PDF- calculus pdf,advanced calculus,differential -Time to conquer Calculus! - Photomath - Calculus


Applied Calculus Math 215

Karl Heinz Dovermann Professor of Mathematics University of Hawaii

July 5,

c Copyright 1999 by the author

All rights reserved

No part of this

publication may be reproduced,

without the prior written permission of the author

Printed in the United States of America

This publication was typeset using AMS-TEX,

the American Mathematical Society’s TEX macro system,

The graphics were produced with the help of Mathematica1

This book is dedicated to my wife Emily (Eun Hee) and my sons Christopher and Alexander

This is a draft which will undergo further changes

Mathematica Version 2

Wolfram Research,


Illinois (1993)

Contents Preface

1 Lines

2 The 2

Derivative Definition of the Derivative

Differentiability as a Local Property

Derivatives of some Basic Functions

Slopes of Secant Lines and Rates of Change Upper and Lower Parabolas

Other Notations for the Derivative

Exponential Growth and Decay

More Exponential Growth and Decay

Differentiability Implies Continuity

Being Close Versus Looking Like a Line

Rules of Differentiation

15 Summary

Definition and Notation 5

Preface These notes are written for a one-semester calculus course which meets three times a week and is,

The course is designed for life science majors who have a precalculus back ground,

and whose primary interest lies in the applications of calculus

We try to focus on those topics which are of greatest importance to them and use life science examples to illustrate them

At the same time,

we try of stay mathematically coherent without becoming technical

To make this feasible,

we are willing to sacrifice generality

There is less of an emphasis on by hand calculations


more complex and demanding problems find their place in a computer lab

In this sense,

we are trying to adopt several ideas from calculus reform

Among them is a more visual and less analytic approach

We typically explore new ideas in examples before we give formal definitions

In one more way we depart radically from the traditional approach to calculus

We introduce differentiability as a local property without using limits

The philosophy behind this idea is that limits are the a big stumbling block for most students who see calculus for the first time,

and they take up a substantial part of the first semester

Though mathematically rigorous,

our approach to the derivative makes no use of limits,

allowing the students to get quickly and without unresolved problems to this concept

It is true that our definition is more restrictive than the ordinary one,

and fewer functions are differentiable in this manuscript than in a standard text

But the functions which we do not recognize as being differentiable are not particularly important for students who will take only one semester of calculus

In addition,

in our opinion the underlying geometric idea of the derivative is at least as clear in our approach as it is in the one using limits

More technically speaking,

instead of the traditional notion of differentiability,

we use a notion modeled on a Lipschitz condition

Instead of an -δ definition we use an explicit local (or global) estimate

For a function to be differentiable at a point x0 one requires that the difference between the iii

function and the tangent line satisfies a Lipschitz condition2 of order 2 in x − x0 for all x in an open interval around x0 ,

instead of assuming that this difference is o(x − x0 )

This approach,

which should be to easy to follow for anyone with a background in analysis,

has been used previously in teaching calculus

The au¨ thor learned about it when he was teaching assistant (Ubungsgruppenleiter) for a course taught by Dr

Bernd Schmidt in Bonn about 20 years ago

There this approach was taken for the same reason,

to find a less technical and efficient approach to the derivative

Schmidt followed suggestions which were promoted and carried out by Professor H

Karcher as innovations for a reformed high school as well as undergraduate curriculum

Professor Karcher had learned calculus this way from his teacher,

Heinz Schwarze

There are German language college level textbooks by K¨ utting and M¨ oller and a high school level book by M¨ uller which use this approach

Calculus was developed by Sir Isaac Newton (1642–1727) and Gottfried Wilhelm Leibnitz (1646–1716) in the 17th century

The emphasis was on differentiation and integration,

and these techniques were developed in the quest for solving real life problems

Among the great achievements are the explanation of Kepler’s laws,

the development of classical mechanics,

and the solutions of many important differential equations

Though very successful,

the treatment of calculus in those days is not rigorous by nowadays mathematical standards

In the 19th century a revolution took place in the development of calculus,

foremost through the work of Augustin-Louis Cauchy (1789–1857) and Karl Weierstrass (1815–1897),

when the modern idea of a function was introduced and the definitions of limits and continuous functions were developed

This elevated calculus to a mature,

mathematically satisfying theory

This also made calculus much more demanding

A considerable,

mathematically challenging setup is required (limits) before one comes to the central ideas of differentiation and integration

A second revolution took place in the first half of the 20th century with the introduction of generalized functions (distributions)

This was stimulated by the development of quantum mechanics in the 1920ies and found is final mathematical form in the work of Laurent Schwartz in the 1950ies

What are we really interested in

? We want to introduce the concepts of differentiation and integration

The functions to which we like to apply these techniques are those of the first period

In this sense,


Trigonometric Series,

Cambridge University Press,

reprinted with corrections and some additions 1968

v need the powerful machine developed in the 19th century

we like to be mathematically rigorous because this is the way mathematics is done nowadays

This is possible through the use of the slightly restrictive notion of differentiability which avoids the abstraction and the delicate,

technically demanding notions of the second period

To support the student’s learning we rely extensively on examples and graphics

Often times we accept computer generated graphics without having developed the background to deduce their correctness from mathematical principles

Calculus was developed together with its applications

Sometimes the applications were ahead,

and sometimes the mathematical theory was

We incorporate applications for the purpose of illustrating the theory and to motivate it

But then we cannot assume that the students know already the subjects in which calculus is applied,

and it is also not our goal to teach them

For this reason the application have to be rather easy or simplified


Chapter 0

A Preview In this introductory course about calculus you will learn about two principal concepts,

differentiation and integration

We would like to explain them in an intuitive manner using examples

In Figure 1 you see the graph of a function

Suppose it represents a function which describes the size of a

Figure 1: Yeast population as a function of time

population of live yeast bacteria in a bun of pizza dough

Abbreviating 1



time by t (say measured in hours) and the size of the population by P (say measured in millions of bacteria),

we denote this function by P (t)

You like to know at what rate the population is changing at some fixed time,

the rate of change is its slope

We like to apply the idea of rate of change or slope also to the function P (t),

although its graph is certainly not a straight line

What can we do

? Let us try to replace the function P (t) by a line L(t),

at least for values of t near t0

The distance between the points (t,

P (t)) and (t,

L(t)) on the respective graphs is E(t) = |P (t) − L(t)|

This is the error which we make by using L(t) instead of P (t) at time t

We will require that this error is “small” in a sense which we will precise soon

If a line L(t) can be found so that the error is small for all t in some open interval around t0 ,

then we call L(t) the tangent line to the graph of P at t0

The slope of the line L(t) will be called the slope of the graph of P (t) at the point (t0 ,

P (t0 )),

or the rate of change of P (t) at the time t = t0

Figure 2: Zoom in on a point

Figure 3: Graph & tangent line

Let us make an experiment

Put the graph under a microscope or,

P (4)) on the graph

This process works for the given example and most other functions treated in these notes

You see the zoom picture in Figure 2

Only under close

you detect that the graph is not a line,

let us ignore this bit of bending and pretend that the shown piece of graph is a line

Actual measurements in the picture let you suggest that the slope of that line should be about −70

This translates into the statement that the population of the live bacteria decreases at a rate of roughly 70 million per hour

In Figure 3 we drew the actual tangent line to the graph of P (t) at t = 4

A calculation based on the expression for P (t),

which you should be able to carry out only after having studied a good part of this manuscript,

shows that the value of the slope of this line is about −67

You may agree,

that the geometric determination of the rate of change was quite accurate

To some extent,

it is up to us to decide the meaning of the requirement • |P (t) − L(t)| is small for all t near t0

One possible requirement1 ,

which it technically rather simple and which we will use,

is: • The exists a positive number A and an open interval (a,

|P (t) − L(t)| ≤ A(t − t0 )2

The inequality in (2) dictates how close we require the graph of P (t) to be to line L(t)

There may,

exist an interval and a number A such that the inequality holds for an appropriate line

If the line,

Its slope is called the derivative of P (t) at t0 ,

and we say that P (t) is differentiable at t0

Remembering that the rate of change of line L(t) is its slope,

we say • If P (t) is a function which is differentiable at t0 ,

the rate at which P (t) changes when t = t0

In a standard treatment a weaker condition,

which depends on the notion of limits,

Our choice of requirement and our decision to avoid limits is based on the desire to keep the technicalities of the discussion at a minimum,

and to make these notes as accessible as possible

Different interpretations of the word ‘small’ lead to different ideas about differentiability

More or fewer functions will be differentiable

The notion of the derivative,

is not effected by the choice of meaning for the word

On the other hand,

the interpretation of the word ‘small’ has to imply the uniqueness of the derivative



In due time we will explain all of this in more detail

You noticed that we need the idea of a line

When you look at (2) and see the square of the variable you can imagine that we need parabolas

So we review and elaborate on lines and parabolas in Chapter 1

We also introduce the,

two most important functions in life science applications,

the exponential function and the logarithm function

Chapter 2 is devoted to the precise definition of the derivative and the exploration of related ideas

Relying only on the definition,

we calculate the derivative for some basic functions

Then we establish the major rules of differentiation,

which allow us to differentiate many more functions

Chapter 3 is devoted to applications

We investigate the ideas of monotonicity and concavity and discuss the 1st and 2nd derivative tests for finding extrema of functions

In many applications of calculus one proceeds as follows

One finds a mathematical formulation for a problem which one encounters in some other context

One formulates the problem so that its solution corresponds to an extremum of its mathematical formulation

Then one resorts to mathematical tools for finding the extrema

Having found the solution for the mathematically formulated problem one draws conclusions about the problem one started out with

Physical principles dictate that the surface area be minimized

You can derive mathematically that the shape of a body which minimizes the surface area,

This is roughly what you see

There is a slight perturbation due to the effect of gravity

This effect is much greater if you take a drop of water,

for which the internal forced are not as strong as the ones in a drop of mercury

Often calculus is used to solve differential equations

These are equations in which a relation between a function and its rate of change is given2

The unknown in the equation is the function

for some simple population models the equation (Malthusian Law) P 0 (t) = aP (t) is asserted

The rate at which the population changes (P 0 (t)) is proportional to the size of the population (P (t))

We solve this and some other population related differential equations

We will use both,

analytical and numerical means

The second principal concept is the one of the integral

Suppose you need to take a certain medication

Your doctor prescribes you a skin patch

Let 2 In more generality,

the relation may also involve the independent variable and higher derivatives

Figure 4: Constant Rate

Figure 5: Amount absorbed

us say that the rate at which the medication is absorbed through the skin is a function R(t),

where R stands for rate and t for time

It is fair to say,

that over some period of time R(t) is constant,

3 mg/hr

The situation is graphed in Figure 4

Over a period of three hours your body absorbs

We multiplied the rate at which the medication is absorbed with the length of time over which this happened

Assuming that you applied the patch at time t = 0,

the three hours would end at time t = 3

An interpretation of the total amount of medication which is absorbed between t = 0 and t = 3 is the area of the rectangle bounded by the line t = 0,

and the graph of the function R(t) =

Its side lengths are 3 and

In Figure 5 you see the function A(t) =

It tells you,

how much medication has been absorbed

Suppose next that the medication is given orally in form of a pill

As the pill dissolves in the stomach,

it sets the medication free so that your body can absorb it

The rate at which the medication is absorbed is proportional to the amount dissolved

As time progresses,

the medication is moved through your digestive system,

and decreasing amounts are available to being absorbed

A function which could represent the rate of absorption as a function of time is shown in Figure 6

We denote it once more by R(t)

Again you may want to find out how much medication has been absorbed within a given time,

say within the first 4 hours after swallowing the pill

Set the time at which you took the pill as time t = 0

It should be reasonable to say (in fact a strong case can be made for this) that the amount of



Figure 6: Time dependent rate

Figure 7: Amount absorbed

medication which has been absorbed between t = 0 and t = T is the area under the graph of R(t) between t = 0 and t = T

We denote this function by A(T )

Using methods which you will learn in this course,

The graph is shown in Figure 7

You may find the value for A(4) in the graph

A numerical calculation yields A(4) = 0

More generally,

one may want to find the area under the graph of a function f (x) between x = a and x = b

To make sense out of this we first need to clarify what we mean when we talk about the area of a region,

in particular if the region is not bounded by straight lines

Next we need to determine the areas of such regions

In fact,

finding the area between the graph of a non-negative function f and the x-axis between x = a and x = b means to integrate f from a to b

Both topics are addressed in the chapter on integration

The ideas of differentiation and integration are related to each other

If we differentiate the function shown in Figure 7 at some time t,

then we get the function in Figure 6 at t

You will understand this after the discussion in Section 4

In this section we also discuss the Fundamental Theorem of Calculus,

which is our principal tool to calculate integrals

The two basic ideas of the rate of change of a function and the area below the graph of a function will be developed into a substantial body of mathematical results that can be applied in many situations

You are expected to learn about them,

so you can understand other sciences where they are applied

Chapter 1

Some Background Material Introduction In this chapter we review some basic functions such as lines and parabolas

In addition we discuss the exponential and logarithm functions for arbitrary bases

In a prior treatment you may only have been exposed to special cases

Remark 1

Calculus (in one variable) is about functions whose domain and range are subsets of,

So we will not repeat this assumption in every statement we make,

unless we really want to emphasize it

Lines in the plane occur in several contexts in these notes,

and they are fundamental for the understanding of almost everything which follows

A typical example of a line is the graph of the function (1

More generally,

one may consider functions of the form (1

where m and b are real numbers

Their graphs are straight lines with slope m and y-intercept (the point where the line intersects the y axis) b

In the example the slope of the line is m = 2 and the y-intercept is b = −3

Even more generally than this,

we have the following definition



Definition 1

A line consists of the points (x,

y) in the x − y-plane which satisfy the equation (1

for some given real numbers a,

where it is assumed that a and b are not both zero

If b = 0,

then we can write the equation in the form x = c/a,

and this means that the solutions of the equation form a vertical line

The value for x is fixed,

and there is no restriction on the value of y

Lines of this kind cannot be obtained if the line is specified by an equation as in (1

The line given by the equation 2x = 3 is shown as the solid line in Figure 1

If a = 0,

then we can write the equation in the form y = c/b,

and this means that the solutions of the equation form a horizontal line,

and there is no restriction on the value of x

The line given by the equation 2y = 5 is shown as the dashed line in Figure 1

If b 6= 0,

then ax + by = c'translates into y = − ab x + cb ,

and the equation describes a line with slope −a/b and y-intercept c/b

Figure 1

Figure 1

Exercise 1

Sketch the lines 5x = 10 and 3y = 5

Exercise 2

Sketch and determine the y-intercept and slope of the lines 3x + 2y = 6 and 2x − 3y = 8

In application,

we are often given the slope of a line and one of its points

Suppose the slope is m and the point on the line is (x0 ,

Then the line is given by the equation y = m(x − x0 ) + y0

Using functional notation,

the line is the graph of the function (1

To see this,

In addition,

you can rewrite the expression for the function in the form y(x) = mx + (−mx0 + b) to see that it describes a line with slope m

Its y-intercept is −mx0 + b

Example 1

The line with slope 3 through the point (1,


we want to find the equation of a line through two distinct,

Assume that x0 6= x1 ,

otherwise the line is vertical

This is slope formula for a line through the point (x0 ,

You should check that y(x1 ) = y1

This means that (x1 ,

In slope intercept form,

the equation of the line is:     y1 − y0 y1 − y0 y(x) = x+ − x0 + y 0

x1 − x 0 x1 − x 0 Example 1

Find the equation of the line through the points (x0 ,

y0 ) = (1,

y1 ) = (3,

Putting the points into the equation of the line,

we find   4 − (−1) 5 7 y(x) = (x − 1) + (−1) = x −

♦ 3−1 2 2 The line is shown in Figure 1



Figure 1

Summarizing the three examples,

we ended up with three different ways to write down the equation of a non-vertical line,

depending on the data which is given to us: • Intercept-Slope Formula: We are given the y-intercept b and slope m of the line

The equation for the line is y = mx + b

• Point-Slope Formula: We are given a point (x0 ,

The equation of the line is y = m(x − x0 ) + y0

• Two-Point Formula: We are given two points (x0

The equation of the line is y(x) =

Exercise 3

Suppose a line has slope 2 and (2,

Using the point (2,

write down the point slope formula for the line and convert it into the slope intercept formula

Find the x and y-intercept for the line and sketch it

Exercise 4

Find the point-slope and intercept-slope formula of a line with slope 5 through the point (−1,

Exercise 5

A line goes through the points (−1,

1) and (2,

Find the two point and slope intercept formula for the line

What is the slope of the line

? Where does the line intersect the coordinate axes

Intersections of Lines Let us discuss intersections of two lines

Consider the lines l1 : ax + by = c'&

They intersect in the point (x0 ,

to find intersection points of two lines we have to solve two equations in two unknowns simultaneously

Example 1

Find the intersection points of the lines 2x + 5y = 7 &


both equations hold if we set x = 1 and y = 1

This means that the lines intersect in the point (1,

As an exercise you may verify that (1,

♦ The lines ax + by = c'and Ax + By = C are parallel to each other if (1

Ab = aB,

and in this case they will be identical,

or they will have no intersection point

Example 1

The lines 2x + 5y = 7 &

To see this,

observe that the second equation is just twice the first equation

A point (x,

y) will satisfy one equation if and only if it satisfies the other one

A point lies on one line if and only if it lies on the other one

So the lines are identical



Example 1

The lines 2x + 5y = 7

are parallel and have no intersection point

To see this,

observe that the first equation,

There are no numbers x and y for which 4x + 10y = 14 and 4x + 10y = 15 at the same time

Thus this system of two equations in two unknowns has no solution,

and the two lines do not intersect

♦ To be parallel also means to have the same slope

If the lines are not vertical (b 6= 0 and B 6= 0),

then the condition says that the slopes −a/b of the line l1 and −A/B of the line l2 are the same

If both lines are vertical,

then we have not assigned a slope to them

If Ab 6= aB,

then the lines are not parallel to each other,

and one can show that they intersect in exactly one point

You saw an example above

If Aa = −bB,

then the lines intersect perpendicularly

Assuming that neither line is vertical (b 6= 0 and B 6= 0),

the equation may be written as a A × = −1

b B This means that the product of the slopes of the first line and −A/B the one of the of one line is the negative reciprocal of the the condition which you have probably seen perpendicularly

of the lines (−a/b is the slope second line) is −1

The slope slope of the other line

This is before for two lines intersecting

Example 1

The lines 3x − y = 1 &

and intersect perpendicularly in (x,

Find the intersection points of the lines l1 (x) = 3x + 4

Sketch the lines and verify your calculation of the intersection point


Exercise 7

Determine the slope for each of the following lines

For each pair of lines,

decide whether the lines are parallel,

Find all intersection points for each pair of lines

l1 : 3x − 2y = 7 l2 : 6x + 4y = 6 l3 : 2x + 3y = 3 l4 : 6x − 4y = 5 Exercise 8

Suppose a line l(x) goes through the point (1,

What is the slope of the line

? Find its slope point formula (use (1,

Sketch the line

Parabolas and Higher Degree Polynomials

A parabola is the graph of a degree 2 polynomial,

and c'are real numbers and a 6= 0

Depending on whether a is positive or negative the parabola will be open up- or downwards

Abusing language slightly,

we say that y(x) is a parabola

We will study parabolas in their own right,

and they will be of importance to us in one interpretation of the derivative

Typical examples of parabolas are the graphs of the functions p(x) = x2 − 2x + 3 and

4 and 1

The first parabola is open upwards,

The x-intercepts of the graph of p(x) = ax2 + bx + c'are also called roots or the zeros of p(x)

To find them we have to solve the quadratic equation ax2 + bx + c'= 0

The solutions of this equation are found with the help of the quadratic formula i p 1 h p(x) = 0 if and only if x = (1

There are three cases to distinguish:



6 1 5 4

Figure 1

Figure 1

• p(x) has two distinct roots if the discriminant is positive

• p(x) has exactly one root if the discriminant is zero

• p(x) has no (real) root if the discriminant is negative

Example 1

Find the roots of the polynomial p(x) = 3x2 − 5x + 2

According to the quadratic formula √  1 3x2 − 5x + 2 = 0 if and only if x = 5 ± 25 − 24

Exercise 9

Find the roots of the following polynomials

Let us find the intersection points for two parabolas,

To find their intersection points we equate p(x) and q(x)

In other words,

we look for the roots of p(x) − q(x) = (a1 − a2 )x2 + (b1 − b2 )x + (c1 − c2 )

The highest power of x in this equation is at most 2 (this happens if (a1 − a2 ) 6= 0),

and this means that it has at most two solutions


Figure 1

Example 1

Find the intersection points of the parabolas p(x) = x2 − 5x + 2

We need to find the solutions of the equation p(x) − q(x) = −x2 − 8x + 7 = 0

According to the quadratic equation,

the solutions are √ √ 1 x = − [8 ± 64 + 28] = −4 ± 23

You see the parabolas in Figure 1

and you can check that our calculation is correct

Find the intersection points for each pair of parabolas from Exercise 9

Graph the pairs of parabolas and verify your calculation

We will study how parabolas intersect in more detail in Section 2

Right now we like to turn our attention to a different matter

In Section 1

The equation (1



In the last term in (1

When we write down the point slope formula of a line with slope m through the point (x0 ,

y = m(x − x0 ) + y0 = m(x − x0 )1 + y0 (x − x0 )0 ,

then we expressed y in powers of (x − x0 )

The mathematical expression for this is that we expanded y in powers of (x − x0 )

We like to do the same for higher degree polynomials

We start out with an example

Example 1

Expand the polynomial (1

Our task is to find numbers A,

y(x) = A(x − 2)2 + B(x − 2) + C

Expanding the expression in (1

comparing the coefficients of y in (1

and C: A = 1 −4A + B = 5 4A − 2B + C = −2 These equations can be solved consecutively,

So y(x) = (x − 2)2 + 9(x − 2) + 12

We expanded y(x) in powers of (x − 2)

Working through this example with general coefficients,

we come up with the following formula: (1

y(x) = ax2 + bx + c'= A(x − x0 )2 + B(x − x0 ) + C


B = 2ax0 + b C = ax20 + bx0 + c'= y(x0 )

In fact,

given any polynomial p(x) and any x0 ,

one can expand p(x) in powers of (x − x0 )

The highest power of x will be the same as the highest power of (x − x0 )

The process is the same as above,

On the computer you can do it in a jiffy

Exercise 11

Expand y(x) = x2 − x + 5 in powers of (x − 1)

Exercise 12

Expand y(x) = −x2 + 4x + 1 in powers of (x + 2)

Exercise 13

Expand y(x) = x3 − 4x2 + 3x − 2 in powers of (x − 1)

Exercise 14

Expand p(x) = x6 − 3x4 + 2x3 − 2x + 7 in powers of (x + 3)

What is the purpose of expanding a parabola in powers of (x − x0 )

? Let us look at an example and see what it does for us

Consider the parabola p(x) = 2x2 − 5x + 7 = 2(x − 2)2 + 3(x − 2) + 5

The last two terms in the expansion form a line: l(x) = 3(x − 2) + 5

This line has an important property: (1

In the sense of the estimate suggested in (2) in the Preview,

we found a line l(x) which is close to the graph of p(x) near x = 2

The constant A in (2) may be taken as 2 (or any number larger than 2),

and the estimate holds for all x in (−∞,

Exercise 15

For each of the following parabolas p(x) and points x0 ,

find a line l(x) and a constant A,

such that |p(x) − l(x)| ≤ A(x − x0 )2

p(x) = 3x2 + 5x − 18 and x0 = 1

p(x) = −x2 + 3x + 1 and x0 = 3

p(x) = x2 + 3x + 2 and x0 = −1



Let us do a higher degree example: Example 1

Let p(x) = x4 − 2x3 + 5x2 − x + 3 and x0 = 2

Find a line l(x) and a constant A,

such that |p(x) − l(x)| ≤ A(x − x0 )2 for all x in the interval I = (1,

(Note that the open interval I contains the point x0 = 2

) Expanding p(x) in powers of (x − 2) we find p(x) = (x − 2)4 + 6(x − 2)3 + 17(x − 2)2 + 27(x − 2) + 21

Set l(x) = 27(x − 2) + 21

Then |p(x) − l(x)| = (x − 2)4 + 6(x − 2)3 + 17(x − 2)2 = (x − 2)2 + 6(x − 2) + 17 (x − 2)2 ≤ (1 + 6 + 17)(x − 2)2 ≤ 24(x − 2)2

In the calculation we used the triangle inequality ((5

If x ∈ (1,

then |x − 2| < 1 and |x − 2|k < 1 for all k ≥ 1

This helps you to verify the second inequality

with A = 24 and l(x) = 27(x − 2) + 21,

we find that |p(x) − l(x)| ≤ A(x − x0 )2 for all x ∈ (1,

Exercise 16

Let p(x) = 2x4 + 5x3 − 5x2 − 3x + 7 and x0 = 5

Find a line l(x) and a constant A,

such that |p(x) − l(x)| ≤ A(x − x0 )2 for all x in the interval I = (4,

Remark 2

The general recipe (algorithm) for what we just did is as follows

Consider a polynomial p(x) = cn xn + cn−1 xn−1 + · · · + c1 x + c0

Pick a point x0 ,

and expand p(x) in powers of x0 : p(x) = Cn (x − x0 )n + Cn−1 (x − x0 )n−1 + · · · + C1 (x − x0 ) + C0

This can always be done,

Set l(x) = C1 (x − x0 ) + C0


Then |p(x) − l(x)| = Cn (x − x0 )n−2 + · · · + C3 (x − x0 ) + C2 (x − x0 )2 ≤ |Cn (x − x0 )n−2 | + · · · + |C3 (x − x0 )| + |C2 | (x − x0 )2 ≤ (|Cn | + |Cn−1 | + · · · + |C2 |) (x − x0 )2 for all x ∈ I = (x0 − 1,

x0 + 1)

The details of the calculation are as follows

To get the equation,

we took |p(x) − l(x)| and factored out (x − x0 )2

To get the first inequality we repeatedly used the triangle inequality,

The last inequality follows as (x − x0 )k < 1 if k ≥ 1

In summary,

for l(x) = C1 (x − x0 ) + C0 and A = (|Cn | + · · · + |C2 |) we have seen that |p(x) − l(x)| ≤ A(x − x0 )2 for all x ∈ (x0 − 1,

x0 + 1)

In the sense of our preview,

and the upcoming discussion about derivatives,

this means • The rate of change of p(x) at the point (x0 ,

Exercise 17

For each of the following polynomials p(x) and points x0 ,

find the rate of change of p(x) when x = x0

p(x) = x2 − 7x + 2 and x0 = 4

p(x) = x4 − x3 + 3x2 − 8x + 4 and x0 = −1

Remark 3

You may have noticed,

that we began to omit labels on the axes of graphs

One reason for this is,

that we displayed more than one function in one graph,

and that means that there is no natural name for the variable associated to the vertical axis

Our general rule is,

that we use the horizontal axis for the independent variable and the vertical one for the dependent one1

This is the rule which almost any mathematical text abides by

In some sciences this rule is reversed

then we suggest that you read Section 5



The Exponential and Logarithm Functions

Previously you have encountered the expression ax ,

where a is a positive real number and x is a rational number

102 = 100,

101/2 =

10−1 =

In particular,

if x = n/m and n and m are natural numbers,

then ax is obtained by taking the n-th power of a and then the m-root of the result

You may also say that y = am/n is the unique solution of the equation y n = am

By convention,

To handle negative exponents,

Exercise 18

Find exact values for  −2 1 43/2 2



Exercise 19

Use your calculator to find approximate values for 34

Until now you may not have learned about irrational (i

not rational) √ √ π 2 exponents as in expressions like 10 or 10

The numbers π and 2 are irrational

We like to give a meaning to the expression ax for any positive number a and any real number x

A new idea is required which does not only rely on arithmetic

If a > 1 (resp

We think of f (x) = ax as a function in the variable x

So far,

this function is defined only for rational arguments (values of x)

The function is monotonic

More precisely,

it is increasing if a > 1 and decreasing if 0 < a < 1

Theorem-Definition 1

Let a be a positive number,

There exists exactly one monotonic function,

called the exponential function with base a and denoted by expa (x),

which is defined for all real numbers x such that expa (x) = ax whenever x is a rational number


∞) as the range2 of the exponential function expa (x)


We will prove this theorem in Section 4

This will be quite easy once we have more tools available

Right now it would be a rather distracting tour-de-force


the exponential function is of great importance and has many applications,

so that we do not want to postpone its introduction

It is common,

and we will follow this convention,

to use the notation ax for expa (x) also if x is not rational

You can see the graph of an exponential function in Figures 1

7 and 1

We used a = 2 and two different ranges for x

In another graph,

you see the graph of an exponential function with a base a smaller than one

We can allowed a = 1 as the base for an exponential function,

and we do not get a very interesting function

The function f (x) = 1 is just a constant function which does not require such a fancy introduction

5 600 2

400 300

1 200 0

Figure 1

Figure 1

Let us illustrate the statement of Theorem 1

Suppose you like to find 2π

You know that π is between the rational numbers 3

14 and 3

Saying that exp2 (x) is increasing just means that 23

Evaluating the outer expressions in this inequality and rounding them down,

81 and 8

In fact,

if r1 and r2 are any two rational numbers,

then due to the monotonicity of the exponential function,



The theorem asserts that there is at least one real number 2π which satisfies these inequalities,

and the uniqueness part asserts that there is only one number with this property,

Figure 1

9: (1/2)x

The arithmetic properties of the exponential function,

also called the exponential laws,

are collected in our next theorem

The theorem just says that the exponential laws,

which you previously learned for rational exponents,

also hold in the generality of our current discussion

You will derive the exponential laws from the logarithm laws later on in this section as an exercise

Theorem 1

For any positive real number a and all real numbers x and y a0 = 1 a1 = a ax ay = ax+y ax /ay = ax−y (ax )y = axy Some of the exponential laws can be obtained easily from the other ones

The second one holds by definition

Assuming the third one,

one may deduce the first and third one

You are invited to carry out these deductions in the following exercises


Exercise 20

Show: If a 6= 0,

Although we did not consider an exponential function with base 0,

This is convenient in some general formulas

If x 6= 0,

Exercise 21

Assume a0 = 1 and ax ay = ax+y

Show ax /ay = ax−y

We need another observation about exponential functions,

the proof of which we also postpone for a while (see Section 4

Theorem 1

Let a and b be positive real numbers and a 6= 1

There exists a unique (i

exactly one) real number x such that ax = b

You may make the uniqueness statement in the theorem more explicit by saying: (1

If ax = ay ,

Let us consider some examples to illustrate the statement in the theorem

We assume that a and b are positive numbers and that a 6= 1

View the expression ax = b

For a given a and b we want to (and the theorem says that we can) find a number x,

The value for x in the last example was obtained from a calculator and is rounded off

Exercise 22

Solve the equation ax = b if

(1) (a,

(3) (a,

(5) (a,

(2) (a,

(4) (a,

(6) (a,



For a given a (a > 0 and a 6= 1) and b > 0 we denote the unique solution of the equation in (1

In other words: Definition 1

If a and b are positive numbers,

then loga (b) is the unique number,

Here are some sample logarithms for the base 2: log2 4 = 2

2 = 1/2

and for the base 10: log10 1 = 0

Your calculator will give you good approximations for at least log10 (x) for any x > 0

Exercise 23

Find logarithms for the base 10: (1) log10 5

Mathematically speaking,

Let us express it this way

Definition 1

Let a be a positive number,

Mapping b to loga (b) defines a function,

called the logarithm function with base a

It is defined for all positive numbers,

and its range is the set of real numbers

Part of the graph of log2 (x) is shown in Figure 1

In Figure 1

We also like to see for every real number y that (1

Setting b = ay in (1

The statement in (1

Taken together,

17) and (1

loga (a ) = x This just means that

for all y > 0 and for all x ∈ (−∞,


Figure 1

Figure 1

Theorem 1

The exponential function expa (x) = ax and the logarithm function loga (y) are inverses3 of each other

Using the same bases,

we obtain the graph of the logarithm function by reflecting the one of the exponential function at the diagonal in the Cartesian plane

This is the general principle by which the graph of a function and its inverse are related

The role of the independent and dependent variables,

and with this the coordinate axes,

The graph of log 2 (x),

When you compare the two graphs,

you need to take into account that the parts of the function shown are not quite the same and that there is a difference in scale

Once you make these adjustments you will see the relation

Theorem 1

Let a be a positive number,

The logarithm function loga is monotonic

It is increasing if a > 1 and decreasing if a < 1

Suppose u and v are positive numbers

If loga (u) = loga (v),

It is a general fact,

that the inverse of an increasing function is increasing,

and the inverse of a decreasing function is decreasing (see Proposition 5

So the monotonicity statements for the logarithm functions follow from the monotonicity properties of the exponential functions (see Theorem 1

and you are encouraged to read it in case you forgot about this concept




loga (u) = loga (v) implies that u = aloga (u) = aloga (v) = v

This verifies the remaining claim in the theorem

Corresponding to the exponential laws in Theorem 1

Some parts of the theorem are proved in Section 4

The other parts are assigned as exercises below

Theorem 1

For any positive real number a 6= 1,

for all positive real numbers x and y,

and any real number z loga (1) = 0 loga (a) = 1 loga (xy) = loga (x) + loga (y) loga (x/y) = loga (x) − loga (y) loga (xz ) = z loga (x) Because the exponential and logarithm functions are inverses of each other,

In the following exercises you are asked to verify this

Exercise 24

Assume the exponential laws and deduce the laws of logarithms

Exercise 25

Assume the laws of logarithms and deduce the exponential laws

To show you how to solve this kind of problem,

we deduce one of the exponential laws from the laws of logarithms

Observe that loga (ax ay ) = loga (ax ) + loga (ay ) = x + y = loga (ax+y )

The first equation follows from the third equation in Theorem 1

and the remaining two equations hold because of the way the logarithm function is defined

Comparing the outermost expressions,

Exercise 26

Assume that loga 1 = 0

loga (xy) = loga (x) + loga (y)

Show that loga (x/y) = loga (x) − loga (y)


The Euler number e as base You may think that f (x) = 10x is the easiest exponential function,

at least you have no problems to find 10n if n is an integer (a whole number)

Later on you will learn to appreciate the use of a different base,

It is an irrational number,

so the decimal expansion does not have a repeating block

Up to 50 decimal places e is (1

A precise definition of e is given in Definition 4

The reason why f (x) = ex is such an interesting function will become clear in Theorem 2

If we talk about the exponential function then we mean the exponential function for this base

The inverse of this exponential function,

the logarithm function for the base e,

is called the natural logarithm function

It also has a very simple derivative,

For reference purposes,

let us state the definitions formally

We graph these two functions on some reasonable intervals to make sure that you have the right picture in mind when we talk about them,

Figure 1

Figure 1

Definition 1

The exponential function is the exponential function for the base e

It is denoted by exp(x) or ex

Its inverse is the natural logarithm function

It is denoted by ln(x)

Leonard Euler (1707–1783),

one of the great mathematicians of the 18th century



Exponential Functions grow fast

Example 1

It is not so apparent from the graph how fast the exponential function grows

You may remember the tale of the ancient king who,

as payment for a lost game of chess,

was willing to put 1 grain of wheat on the first square on the chess board,

doubling the number of grains with each square

The chess board has 64 squares,

and that commits him to 263 grains on the 64th square for a total of 264 − 1 = 18,

615 grains

In mathematical notation,

you say that he puts f (n) = 2n−1 grains on the n-th square of the chess board

let us graph the function f (x) = 2x for 0 ≤ x ≤ 63,

On the given scale in the graph,

even an already enormous number like 254 ,

cannot be distinguished from 0

10 18 6

10 18 4

10 18 2

Figure 1

It is difficult to imagine how large these numbers are

The amount of grain which the king has to put on the chess board suffices to feed the current world population (of about 6 billion people) for thousands of years


Figure 1

Figure 1

Example 1

A different way of illustrating the growth of an exponential function is to compare it with the growth of a polynomial

In Figures 1

15 and 1

In each figure,

the graph of f is shown as a solid line,

and the one of p as a dashed line

In the first figure you see that,

the polynomial p is substantially larger than the exponential function f

In the second figure you see how the exponential function has overtaken the polynomial and begins to grow a lot faster

Other Bases Finally,

let us relate the exponential and logarithm functions for different bases

The result is,

for any positive number a (a 6= 1),

Theorem 1

This is seen quite easily

The first identity is obtained in the following way: ax = ex ln a

To see the second identity,

use eln x = x = aloga x = (eln a )loga x = eln a loga x

This means that ln x = (ln a)(loga x),



Exponential Growth Consider a function of the form f (t) = Ceat

The constants C and a,

and with this the function f (t) itself,

can be determined if we give the value of f at two points

We call a the growth rate5

We say that a function f grows exponentially if it has the form in (1

Example 1

Suppose the function f (t) grows exponentially,

Find the function f ,

and the time t0 for which f (t0 ) = 10

Solution: By assumption,

the function is of the form f (t) = Ceat

Substituting t = 0,

we find 3 = f (0) = Cea·0 = Ce0 = C

After having found C = 3,

we substitute t = 5 into the expression of f (t): 7 = f (5) = 3e5a

From this we deduce,

using arithmetic and the fact that the natural logarithm function is the inverse of the exponential function,

In particular,

the growth rate of the function is (approximately)


We calculate: e

Some texts call this number a the growth constant,

others the relative growth rate


the rate of change of f (t) at time t0 is af (t0 ),

so that the name relative growth rate (i

relative to the value to f (t)) is quite appropriate

you may get tired of having to say relative all the time,

and with the exact meaning understood,

you are quite willing to drop this adjective


Exercise 27

Suppose the function f (t) grows exponentially,

Find the function f ,

and the time t0 for which f (t0 ) = 10

Exercise 28

Suppose the function f (t) grows exponentially,

Show that f (t + T ) = 2f (t) for any t

Exercise 29

Suppose f (t) describes a population of e-coli bacteria in a Petrie dish

You assume that the population grows exponentially

At time t = 0 you start out with a population of 800 bacteria

After three hours the population is 1900

What is the relative growth rate for the population

? How long did it take for the population to double

How long does it take until the population has increased by a factor 4

Some problems remain unresolved in this section

We still have justify our characterization of the exponential function in Theorem 1

We still have to prove two of the laws of logarithms from Theorem 1

and we have to define the Euler number e

All of this will be done in Sections 4

Use of Graphing Utilities

A word of caution is advised

We are quite willing to use graphing utilities,

We use these graphs to illustrate the ideas and concepts under discussion

They allow you to visualize situations and help you to understand them

For a number of reasons,

no graphing utility is perfect and we cannot uncritically accept their output

When one of the utilities is pushed to the limit errors occur

Given any computer and any software,

with some effort you can produce erroneous graphs

That is not their mistake,

it only says that their abilities are limited

In Figures 1

17 and 1

Once we instructed the program to use the expression (x+1)6 to produce the graph,

and then we asked it to use the expanded expression

The outcome is remarkably different

? The program makes substantial round-off errors in the calculation

Which one is the correct graph