s rule twice before we get an answer.
Example 3 .Evaluate lim x →π 1 + cos x (x − π)2
.Applying l’H ˆopital’s rule gives − sin x 2(x − π) ,
which still produces 0/0 when we plug in x = π. Going again, we get
− cos x
2 =
1
2 .
This works because of the following generalization of L’Hˆopital’s rule
Theorem: L’Hˆopital’s rule (first generalization) If u and v are functions with u(a) = 0 and v(a) = 0, and the derivatives u0(a) and v0(a) are defined, then
lim x→a u v = limx→a u0(x) v0(x) .
The difference from the original form of the theorem is that we no longer require v0(a) 6= 0, and the right-hand side has a limit. In cases where v0(a) 6= 0, the original form would have been good enough, but the general form also works, since the limit on the right-hand side can be evaluated simply by plugging in. We will prove this more general form of the rule in section 6.3.4, p. 142.
6.3.2 The indeterminate form ∞/∞
Consider an example like this: lim
x→0
1 + 1/x
This is an indeterminate form like ∞/∞ rather than the 0/0 form for which we’ve already proved l’Hˆopital’s rule. L’Hˆopital’s rule applies to examples like this as well. This can be proved by rewriting an expression like lim u/v, where both u and v blow up, in terms of new variables U = 1/u and V = 1/v. The result is to reduce the ∞/∞ form to the 0/0 form. The proof is carried through in section 6.3.4, p. 142. Example 4 .Evaluate lim x →0 1 + 1/x 1 + 2/x .
. Both the numerator and the denominator go to infinity. Differ- entiation of the top and bottom gives (−x−2)/(−2x−2) = 1/2. We can see that the reason the rule worked was that (1) the constant terms were irrelevant because they become negligible as the 1/x terms blow up; and (2) differentiating the blowing-up 1/x terms makes them into the same x−2on top and bottom, which cancel. Note that we could also have gotten this result without l’H ˆopital’s rule, simply by multiplying both the top and the bottom of the orig- inal expression by x in order to rewrite it as (x + 1)/(x + 2).
6.3.3 Limits at infinity
It is straightforward to prove a variant of l’Hˆopital’s rule that allows us to do limits at infinity. We use a change of variable to change a limit like limx→∞u(x)/v(x) to a new limit stated in terms
of a variable X = 1/x. The proof is left as an exercise (problem z1, p. 145). The result is that l’Hˆopital’s rule is equally valid when the limit is at ±∞ rather than at some real number a.
Acme or Glutco? Example 5
.You have some money, and two choices of what to invest it in. A share in Acme, Inc., costs $7, and returns a dividend of $1 per year. A share of Glutco costs $30 and gives a dividend of $2 per year. If we want to compare the long-term value of the two investments, a natural way to do it is with the limit
lim
t→∞
−7 + t
−30 + 2t .
The top represents the net return on Acme, the bottom Glutco. If this limit is greater than 1, then Acme is the better long-term investment. What is the value of this limit?
.Differentiation of the top gives 1, and differentiation of the bot- tom gives 2. The limit is therefore 1/2, and you’re wiser to invest in Glutco. The interpretation is that the constant terms are irrele- vant, and in the long run the competition between the numerator and denominator is determined by which one grows faster.