j´anos fodor - Állatorvostudományi egyetemfunctions whose graphs are horizontal lines are the...

Szent Istvan University, Faculty of Veterinary ScienceDepartment of Biomathematics and Informatics

Biomathematics 4

Functions II.Janos Fodor

Copyright c© [email protected] Revision Date: September 11, 2006 Version 1.25

mailto:[email protected]

Table of Contents

1 Important function classes 3

1.1 Linear functions . . . . . . . . . . . 4

1.2 Power functions . . . . . . . . . . . 11

• Application of Power Functions:How big can a cell be? . . . . . . 19

1.3 Polynomial and Rational Functions . 32

1.4 Exponential Functions . . . . . . . . 39

• The Exponential Function with Basee . . . . . . . . . . . . . . . . . . 47

Table of Contents (cont.) 3

1.5 Logarithmic Functions . . . . . . . . 52

1.6 Periodic Functions . . . . . . . . . . 60

• The Sine Function . . . . . . . . 63

• The Cosine Function . . . . . . . 66

• The Tangent Function . . . . . . 68

• The Cotangent Function . . . . . 69

Toc JJ II J I Back J Doc Doc I

Section 1: Important function classes 4

1. Important function classes

We consider some important particular classes offunctions, namely:

• linear functions,

• power functions,

• polynomials and rational functions,

• exponential functions,

• logarithmic functions,

• periodic functions.



1.1. Linear functions

Linear FunctionA function f is called linear if it can be written inthe form:

f(x) = mx + b,

where m and b are real numbers.

Example. The pressure y measured x meters belowthe sea level is y = 0.1x + 1 atmosphere. Thus, y

is a linear function of x with m = 0.1, b = 1.



The graph of a linear function is a line. If m = 0then f(x) = b, and the graph of f is a horizontalline.

Functions whose graphs are horizontal lines are theconstant functions.

Note that vertical lines are not graphs of functions.



Vertical lines always have the form x = c, where c

is some constant (x = 3, for example).

The domain of a linear function is always the setof all real numbers R. If a linear function is notconstant, its range is also R. If it is constant, e.g.,f(x) = b, then Rf = {b}.

The constants b and m in the form f(x) = mx + b

give us important information about the line we wishto graph.

We can see, by choosing x = 0, that the line passes



through the y-axis at the point (0, b). In otherwords: the y-intercept of f is (0, b) (we simplysay the y-intercept of f is b).

The second constant m, which is the coefficient ofx, tells us the steepness or slope of the line.

Let P1(x1, y1) and P2(x2, y2) be points on a line.Then

Slope =vertical change (rise)

horizontal change (run)=

y2 − y1

x2 − x1.



If the horizontal change is 0, then the line is verticaland has no slope. If the vertical change is 0, thenthe line is horizontal and has zero slope.



Geometric Interpretation of Slope



How can we tell when two nonvertical lines are par-allel or perpendicular to each other? The followingtheorem provides a convenient test.

Parallel and Perpendicular LinesLet f1(x) = m1x+ b1, f2(x) = m2x+ b2 be linearfunctions. Their graphs are

• parallel if and only if m1 = m2 ;

• perpendicular if and only if m1 ·m2 = −1 .



1.2. Power functions

Power FunctionA function f is called a power function if it canbe written in the form

f(x) = xp,

where p is any real number.

For example, f(x) = x2, g(x) = x−3, h(x) = x1/2

are all power functions.



So are y =1

x2 and y = 3√

x since they can be

rewritten as y = x−2 and y = x1/3, respectively.

Note that sometimes functions f(x) = c · xn arealso called power functions (where c 6= 0 constant).

Consider two well-known examples now.



2x3 and 3x2



When n is a positive integer, we see in the nextfigure that power functions are relatively shallownear the origin, but go steeply beyond |x| = 1,i.e. for x larger than 1 or smaller than −1.

For larger values of the power n, the graph of thepower function y = xn gets flatter close to the originand steeper for |x| > 1.

Symmetry properties of power functions depend onwhether n is even (the graphs are all symmetricabout the y axis; a function with this property iscalled an even function) or n is odd (the graphs



are all symmetric about the origin; a function withthis type of symmetry is called an odd function).

In the next figure we can see graphs of a few of theeven (y = x2; y = x4; y = x6 ) and odd (y = x;y = x3; y = x5 ) power functions.



Power functions with integer powers satisfy an im-portant relationship with respect to one another:

For larger powers, the function y = xn, gets flatter(and smaller) close to x = 0 and steeper (andlarger) for large values of x.

For example, at x = 0.1, the function y = f(x) =x2 takes on a larger value (f(0.1) = 0.01) than thefunction y = g(x) = x4 (g(0.1) = 0.0001). Atx = 2, the roles are reversed. (f(2) = 4, whereasg(2) = 16.)



We say that the low powers dominate close to x =0, while the higher powers dominate for large x.This will have important implications on the relativeeffects of terms of various powers in a polynomial.



• Application of Power Functions: How big can a cell be?

Most applications of power functions in biology arerelated to processes of the surface or the volumeof organisms.

Now we try to answer the following questions:

1. What determines the size of a cell and why somesize limitations exist?

2. Why should animals be made of millions of tinycells, instead of just a few hundred large ones?



While these questions seem extremely complicated,a relatively simple mathematical argument can helpin finding the answers.

We will formulate a mathematical model.

A model is just a representation of a real situationwhich simplifies things by representing the mostimportant aspects, while neglecting or idealizingthe other aspects.



Our model is based on the following assumptions:

1. The cell is spherical.

2. The cell absorbs oxygen and nutrients from theenvironment through its surface. We will as-sume that the rate at which nutrients (or oxy-gen) are absorbed is proportional to the surfacearea, S, of the cell.

3. The rate at which nutrients are consumed inmetabolism (i.e. used up) is proportional tothe volume, V , of the cell.



Model of a single cell.

We define the following quantities relevant to a sin-gle cell:



A = net rate of absorption of nutrients per unit time.

C = net rate of consumption of nutrients per unittime.

V = cell volume.

S = cell surface area.

r = radius of the cell.

We now rephrase the assumptions mathematically.

By assumption (2), A is proportional to S. Thismeans that

A = k1S ,



where k1 > 0 is a constant of proportionality.

The value of this constant would depend on the per-meability of the cell membrane, how many poresor channels it contains, and/or any active transportmechanisms that help transfer substances across thecell surface into its interior.

By assumption (3), C is proportional to V , so that

C = k2V ,

where k2 > 0 is a second proportionality constant.The value of k2 would depend on the rate of me-tabolism of the cell, i.e. how quickly it consumes



nutrients in carrying out its activities.

By the first assumption, the surface area and volumeof the cell are:

V =4

3πr3 , S = 4πr2 .

Putting these facts together leads to the followingrelationships between nutrient absorption, consump-tion, and cell radius:

A = k1(4πr2) = (4πk1)r2 ,



C = k2

(4

3πr3

)=

(4

3πk2

)r3.

Note that A and C are now quantities that dependon the radius of the cell.

In order for the cell to survive, the overall rate ofconsumption of nutrients should balance (be equalto) the overall rate of absorption, i.e. C = A:(

4

3πk2

)r3 = (4πk1)r

2 .

One solution is r = 0 (not interesting). If r 6= 0,



then we can cancel a factor of r2 from both sidesto obtain the value of the radius r at which nutrientbalance occurs:

r = 3k1

k2.

We know: for large values of r, higher powers dom-inate; for small r lower powers dominate.

Since at r = 3k1

k2the two functions are equal, it

follows that



• for smaller cell sizes the absorption A ≈ r2 isthe dominant process;

• for large cells, the consumption C ≈ r3 is higherthan absorption.

We conclude:

Cells larger than the critical size r = 3k1

k2will be

unable to keep up with the nutrient demand, andwill not survive.



Thus, the size of the cell has strong implicationson its ability to absorb oxygen and nutrients quicklyenough to feed itself. For these reasons, cells largerthan some maximal size (roughly 1 mm in diameter)rarely occur.

Furthermore, organisms that are bigger than thissize cannot rely on simple diffusion to carry oxygento their parts—they must develop a circulatory sys-tem to allow more rapid dispersal of such life-givingsubstances or else they will perish.

Further example:



Example.

For plants and animals with a shape that is morecomplicated than the sphere, there is still an easygeometric relation between the volume and the sur-face of the body. The volume is a cubic and thesurface is a quadratic function of the linear dimen-sion (such as length, height) of the body. Therefore,the size of the animals can only vary within a certaininterval.

As an illustration, take a giant mouse, in shape likean ordinary one but with a linear dimension (e.g.length) ten times greater. From the above facts



it follows that the mass of the giant mouse is athousand times greater than an ordinary one, whilethe surface of its lung is only a hundred timesgreater. This mouse can hardly survive.



1.3. Polynomial and Rational Functions

Linear (y = mx + b) and quadratic functions (y =ax2+bx+c) are special types of more general math-ematical functions called polynomial functions orsimply polynomials. Examples of polynomial func-tions:

f(x) = 5x3+2x2−3x+5, g(x) = x2−1, h(x) = 6.

Note that polynomials of degree higher than 2 areoccasionally used in biology.



Polynomial FunctionA function f is a polynomial function of degreen if

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0,

where a0, a1, . . . , an are real numbers, n is a non-negative integer, and an 6= 0.

Example.

(a) f(x) = x is a polynomial.

(b) f(x) = 1x is not a polynomial since 1

x = x−1,



and −1 is negative.

(c) f(x) = 16x is a polynomial (n = 1, an = 1

6).

(d) f(x) =√

x + 3x2 is not a polynomial since√x = x1/2 and 1/2 is not an integer.

We postpone the graphing of polynomial functionsuntil we discuss curve sketching as application ofdifferential calculus later on. The following figure isonly an illustration.



Three polynomials of degree 3.



Behaviour for large x: All polynomials are un-bounded as x → +∞ and as x → −∞. In fact,for large enough values of x, we have seen that thepower function y = f(x) = xn with the largestpower, n, dominates over other power functions withsmaller powers. For

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0

the highest power term will dominate for large x.Thus for large x (whether positive or negative)

f(x) ≈ anxn (whenever |x| is large).



Behaviour for small x: Close to the origin, wehave seen that power functions with smallest powersdominate. This means that for x ≈ 0 the polynomialis governed by the behaviour of the smallest (non-zero coefficient) power, i.e,

f(x) ≈ a1x + a0 (for small x).



Now we define rational functions (remember the def-inition of rational numbers).

Rational FunctionsA function is called rational if it is the quotient oftwo polynomial functions.

For example, the function

f(x) =x2 − 1

x2 − 3x + 2is a rational function. You will have the chance tomeet more rational functions in later subjects.



1.4. Exponential Functions

Exponential functions can be used as models for cer-tain types of growth or decay.

Example: Growth of a foal.

We have a foal with weight 50 kg. The weight in-

creases at a rate of 20% during consecutive time

intervals of equal length. Then the weights at the

end of 0, 1, 2, . . . time intervals are

50, 50

(1 +

20

100

), 50

(1 +

20

100

)2

, 50

(1 +

20

100

)3

, . . .



In general, if the initial weight is c and the rate of

growth is p then the weights at the end of 0, 1, 2, . . .

time intervals are

c, c(1 +

p

100

), c

(1 +

p

100

)2

, c(1 +

p

100

)3

, . . .

If b := 1 + p100 then the weight after x time interval

isc · bx (x = 0, 1, 2, . . .).

An animal does not grow in steps, it grows con-tinuously. Does the previous expression have anymeaning if x is a real number?



Mathematically speaking, we try to replace the do-main {0, 1, 2, . . .} by the set R of all real numbers.

At this stage you should take it for granted thatthis is possible. The corresponding function is calledexponential.



Exponential Function The equation

f(x) = bx (b > 0, b 6= 1)

defines an exponential function for each differ-ent constant b, called the base. The independentvariable x may assume any real value.

Thus, the domain of f is the set of all real numbers,and it can be shown that the range of f is the setof all positive real numbers.



Sometimes functions defined by

f(x) = c · ax,

where a 6= 1 is a positive real number, c 6= 0 is areal number, are also called exponential functions.

It is useful to compare the graphs of y = 2x andy = (1/2)x = 2−x by plotting both on the samecoordinate system.

The graph of f(x) = bx for b > 1 looks very muchlike the graph of the particular case y = 2x, and thegraph of f(x) = bx for 0 < b < 1 looks very muchlike the graph of y = (1/2)x.



Basic Properties of the Graph of f(x) = bx,b > 0, b 6= 1

1. All graphs pass through the point (0, 1).

2. All graphs are continuous, with no holes orjumps.

3. The x axis is a horizontal asymptote.

4. If b > 1, then bx increases as x increases.

5. If 0 < b < 1, then bx decreases as x increases.

6. The function f is one-to-one.



• The Exponential Function with Base e

The following expression is important to the studyof calculus (m is a positive integer):(

1 +1

m

)m

.

Interestingly, by calculating the value of the expres-sion for larger and larger values of m (see Tablebelow), it appears that [1 + (1/m)]m approaches anumber close to 2.7183.



It can be shown that as m “increases without bound”,the value of [1 + (1/m)]m approaches an irrationalnumber that we call e. Just as irrational numberssuch as π and

√2 have unending, nonrepeating dec-

imal representations (see Chapter 1), e also has anunending, nonrepeating decimal representation. To12 decimal places,



The constant e turns out to be an ideal base for anexponential function. This is why you will see e usedextensively in expressions and formulas that modelreal-world phenomena.

Since e is a positive number different from 1, for anyreal number x the equation f(x) = ex defines theexponential function with base e. The exponentialfunction with base e is used so frequently that it isoften referred to as the exponential function. Thegraphs of y = ex and y = e−x are shown in the nextfigure.



1.5. Logarithmic Functions

Consider again the example of growth of a foal. Wecan ask: at what time of growth does the weightreach 86.4 kg?

That is, the value of y (the dependent variable) isgiven; find the corresponding value of the indepen-dent variable x:

50 · 1.2x = 86.4.

A new class of functions is required, called logarith-mic functions, as inverses of exponential functions.



If we start with the exponential function, f definedby y = 2x and interchange the variables x and y, weobtain the inverse of f , denoted by f−1 and definedby the equation x = 2y.

The graphs of f , f−1, and the line y = x are shownin the next figure. This new function is given thename logarithmic function with base 2.



Since we cannot solve the equation x = 2y for y

using the algebraic properties discussed so far, weintroduce a new symbol to represent this inverse



function:

y = log2 x (read: “log to the base 2 of x”).

Thus,

y = log2 x is equivalent to x = 2y.

In general, we define the logarithmic function withbase b to be the inverse of the exponential functionwith base b (b > 0, b 6= 1), where

y = logb x is equivalent to x = by.



Logarithmic FunctionThe function f defined for x > 0 by

f(x) := logb x,

where b > 0, b 6= 1, is called the logarithmicfunction with base b.

The domain of a logarithmic function is the setof all positive real numbers and its range is theset of all real numbers. Thus, log10 3 is defined,but log10 0 and log10(−5) are not defined. Typicallogarithmic curves are shown in the next figure.



Properties of Logarithmic Functions

If b, M , N are positive real numbers, b 6= 1, and p and x are real

numbers, then:

1. logb 1 = 0 5. logb MN = logb M + logb N

2. logb b = 1 6. logbMN = logb M − logb N

3. logb(bx) = x 7. logb(Mp) = p logb M

4. blogb x = x (x > 0) 8. logb M = logb N if and only if M = N .



Common and Natural Logarithms

Of all possible logarithmic bases, the base e and thebase 10 are used almost exclusively.

Common logarithms: logarithms with base 10.

Natural logarithms: logarithms with base e.



1.6. Periodic Functions

Take a look at the following graph, which shows theapproximate average daily high temperature in NewYork’s Central Park:



Each year, the pattern repeats over and over, result-ing in the following graph.

This is an example of cyclical or periodic behavior.

Periodic functions describe processes that have pa-rameters of dynamics repeating in time. For in-



stance, the period of the average daily temperatureis 1 year and the period of the parameters of theheart functions is a fraction of one second.

A function f is called periodic if there exists apositive number T such that

f(x + T ) = f(x)

for all real number x. The smallest T with thisproperty is called the period of f .



We model cyclical behavior using the sine, the co-sine, the tangent and the cotangent functions.

• The Sine Function

The sine of a real number t is given by the y-coordinate (height) of the point P in the followingdiagram, in which t is the distance of the arc shown.



sin(t) = y-coordinate of the point P

The period of sin is 2π. The graph of sin:



• The Cosine Function

cos t = the x-coordinate of the point P shown



Fundamental Trigonometric Identity:sin2 t + cos2 t = 1.

The period of cos is 2π. The graph of cos, as youmight expect, is almost identical to that of the sinefunction, except for a ”phase shift” (see the figure).



• The Tangent Function

tan x :=sin x

cos x. The period of tan is π.



• The Cotangent Function

cot x :=cos x

sin x. The period of cot is π.


j´anos fodor - Állatorvostudományi egyetemfunctions whose graphs are horizontal lines are the...

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