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370
Guided Projects
Guided Project 11: Numerical differentiation
Topics and skills: Derivatives, calculator
While the rules of differentiation allow us to compute the derivative of just about any function, there are
practical situations in which these rules cannot be used. For example, in some applications, a relationship
between two variables may be given as a set of data points, but not as a formula. In situations like this, the rate
of change of one variable with respect to the other (that is, the derivative) might be needed, but the rules do not
apply to sets of data. This project focuses on methods for approximating the derivative of a function at a
particular point.
Backward and Forward Difference Quotients
Assuming the limit exists, the definition of the derivative f ' (a) = lim
h→0
f ( a + h) − f ( a )
implies that
h
f ( a + h) − f ( a )
f ' (a) ≈
,
(1)
h
for h near 0. If h > 0, then (1) is referred to as a forward difference quotient and if h < 0, (1) is a backward
difference quotient. The geometry of these formulas is shown in Figure 1.
y
y
y f(x)
(a, f(a))
f'(a) ⬇
(a h) a
0
(a, f(a))
y f(x)
f(a h) f(a)
h
a (a h)
0
x
x
Forward difference: h 0
Backward difference: h 0
Figure 1
1.
2.
Why do you think (1) is called the forward difference quotient if h > 0 and a backward difference quotient
if h < 0?
Let f ( x ) = x .
a. Find the exact value of f ' (4) .
4+h −2
. Therefore we estimate f ' (4) by calculating
h
4+h −2
4+h −2
for values of h near 0. Complete columns 2 and 5 of Table 1 and describe how
h
h
behaves as h approaches 0.
Error
h
h
4+h −2
4 + h − 2 Error
b. By equation (1), f ' (4) ≈
f (4 + h) − f (4)
=
h
h
0.1
0.01
0.001
0.0001
3.
h
–0.1
–0.01
–0.001
–0.0001
Table 1
The accuracy of an approximation is given by
Error = |exact value – approximate value|.
Use the exact value of f ' (4) in part (a) to complete columns 3 and 6 in Table 1. Describe the behavior of
the errors as h approaches 0.
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Numerical differentiation
371
Centered Difference Quotients
Another formula that is used to approximate the derivative of a function at a point is the centered difference
f ( a + h) − f ( a − h)
f ' (a) ≈
.
(2)
quotient (CDQ)
2h
4. Again consider f ( x ) = x .
a. Graph f near the point (4, 2) and let h = 1/ 2 in the centered difference quotient. Show the line whose
slope is computed by the centered difference quotient and explain why the formula approximates f ' (4).
5.
b. Use the centered difference formula to approximate f ' (4) by completing Table 2.
h
Approximation
Error
0.1
0.01
0.001
0.0001
Table 2
c. Explain why it is not necessary to use negative values of h in Table 2.
d. Compare the accuracy of the derivative estimates in part (b) with those found in Steps 2 and 3.
Use the CDQ (2) and a table similar to Table 2 to find a good approximation to f ' (0) for f(x) = (1 + x)–1.
6.
Use the CDQ (2) and a table similar to Table 2 to find a good approximation to f ' (π 6) for f(x) = sin x.
7.
Table 3 gives the distance f(t) fallen by a smokejumper t seconds after she opens her chute.
a. Use the forward difference quotient (1) with h = 0.5 to estimate the velocity of the skydiver at t = 2 s.
b. Repeat part (a) using the centered difference quotient (2).
t (seconds)
f(t) (feet)
0
0
0.5
4
1.0
15
1.5
33
2.0
55
2.5
81
3.0
109
3.5
138
4.0
169
Table 3
Computer Rounding Error
Using difference approximations to approximate derivatives with a computer or calculator is prone to rounding
errors. These errors occur when a calculator rounds a number before using it in an arithmetic calculation. Such
rounding may lead to remarkably inaccurate results.
8. Consider the function f ( x ) = x10 .
a. Use analytical methods to find the exact value of f ' (1) .
b. Use the forward difference quotient to approximate f ' (1) using values of h = 10–2, 10–3, and 10–4. What
do you observe?
c. Compute approximations to f ' (1) using h = 10− n , for n = 5, 6, 7, …, 15. What do you observe?
In Step 8c, you should find that for small enough values of h, the approximations to f ' (1) eventually are 0,
which is clearly a bad estimate. Here is why this error occurs. Suppose h = 10–14. The calculator rounds
f (1 + 10−14 ) to 1 and therefore the forward difference quotient becomes
f (1 + 10−14 ) − f (1)
, which is
10−14
1− 1
or 0.
10−14
d. The remedy to rounding errors in this situation is to use small—but not too small—values of h. Based on
the approximations computed in parts (b) and (c), what is a good approximation to f ' (1)?
estimated to equal
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372
Guided Projects
Guided Project 12: Enzyme kinetics
Topics and skills: Graphing, derivatives
Enzymes are catalysts that facilitate the biochemical reactions that occur within all living organisms. One of the
fundamental laws of enzyme kinetics was proposed by Leonor Michaelis and Maud Menten in 1913. The law
has been supported by laboratory experiments and explained through mathematical modeling. Today MichaelisMenten kinetics are used in many biological models.
An enzyme molecule is designed to “fit” another molecule called a substrate. The substrate (S) and enzyme (E)
form an intermediate complex (ES), which then dissociates to form the final end-product of the reaction (P) and
the original enzyme (which can be re-used; Figure 1). An important question concerns the rate at which product
molecules are formed. Under certain assumptions, Michelis-Menton kinetics relates the rate of production of P
to the amount of substrate present.
Figure 1
1. We let R be the rate of production of the final
product P and we let s be the concentration of
the substrate initially present. Both s and P are
measured in units such as micro-moles (µ M),
while R is measured in µ M/s. The MichaelisMenton law says that
Vs
R( s) =
,
K +s
where V > 0 and K > 0 are constants that are
specific to each enzyme. Let K = 5 µ M and V =
10 µ M/s and graph R as a function of s. Verify
that the graph shown in Figure 2 is correct.
Figure 2
2.
Compute R' ( s ) and confirm that R is an increasing function.
3.
What does the function mean biologically? As the initial concentration of the substrate s increases, what is
the effect on the production rate of P?
4.
Now let’s interpret the constants V and K. Evaluate lim R ( s ). Explain why V is the maximum production
s →∞
rate. Is there any value of s for which the production rate equals V?
5.
How does the shape of the graph of R change if V increases? if V decreases?
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Enzyme kinetics
373
6.
The constant K has the same units as s. Evaluate R(K), the rate of production when s = K. Show that K is
the initial enzyme concentration that gives a production rate of V/2 (half of the maximum rate).
7.
How does the shape of the graph of R change if K increases? if K decreases? Explain why a small value of
K means the enzyme has a high affinity for the substrate.
8.
Evaluate R' (0), the slope of the curve at the origin. How does the slope change if K is increased with V
fixed? How does the slope change if V is decreased with K fixed?
9.
Suppose an enzyme is known to obey Michaelis-Menton kinetics, but the parameters V and K are not
known. Now suppose that two data points
(s1, R1) = (2 µM, 0.5 µM/s) and (s2, R2) = (5 µM, 1 µM/s)
are measured. By substituting the data points into the Michaelis-Menton law, find values of K and V that fit
the data.
10. Graph the production rate function R for the parameter values found in Step 9. Also plot the two data
points. How well does the curve fit the two data points?
11. Occasionally the Michaelis-Menten law is graphed with ln s on the horizontal axis and R on the vertical
axis (a semilog plot). Make a semilog plot of the Michaelis-Menten law with K = 5 µM and V = 10 µM/s.
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374
Guided Projects
Guided Project 13: Elasticity in economics
Topics and skills: Derivatives
Economists apply the term elasticity to supply, demand, income, capital, labor, and many other variables in
systems with input and output. In a few words, elasticity describes how changes in the input to a system are
related to changes in the output. Because elasticity involves change, it also involves derivatives.
In this project we investigate the idea of elasticity as it applies to demand functions. It’s a common experience
that as the price of an item increases, the number of sales of that item generally decreases. This relationship is
expressed in a demand function.
1.
Suppose a gas station has the linear
demand function D(p) = 1200 –
200p (Figure 1). According to this
function, how many gallons of gas
can the gas station owners expect to
sell per month if the price is set at
$4 per gallon?
Figure 1
2.
Evaluate D' ( p) and show that the demand function is decreasing. Explain why demand functions are
usually decreasing functions.
3.
Suppose the price of a gallon of gasoline (Steps 1 and 2) increases from $3.50 to $4.00 per gallon; call this
change ∆p. What is the resulting change in the number of gallons sold; call it ∆D? (Note that the change is
a decrease, so it should be negative.)
4.
Now express the answer to Step 3 in terms of percentages: What is the percent change in price, ∆p/p, when
it increases from $3.50 to $4.00 per gallon? What is the resulting percent change in the number of gallons
sold, ∆D/D? (Note that the percent change is negative.)
5.
The elasticity in the demand is the ratio of the percent change in demand to the percent change in price; that
∆D / D
. Compute the elasticity for the changes in Steps 3 and 4 (it should be negative).
is, E =
∆p / p
6.
The elasticity is simplified by considering small changes in p and D. In this case we use the definition of
the derivative and write
∆D / D
∆D ⎛ p ⎞ dD p
E = lim
= lim
.
⎜ ⎟=
∆ p → 0 ∆p / p
∆p →0 ∆p ⎝ D ⎠
dp D
Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is
p
E ( p) = −
.
6− p
7.
The elasticity may be interpreted as the percent change in the demand that results for every one percent
change in the price. For example if E(p) = –2, a one-percent increase in price produces a two-percent
decrease in demand. If the price of gasoline is p = $4.50 and there is a 3.5% increase in the price, what is
the elasticity and the corresponding percent change in the number of gallons sold?
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Elasticity in economics
8.
9.
375
Graph the gasoline demand elasticity function for 0 ≤ p < 6.
When –∞ < E < –1, the demand is said to be elastic. When –1 < E < 0, the demand is said to be inelastic.
When E = –∞, the demand is perfectly elastic and when E = 0 the demand is perfectly inelastic. Essential
goods such as basic foods tend to have inelastic demands; discretionary items, such as electronic equipment
have elastic demands. Explain the meaning of these terms in this context.
10. For what prices is the gasoline demand function elastic and inelastic?
11. The demand for processed pork in Canada is described by the function D(p) = 286 – 20p1. Graph the
demand function, compute the elasticity, and graph the elasticity. For what prices is the demand function
elastic and inelastic?
12. Show that the general linear demand function D(p) = a – bp, where a and b are positive real numbers, has a
decreasing elasticity for 0 ≤ p < a/b. Show that for the general linear demand function, E = –1 when
p = a/2b.
13. Not all demand functions are linear. Compute the elasticity for the exponential demand function
D(p) =ae–bp, where a and b are positive real numbers.
14. Show that the demand function D(p) =a/pb, where a and b are positive real numbers, has a constant
elasticity for all positive prices.
1
Microeconomics, J. Perloff, Addison Wesley
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376
Guided Projects
Guided Project 14: Pharmacokinetics—drug metabolism
Topics and skills: Exponential functions, graphing
Medications are commonly administered in one of two ways: injection (a shot or bolus) and infusion (an
intravenous line). It turns out that the concentration of most drugs in the blood (for example, antibiotics,
sedatives, anesthesia, and anti-inflammatory drugs) is well approximated by exponential functions. This is true
because most common drugs are eliminated from the blood by an exponential decay process, where every drug
has a known half-life (the half-life also varies among individuals, so each drug has a mean or population halflife).
Injection
Suppose m0 milligrams of a drug are put in the blood by an injection. The amount of drug t hours after the
injection is given by m(t) = m0e–kt, for t ≥ 0, where k is the rate constant, which is related to the half-life. We
also treat oral administration of drugs (swallowing pills) as an injection, although the model is less accurate
because the drug must be absorbed into the blood through the stomach.
1.
2.
3.
4.
5.
6.
Ibuprofen has a short half-life of 1.5 hours. Find the rate constant k for ibuprofen and write the function that
gives the drug level after t hours. Graph the drug function m with m0 = 400 for 0 ≤ t ≤ 10 hours. How much
drug remains in the blood 4 hr and 8 hr after a 400-mg dose is taken?
Show that if the half-life T1/2 of a drug is known, then its rate constant is k = (ln 2)/T1/2.
How many hours after a taking a dose of ibuprofen does the amount of drug in the blood reach 1% of the
amount of the initial dose?
The sedative diazepam has a half-life of 7 hr. Find the drug function m for diazepam. Graph the drug
function with m0 = 5 for 0 ≤ t ≤ 48 hours. How much drug remains in the blood 12 hr and 24 hr after a 5-mg
dose is taken?
The antibiotic tetracycline has a half-life of 9 hours. Suppose a doctor wants a patient to have 100 mg of
tetracycline in the blood 18 hours after an injection. What initial dose meets this requirement?
Twelve hours after a 200-mg dose of a drug is injected, the drug level in the blood is 75 mg. What is the
(approximate) half-life of the drug?
Infusion
An intravenous line provides a continuous flow of a drug directly into the blood. Assuming no initial drug in the
blood, the amount of drug in the blood t hours after the dosing begins is m(t) = (A/k)(1 – e–kt), for t ≥ 0, where k
is the rate constant (again related to the half-life) and A is the rate at which drug flows into the blood (in units of
mg/hr).
7.
8.
Suppose an antibiotic with a half-life of 12 hr is given to a patient intravenously at a rate of A = 50 mg/hr.
Find the rate constant k and graph the drug function m for 0 ≤ t ≤ 48.
What is the steady-state level of the antibiotic in Step 7? That is, evaluate lim m(t ). Verify that the limit is
t →∞
consistent with the graph in Step 7.
9. In general, what is the steady-state level of a drug delivered by infusion in terms of A and k? In general, at
what time does the drug level reach 90% of the steady-state level, in terms of A and k?
10. Based on a patient’s weight, a doctor targets a steady state level of tetracycline of 100 mg through infusion.
What infusion rate A should be used? The half-life of tetracycline is 9 hr.
11. In Step 10, at what time does the drug level reach 90% of the steady-state level? At that time, how much
drug has actually been delivered?
12. Suppose a patient has been on infusion of tetracycline for 72 hours with infusion rate as found in Step 10,
when the delivery is terminated. How long does it take for the drug level in the blood to reach 2 mg?
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