lecture notes on calculus of variations and what calculus is used for in real life. and what calculus is needed for physics pdf free download
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1MATH 221 1st SEMESTER CALCULUS
LECTURE NOTES VERSION 2.0 (fall 2009)
This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting
from an extensive collection of notes and problems compiled by Joel Robbin. The LT X and Python les
which were used to produce these notes are available at the following web site
They are meant to be freely available in the sense that \free software" is free. More precisely:
Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or
modify this document under the terms of the GNU Free Documentation License, Version
1.2 or any later version published by the Free Software Foundation; with no Invariant
Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is
included in the section entitled "GNU Free Documentation License".Contents 3. Exercises 64
4. Finding sign changes of a function 65
5. Increasing and decreasing functions 66
6. Examples 67
Chapter 1. Numbers and Functions 5
7. Maxima and Minima 69
1. What is a number? 5
8. Must there always be a maximum? 71
2. Exercises 7
9. Examples functions with and without maxima or
3. Functions 8
4. Inverse functions and Implicit functions 10
10. General method for sketching the graph of a
5. Exercises 13
11. Convexity, Concavity and the Second Derivative 74
Chapter 2. Derivatives (1) 15
12. Proofs of some of the theorems 75
1. The tangent to a curve 15
13. Exercises 76
2. An example tangent to a parabola 16
14. Optimization Problems 77
3. Instantaneous velocity 17
15. Exercises 78
4. Rates of change 17
5. Examples of rates of change 18
Chapter 6. Exponentials and Logarithms (naturally) 81
6. Exercises 18
1. Exponents 81
2. Logarithms 82
Chapter 3. Limits and Continuous Functions 21
3. Properties of logarithms 83
1. Informal denition of limits 21
4. Graphs of exponential functions and logarithms 83
2. The formal, authoritative, denition of limit 22
5. The derivative of a and the denition of e 84
3. Exercises 25
6. Derivatives of Logarithms 85
4. Variations on the limit theme 25
7. Limits involving exponentials and logarithms 86
5. Properties of the Limit 27
8. Exponential growth and decay 86
6. Examples of limit computations 27
9. Exercises 87
7. When limits fail to exist 29
8. What's in a name? 32
Chapter 7. The Integral 91
9. Limits and Inequalities 33
1. Area under a Graph 91
10. Continuity 34
2. When f changes its sign 92
11. Substitution in Limits 35
3. The Fundamental Theorem of Calculus 93
12. Exercises 36
4. Exercises 94
13. Two Limits in Trigonometry 36
5. The indenite integral 95
14. Exercises 38
6. Properties of the Integral 97
7. The denite integral as a function of its integration
Chapter 4. Derivatives (2) 41
1. Derivatives Dened 41
8. Method of substitution 99
2. Direct computation of derivatives 42
9. Exercises 100
3. Dierentiable implies Continuous 43
4. Some non-dierentiable functions 43
Chapter 8. Applications of the integral 105
5. Exercises 44
1. Areas between graphs 105
6. The Dierentiation Rules 45
2. Exercises 106
7. Dierentiating powers of functions 48
3. Cavalieri's principle and volumes of solids 106
8. Exercises 49
4. Examples of volumes of solids of revolution 109
9. Higher Derivatives 50
5. Volumes by cylindrical shells 111
10. Exercises 51
6. Exercises 113
11. Dierentiating Trigonometric functions 51
7. Distance from velocity, velocity from acceleration 113
12. Exercises 52
8. The length of a curve 116
13. The Chain Rule 52
9. Examples of length computations 117
14. Exercises 57
10. Exercises 118
15. Implicit dierentiation 58
11. Work done by a force 118
16. Exercises 60
12. Work done by an electric current 119
Chapter 5. Graph Sketching and Max-Min Problems 63
Chapter 9. Answers and Hints 121
1. Tangent and Normal lines to a graph 63
2. The Intermediate Value Theorem 63 GNU Free Documentation License 125
31. APPLICABILITY AND DEFINITIONS 125
2. VERBATIM COPYING 125
3. COPYING IN QUANTITY 125
4. MODIFICATIONS 125
5. COMBINING DOCUMENTS 126
6. COLLECTIONS OF DOCUMENTS 126
7. AGGREGATION WITH INDEPENDENT WORKS 126
8. TRANSLATION 126
9. TERMINATION 126
10. FUTURE REVISIONS OF THIS LICENSE 126
11. RELICENSING 126
Numbers and Functions
The subject of this course is \functions of one real variable" so we begin by wondering what a real number
\really" is, and then, in the next section, what a function is.
1. What is a number?
1.1. Dierent kinds of numbers. The simplest numbers are the positive integers
1; 2; 3; 4;
the number zero
and the negative integers
; 4; 3; 2; 1:
Together these form the integers or \whole numbers."
Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number.
These are the so called fractions or rational numbers such as
1 1 2 1 2 3 4
; ; ; ; ; ; ;
2 3 3 4 4 4 3
1 1 2 1 2 3 4
; ; ; ; ; ; ;
2 3 3 4 4 4 3
By denition, any whole number is a rational number (in particular zero is a rational number.)
You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a
rational number (provided you don't try to divide by zero).
One day in middle school you were told that there are other numbers besides the rational numbers, and
the rst example of such a number is the square root of two. It has been known ever since the time of the
greeks that no rational number exists whose square is exactly 2, i.e. you can't nd a fraction such that
= 2; i.e. m = 2n :
Nevertheless, if you compute x for some values ofx between 1 and 2, and check if you x x
get more or less than 2, then it looks like there should be some number x between 1:4 and
1:5 whose square is exactly 2. So, we assume that there is such a number, and we call it
the square root of 2, written as 2. This raises several questions. How do we know there
1:4 1:96 2
really is a number between 1:4 and 1:5 for which x = 2? How many other such numbers
1:5 2:25 2
are we going to assume into existence? Do these new numbers obey the same algebra rules
(like a +b =b +a) as the rational numbers? If we knew precisely what these numbers (like
2) were then we could perhaps answer such questions. It turns out to be rather dicult to give a precise
description of what a number is, and in this course we won't try to get anywhere near the bottom of this
issue. Instead we will think of numbers as \innite decimal expansions" as follows.
One can represent certain fractions as decimal fractions, e.g.
= = 11:16:
Not all fractions can be represented as decimal fractions. For instance, expanding into a decimal fraction
leads to an unending decimal fraction
= 0:333 333 333 333 333
It is impossible to write the complete decimal expansion of because it contains innitely many digits.
But we can describe the expansion: each digit is a three. An electronic calculator, which always represents
numbers as nite decimal numbers, can never hold the number exactly.
Every fraction can be written as a decimal fraction which may or may not be nite. If the decimal
expansion doesn't end, then it must repeat. For instance,
= 0:142857 142857 142857 142857 :::
Conversely, any innite repeating decimal expansion represents a rational number.
A real number is specied by a possibly unending decimal expansion. For instance,
2 = 1:414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 9:::
Of course you can never write all the digits in the decimal expansion, so you only write the rst few digits
and hide the others behind dots. To give a precise description of a real number (such as 2) you have to
explain how you could in principle compute as many digits in the expansion as you would like. During the
next three semesters of calculus we will not go into the details of how this should be done.
1.2. A reason to believe in 2. The Pythagorean theorem says that the hy-
potenuse of a right triangle with sides 1 and 1 must be a line segment of length 2. In
middle or high school you learned something similar to the following geometric construction
of a line segment whose length is 2. Take a square with side of length 1, and construct
a new square one of whose sides is the diagonal of the rst square. The gure you get
consists of 5 triangles of equal area and by counting triangles you see that the larger
square has exactly twice the area of the smaller square. Therefore the diagonal of the smaller square, being
the side of the larger square, is 2 as long as the side of the smaller square.
Why are real numbers called real? All the numbers we will use in this rst semester of calculus are
\real numbers." At some point (in 2nd semester calculus) it becomes useful to assume that there is a number
whose square is 1. No real number has this property since the square of any real number is positive, so
it was decided to call this new imagined number \imaginary" and to refer to the numbers we already have
(rationals, 2-like things) as \real."
1.3. The real number line and intervals. It is customary to visualize the real numbers as points
on a straight line. We imagine a line, and choose one point on this line, which we call the origin. We also
decide which direction we call \left" and hence which we call \right." Some draw the number line vertically
and use the words \up" and \down."
To plot any real number x one marks o a distance x from the origin, to the right (up) if x 0, to the
left (down) if x 0.
The distance along the number line between two numbers x and y isjx yj. In particular, the
distance is never a negative number.
3 2 1 0 1 2 3
Figure 1. To draw the half open interval 1; 2) use a lled dot to mark the endpoint which is included
and an open dot for an excluded endpoint.
2 1 0 1 2
Figure 2. To nd 2 on the real line you draw a square of sides 1 and drop the diagonal onto the real line.
Almost every equation involving variables x, y, etc. we write down in this course will be true for some
values of x but not for others. In modern abstract mathematics a collection of real numbers (or any other
kind of mathematical objects) is called a set. Below are some examples of sets of real numbers. We will use
the notation from these examples throughout this course.
The collection of all real numbers between two given real numbers form an interval. The following
notation is used
(a;b) is the set of all real numbers x which satisfy axb.
a;b) is the set of all real numbers x which satisfy axb.
(a;b is the set of all real numbers x which satisfy axb.
a;b is the set of all real numbers x which satisfy axb.
If the endpoint is not included then it may be1 or 1. E.g. ( 1; 2 is the interval of all real numbers
(both positive and negative) which are 2.
1.4. Set notation. A common way of describing a set is to say it is the collection of all real numbers
which satisfy a certain condition. One uses this notation
A = xjx satises this or that condition
Most of the time we will use upper case letters in a calligraphic font to denote sets. (A,B,C,D, . . . )
For instance, the interval (a;b) can be described as
(a;b) = xjaxb
B = xjx 1 0
consists of all real numbers x for which x 1 0, i.e. it consists of all real numbers x for which either x 1
or x 1 holds. This set consists of two parts: the interval ( 1; 1) and the interval (1;1).
You can try to draw a set of real numbers by drawing the number line and coloring the points belonging
to that set red, or by marking them in some other way.
Some sets can be very dicult to draw. For instance,
C = xjx is a rational number
can't be accurately drawn. In this course we will try to avoid such sets.
Sets can also contain just a few numbers, like
D =f1; 2; 3g
which is the set containing the numbers one, two and three. Or the set
E = xjx 4x + 1 = 0
which consists of the solutions of the equation x 4x + 1 = 0. (There are three of them, but it is not easy
to give a formula for the solutions.)
IfA andB are two sets then the union ofA andB is the set which contains all numbers that belong
either toA or toB. The following notation is used
AB = xjx belongs toA or toB or both.
7Similarly, the intersection of two setsA andB is the set of numbers which belong to both sets. This
notation is used:
A\B = xjx belongs to bothA andB.
1. What is the 2007 digit after the period in the expan- 4. SupposeA andB are intervals. Is it always true that
sion of ? A\B is an interval? How aboutAB?
5. Consider the sets
2. Which of the following fractions have nite decimal
M = xjx 0 andN = yjy 0 :
2 3 276937
Are these sets the same?
a = ; b = ; c = :
3 25 15625
6. Group Problem.
3. Draw the following sets of real numbers. Each of these
Write the numbers
sets is the union of one or more intervals. Find those
x = 0:3131313131:::; y = 0:273273273273:::
intervals. Which of thee sets are nite?
and z = 0:21541541541541541:::
A = xjx 3x + 2 0
B = xjx 3x + 2 0 as fractions (i.e. write them as , specifying m and n.)
C = xjx 3x 3
(Hint: show that 100x = x + 31. A similar trick
D = xjx 5 2x works for y, but z is a little harder.)
E = tjt 3t + 2 0
7. Group Problem.
F = j 3 + 2 0
Is the number whose decimal expansion after the
G = (0; 1) (5; 7
period consists only of nines, i.e.
H = f1gf2; 3g \ (0; 2 2)
x = 0:99999999999999999:::
Q = j sin =
R = 'j cos' 0 an integer?
Wherein we meet the main characters of this semester
3.1. Denition. To specify a function f you must
(1) give a rule which tells you how to compute the value f(x) of the function for a given real number
(2) say for which real numbers x the rule may be applied.
The set of numbers for which a function is dened is called its domain. The set of all possible numbers f(x)
as x runs over the domain is called the range of the function. The rule must be unambiguous: the same
xmust always lead to the same f(x).
For instance, one can dene a function f by putting f(x) = x for all x 0. Here the rule dening f is
\take the square root of whatever number you're given", and the function f will accept all nonnegative real
The rule which species a function can come in many dierent forms. Most often it is a formula, as in
the square root example of the previous paragraph. Sometimes you need a few formulas, as in
2x for x 0
g(x) = domain of g = all real numbers.
x for x 0
Functions which are dened by dierent formulas on dierent intervals are sometimes called piecewise
3.2. Graphing a function. You get the graph of a function f by drawing all points whose coordi-
nates are (x;y) where x must be in the domain of f and y =f(x).
8range of f
domain of f
Figure 3. The graph of a function f. The domain of f consists of all x values at which the function is
dened, and the range consists of all possible values f can have.
Figure 4. A straight line and its slope. The line is the graph of f(x) =mx +n. It intersects the y-axis
at height n, and the ratio between the amounts by which y and x increase as you move from one point
to another on the line is =m:
3.3. Linear functions. A function which is given by the formula
f(x) =mx +n
where m and n are constants is called a linear function. Its graph is a straight line. The constants m
and n are the slope and y-intercept of the line. Conversely, any straight line which is not vertical (i.e. not
parallel to the y-axis) is the graph of a linear function. If you know two points (x ;y ) and (x ;y ) on the
0 0 1 1
line, then then one can compute the slope m from the \rise-over-run" formula
m = :
This formula actually contains a theorem from Euclidean geometry, namely it says that the ratio (y y ) :
(x x ) is the same for every pair of points (x ;y ) and (x ;y ) that you could pick on the line.
1 0 0 0 1 1
3.4. Domain and \biggest possible domain. " In this course we will usually not be careful about
specifying the domain of the function. When this happens the domain is understood to be the set of all x
for which the rule which tells you how to compute f(x) is meaningful. For instance, if we say that h is the
h(x) = x
y =x x
Figure 5. The graph of y =x x fails the \horizontal line test," but it passes the \vertical line test."
The circle fails both tests.
then the domain of h is understood to be the set of all nonnegative real numbers
domain of h = 0;1)
since x is well-dened for all x 0 and undened for x 0.
A systematic way of nding the domain and range of a function for which you are only given a formula is
The domain of f consists of all x for which f(x) is well-dened (\makes sense")
The range of f consists of all y for which you can solve the equation f(x) =y.
3.5. Example nd the domain and range of f(x) = 1=x . The expression 1=x can be computed
for all real numbers x except x = 0 since this leads to division by zero. Hence the domain of the function
f(x) = 1=x is
\all real numbers except 0" = xjx =6 0 = ( 1; 0) (0;1):
To nd the range we ask \for which y can we solve the equation y =f(x) for x," i.e. we for which y can you
solve y = 1=x for x?
If y = 1=x then we must have x = 1=y, so rst of all, since we have to divide by y, y can't be zero.
Furthermore, 1=y =x says thaty must be positive. On the other hand, ify 0 theny = 1=x has a solution
(in fact two solutions), namely x =1= y. This shows that the range of f is
\all positive real numbers" =fxjx 0g = (0;1):
3.6. Functions in \real life. " One can describe the motion of an object using a function. If some
object is moving along a straight line, then you can dene the following function: Let x(t) be the distance
from the object to a xed marker on the line, at the time t. Here the domain of the function is the set of all
times t for which we know the position of the object, and the rule is
Given t, measure the distance between the object and the marker at time t.
There are many examples of this kind. For instance, a biologist could describe the growth of a cell by
dening m(t) to be the mass of the cell at time t (measured since the birth of the cell). Here the domain is
the interval 0;T , where T is the life time of the cell, and the rule that describes the function is
Given t, weigh the cell at time t.
3.7. The Vertical Line Property. Generally speaking graphs of functions are curves in the plane but
they distinguish themselves from arbitrary curves by the way they intersect vertical lines: The graph of
a function cannot intersect a vertical line \x = constant" in more than one point. The reason
why this is true is very simple: if two points lie on a vertical line, then they have the same x coordinate, so if
they also lie on the graph of a function f, then their y-coordinates must also be equal, namely f(x).
3.8. Examples. The graph off(x) =x x \goes up and down," and, even though it intersects several
horizontal lines in more than one point, it intersects every vertical line in exactly one point.
The collection of points determined by the equation x +y = 1 is a circle. It is not the graph of a
function since the vertical line x = 0 (the y-axis) intersects the graph in two points P (0; 1) and P (0; 1).
See Figure 6.
4. Inverse functions and Implicit functions
For many functions the rule which tells you how to compute it is not an explicit formula, but instead an
equation which you still must solve. A function which is dened in this way is called an \implicit function."
4.1. Example. One can dene a functionf by saying that for eachx the value off(x) is the solutiony
of the equation
x + 2y 3 = 0:
In this example you can solve the equation for y,
y = :
Thus we see that the function we have dened is f(x) = (3 x )=2.
Here we have two denitions of the same function, namely
(i) \y =f(x) is dened by x + 2y 3 = 0," and
(ii) \f is dened by f(x) = (3 x )=2."
The rst denition is the implicit denition, the second is explicit. You see that with an \implicit function"
it isn't the function itself, but rather the way it was dened that's implicit.
4.2. Another example: domain of an implicitly dened function. Dene g by saying that for
any x the value y =g(x) is the solution of
x +xy 3 = 0:
Just as in the previous example one can then solve for y, and one nds that
g(x) =y = :
Unlike the previous example this formula does not make sense when x = 0, and indeed, for x = 0 our rule for
g says that g(0) =y is the solution of
0 + 0y 3 = 0; i.e. y is the solution of 3 = 0:
That equation has no solution and hence x = 0 does not belong to the domain of our function g.
x +y = 1
y = + 1 x
y = 1 x
Figure 6. The circle determined byx +y = 1 is not the graph of a function, but it contains the graphs
of the two functions h (x) = 1 x and h (x) = 1 x .
114.3. Example: the equation alone does not determine the function. Deney =h(x) to be the
x +y = 1:
If x 1 or x 1 then x 1 and there is no solution, so h(x) is at most dened when 1x 1. But
when 1x 1 there is another problem: not only does the equation have a solution, but it even has two
x +y = 1 () y = 1 x or y = 1 x :
The rule which denes a function must be unambiguous, and since we have not specied which of these two
solutions is h(x) the function is not dened for 1x 1.
One can x this by making a choice, but there are many possible choices. Here are three possibilities:
h (x) = the nonnegative solution y of x +y = 1
h (x) = the nonpositive solution y of x +y = 1
h (x) when x 0
h (x) =
h (x) when x 0
4.4. Why use implicit functions? In all the examples we have done so far we could replace the
implicit description of the function with an explicit formula. This is not always possible or if it is possible the
implicit description is much simpler than the explicit formula. For instance, you can dene a function f by
saying that y =f(x) if and only if
(1) y + 3y + 2x = 0:
This means that the recipe for computing f(x) for any given x is \solve the equation y + 3y + 2x = 0."
E.g. to compute f(0) you set x = 0 and solve y + 3y = 0. The only solution is y = 0, so f(0) = 0. To
compute f(1) you have to solve y + 3y + 2 1 = 0, and if you're lucky you see that y = 1 is the solution,
and f(1) = 1.
In general, no matter whatx is, the equation (1) turns out to have exactly one solution y (which depends
on x, this is how you get the function f). Solving (1) is not easy. In the early 1500s Cardano and Tartaglia
discovered a formula for the solution. Here it is:
y =f(x) = x + 1 +x x + 1 +x :
The implicit description looks a lot simpler, and when we try to dierentiate this function later on, it will be
much easier to use \implicit dierentiation" than to use the Cardano-Tartaglia formula directly.
4.5. Inverse functions. If you have a function f, then you can try to dene a new function f , the
so-called inverse function of f, by the following prescription:
(2) For any given x we say that y =f (x) if y is the solution to the equation f(y) =x.
So to nd y =f (x) you solve the equation x =f(y). If this is to dene a function then the prescription
(2) must be unambiguous and the equation f(y) =x has to have a solution and cannot have more than one
To see the solution and its history visit
12The graph of f
The graph of f
a b c f(a) f(b) f(c)
Figure 7. The graph of a function and its inverse are mirror images of each other.
4.6. Examples. Consider the function f with f(x) = 2x + 3. Then the equation f(y) =x works out to
2y + 3 =x
and this has the solution
y = :
So f (x) is dened for all x, and it is given by f (x) = (x 3)=2.
Next we consider the function g(x) =x with domain all positive real numbers. To see for which x the
inverse g (x) is dened we try to solve the equation g(y) =x, i.e. we try to solve y =x. If x 0 then this
equation has no solutions since y 0 for all y. But if x 0 then y x does have a solution, namely y = x.
So we see that g (x) is dened for all nonnegative real numbers x, and that it is given by g (x) = x.
4.7. Inverse trigonometric functions. The familiar trigonometric functions Sine, Cosine and Tangent
have inverses which are called arcsine, arccosine and arctangent.
y =f(x) x =f (y)
y = sinx ( =2x=2) x = arcsin(y) ( 1y 1)
y = cosx (0x) x = arccos(y) ( 1y 1)
y = tanx ( =2x=2) x = arctan(y)
1 1 1
The notations arcsiny = sin y, arccosx = cos x, and arctanu = tan u are also commonly used for
the inverse trigonometric functions. We will avoid the sin y notation because it is ambiguous. Namely,
everybody writes the square of siny as
siny = sin y:
Replacing the 2's by 1's would lead to
arcsiny = sin y = siny = ; which is not true
8. The functions f and g are dened by 10. Find a formula for the function f which is dened by
f(x) =x and g(s) =s :
y =f(x) () x y y = 6:
Are f and g the same functions or are they dierent?
What is the domain of f?
9. Find a formula for the function f which is dened by
11. Let f be the function dened by y =f(x) () y is
the largest solution of
y =f(x) () x y +y = 7:
What is the domain of f? y = 3x 2xy:
13Find a formula for f. What are the domain and range of for all real numbers x.
12. Find a formula for the function f which is dened by
(a) f(1) (b) f(0) (c) f(x)
y =f(x) () 2x + 2xy +y = 5 and y x:
(d) f(t) (e) f(f(2))
Find the domain of f.
where x and t are arbitrary real numbers.
13. Use a calculator to compute f(1:2) in three deci-
What are the range and domain of f?
mals where f is the implicitly dened function fromx4.4.
21. Does there exist a function f which satises
(There are (at least) two dierent ways of nding f(1:2))
f(x ) =x + 1
14. Group Problem.
(a) True or false:
for all real numbers x?
for all x one has sin arcsinx =x?
(b) True or false:
for all x one has arcsin sinx =x?
The following exercises review precalculus material in-
15. On a graphing calculator plot the graphs of the follow-
volving quadratic expressions ax +bx +c in one way or
ing functions, and explain the results. (Hint: rst do the
22. Explain how you \complete the square" in a quadratic
f(x) = arcsin(sinx); 2x 2
expression like ax +bx.
g(x) = arcsin(x) + arccos(x); 0x 1
23. Find the range of the following functions:
h(x) = arctan ; jxj=2
cosx f(x) = 2x + 3
k(x) = arctan ; jxj=2
g(x) = 2x + 4x
l(x) = arcsin(cosx); x
h(x) = 4x +x
m(x) = cos(arcsinx); 1x 1
k(x) = 4 sinx + sin x
`(x) = 1=(1 +x )
16. Find the inverse of the function f which is given by
f(x) = sinx and whose domain is x 2. Sketch 2
m(x) = 1=(3 + 2x +x ):
the graphs of both f and f .
17. Find a number a such that the function f(x) = 24. Group Problem.
sin(x +=4) with domain axa + has an inverse.
For each real number a we dene a line ` with
Give a formula for f (x) using the arcsine function. 2
equation y =ax +a .
18. Draw the graph of the function h fromx4.3.
(a) Draw the lines corresponding to a =
2; 1; ; 0; ; 1; 2.
19. A function f is given which satises 2 2
(b) Does the point with coordinates (3; 2) lie on one
f(2x + 3) =x
or more of the lines ` (where a can be any number, not
for all real numbers x.
just the ve values from part (a))? If so, for which values
of a does (3; 2) lie on ` ?
(a) f(0) (b) f(3) (c) f(x)
(c) Which points in the plane lie on at least one of
the lines ` ?.
(d) f(y) (e) f(f(2))
where x and y are arbitrary real numbers.
25. For which values of m and n does the graph of
f(x) = mx +n intersect the graph of g(x) = 1=x in
What are the range and domain of f?
exactly one point and also contain the point ( 1; 1)?
20. A function f is given which satises
26. For which values of m and n does the graph of
f = 2x 12:
f(x) =mx +n not intersect the graph of g(x) = 1=x?
x + 1
To work with derivatives you have to know what a limit is, but to motivate why we are going to study
limits let's rst look at the two classical problems that gave rise to the notion of a derivative: the tangent to
a curve, and the instantaneous velocity of a moving object.
1. The tangent to a curve
Suppose you have a function y =f(x) and you draw its graph. If you want to nd the tangent to the
graph of f at some given point on the graph of f, how would you do that?
Figure 1. Constructing the tangent by letting QP
Let P be the point on the graph at which want to draw the tangent. If you are making a real paper and
ink drawing you would take a ruler, make sure it goes through P and then turn it until it doesn't cross the
graph anywhere else.
If you are using equations to describe the curve and lines, then you could pick a point Q on the graph
and construct the line through P and Q (\construct" means \nd an equation for"). This line is called a
\secant," and it is of course not the tangent that you're looking for. But if you choose Q to be very close toP
then the secant will be close to the tangent.
15So this is our recipe for constructing the tangent through P : pick another point Q on the graph, nd the
line through P and Q, and see what happens to this line as you take Q closer and closer to P . The resulting
secants will then get closer and closer to some line, and that line is the tangent.
We'll write this in formulas in a moment, but rst let's worry about how close Q should be to P. We
can't set Q equal to P, because then P and Q don't determine a line (you need two points to determine a
line). If you choose Q dierent from P then you don't get the tangent, but at best something that is \close"
to it. Some people have suggested that one should take Q \innitely close" to P , but it isn't clear what that
would mean. The concept of a limit is meant to solve this confusing problem.
2. An example tangent to a parabola
To make things more concrete, suppose that the function we had was f(x) =x , and that the point was
(1; 1). The graph of f is of course a parabola.
Any line through the point P (1; 1) has equation
y 1 =m(x 1)
where m is the slope of the line. So instead of nding the equation of the secant and tangent lines we will
nd their slopes.
Let Q be the other point on the parabola, with coordinates (x;x ). We can
\move Q around on the graph" by changing x. Whatever x we choose, it must be
dierent from 1, for otherwise P and Q would be the same point. What we want to
nd out is how the line throughP andQ changes ifx is changed (and in particular, if
x is chosen very close to a). Now, as one changes x one thing stays the same, namely,
the secant still goes through P. So to describe the secant we only need to know its
slope. By the \rise over run" formula, the slope of the secant line joining P and Q is
m = where y =x 1 and x =x 1:
By factoring x 1 we can rewrite the formula for the slope as follows
y x 1 (x 1)(x + 1)
(3) m = = = =x + 1:
x x 1 x 1
As x gets closer to 1, the slope m , being x + 1, gets closer to the value 1 + 1 = 2. We say that
the limit of the slope m as Q approaches P is 2.
lim m = 2;
or, since Q approaching P is the same as x approaching 1,
(4) limm = 2:
So we nd that the tangent line to the parabola y =x at the point (1; 1) has equation
y 1 = 2(x 1); i.e. y = 2x 1:
A warning: you cannot substitute x = 1 in equation (3) to get (4) even though it looks like that's what we
did. The reason why you can't do that is that when x = 1 the point Q coincides with the point P so \the
line through P and Q" is not dened; also, if x = 1 then x = y = 0 so that the rise-over-run formula for
the slope gives
m = = = undened.
It is only after the algebra trick in (3) that setting x = 1 gives something that is well dened. But if the
intermediate steps leading to m =x + 1 aren't valid for x = 1 why should the nal result mean anything
for x = 1?
16Something more complicated has happened. We did a calculation which is valid for all x6= 1, and later
looked at what happens if x gets \very close to 1." This is the concept of a limit and we'll study it in more
detail later in this section, but rst another example.
3. Instantaneous velocity
If you try to dene \instantaneous velocity" you will again end up trying to divide zero by zero. Here is
how it goes: When you are driving in your car the speedometer tells you how fast your are going, i.e. what
your velocity is. What is this velocity? What does it mean if the speedometer says \50mph"?
s = 0 Time =t Time =t + t
s(t) s =s(t + t) s(t)
We all know what average velocity is. Namely, if it takes you two hours to cover 100 miles, then your
average velocity was
= 50 miles per hour:
time it took
This is not the number the speedometer provides you it doesn't wait two hours, measure how far you went
and compute distance=time. If the speedometer in your car tells you that you are driving 50mph, then that
should be your velocity at the moment that you look at your speedometer, i.e. \distance traveled over time
it took" at the moment you look at the speedometer. But during the moment you look at your speedometer
no time goes by (because a moment has no length) and you didn't cover any distance, so your velocity at that
moment is , i.e. undened. Your velocity at any moment is undened. But then what is the speedometer
To put all this into formulas we need to introduce some notation. Let t be the time (in hours) that has
passed since we got onto the road, and let s(t) be the distance we have covered since then.
Instead of trying to nd the velocity exactly at time t, we nd a formula for the average velocity during
some (short) time interval beginning at time t. We'll write t for the length of the time interval.
At time t we have traveled s(t) miles. A little later, at time t + t we have traveled s(t + t). Therefore
during the time interval from t to t + t we have moved s(t + t) s(t) miles. Our average velocity in that
time interval is therefore
s(t + t) s(t)
miles per hour.
The shorter you make the time interval, i.e. the smaller you choose t, the closer this number should be to
the instantaneous velocity at time t.
So we have the following formula (denition, really) for the velocity at time t
s(t + t) s(t)
(5) v(t) = lim :
4. Rates of change
The two previous examples have much in common. If we ignore all the details about geometry, graphs,
highways and motion, the following happened in both examples:
We had a function y =f(x), and we wanted to know how much f(x) changes if x changes. If you change
x to x + x, then y will change from f(x) to f(x + x). The change in y is therefore
y =f(x + x) f(x);
and the average rate of change is
y f(x + x) f(x)
(6) = :
17This is the average rate of change of f over the interval from x tox + x. To dene the rate of change of
the function f at x we let the length x of the interval become smaller and smaller, in the hope that the
average rate of change over the shorter and shorter time intervals will get closer and closer to some number.
If that happens then that \limiting number" is called the rate of change of f at x, or, the derivative off at
x. It is written as
f(x + x) f(x)
(7) f (x) = lim :
Derivatives and what you can do with them are what the rst half of this semester is about. The description
we just went through shows that to understand what a derivative is you need to know what a limit is. In the
next chapter we'll study limits so that we get a less vague understanding of formulas like (7).
5. Examples of rates of change
5.1. Acceleration as the rate at which velocity changes. As you are driving in your car your
velocity does not stay constant, it changes with time. Suppose v(t) is your velocity at time t (measured
in miles per hour). You could try to gure out how fast your velocity is changing by measuring it at one
moment in time (you get v(t)), then measuring it a little later (you get v(t))). You conclude that your
velocity increased by v =v(t + t) v(t) during a time interval of length t, and hence
v v(t + t) v(t)
average rate at which
= = :
your velocity changed
This rate of change is called your average acceleration (over the time interval from t to t + t). Your
instantaneous acceleration at time t is the limit of your average acceleration as you make the time interval
shorter and shorter:
v(t + t) v(t)
facceleration at time tg =a = lim :
th the average and instantaneous accelerations are measured in \miles per hour per hour," i.e. in
(mi=h)=h = mi=h :
Or, if you had measured distances in meters and time in seconds then velocities would be measured in meters
per second, and acceleration in meters per second per second, which is the same as meters per second , i.e.
\meters per squared second."
5.2. Reaction rates. Think of a chemical reaction in which two substances A and B react to form
AB according to the reaction
A + 2B AB :
If the reaction is taking place in a closed reactor, then the \amounts" of A and B will be decreasing, while the
amount of AB will increase. Chemists write A for the amount of \A" in the chemical reactor (measured in
moles). Clearly A changes with time so it denes a function. We're mathematicians so we will write \A(t)"
for the number of moles of A present at time t.
To describe how fast the amount of A is changing we consider the derivative of A with respect to time,
A(t + t) A(t)
A (t) = lim :
This quantity is the rate of change of A. The notation \A (t)" is really only used by calculus professors. If
you open a paper on chemistry you will nd that the derivative is written in Leibniz notation:
More on this inx1.2
How fast does the reaction take place? If you add more A or more B to the reactor then you would expect
that the reaction would go faster, i.e. that more AB is being produced per second. The law of mass-action
18kinetics from chemistry states this more precisely. For our particular reaction it would say that the rate at
which A is consumed is given by
=k A B ;
in which the constant k is called the reaction constant. It's a constant that you could try to measure by
timing how fast the reaction goes.
27. Repeat the reasoning inx2 to nd the slope at the 31. Look ahead at Figure 3 in the next chapter. What is
1 1 2
point ( ; ), or more generally at any point (a;a ) on the derivative of f(x) =x cos at the points A and B
2 4 x
on the graph?
the parabola with equation y =x .
28. Repeat the reasoning inx2 to nd the slope at the
32. Suppose that some quantity y is a function of some
1 1 3
point ( ; ), or more generally at any point (a;a ) on
other quantity x, and suppose that y is a mass, i.e. y
the curve with equation y =x . is measured in pounds, and x is a length, measured in
feet. What units do the increments y and x, and the
29. Group Problem.
derivative dy=dx have?
Should you trust your calculator?
33. A tank is lling with water. The volume (in gallons)
Find the slope of the tangent to the parabolay =x
of water in the tank at time t (seconds) is V (t). What
at the point ( ; ) (You have already done this: see
units does the derivative V (t) have?
34. Group Problem.
Instead of doing the algebra you could try to compute
the slope by using a calculator. This exercise is about
LetA(x) be the area of an equilateral triangle whose
how you do that and what happens if you try (too hard).
sides measure x inches.
Compute for various values of x:
x (a) Show that has the units of a length.
x = 0:1; 0:01; 0:001; 10 ; 10 :
(b) Which length does represent geometrically?
Hint: draw two equilateral triangles, one with side x and
As you choose x smaller your computed ought to
another with side x + x. Arrange the triangles so that
get closer to the actual slope. Use at least 10 decimals
they both have the origin as their lower left hand corner,
and organize your results in a table like this:
and so there base is on the x-axis.
x f(a) f(a + x) y y=x
0:1 . . . . . . . . . . . . 35. Group Problem.
0:01 . . . . . . . . . . . .
LetA(x) be the area of a square with sidex, and let
0:001 . . . . . . . . . . . .
L(x) be the perimeter of the square (sum of the lengths
10 . . . . . . . . . . . .
of all its sides). Using the familiar formulas for A(x) and
10 . . . . . . . . . . . . 1
L(x) show that A (x) = L(x).
Look carefully at the ratios y=x. Do they look like
Give a geometric interpretation that explains why
they are converging to some number? Compare the values
A L(x)x for small x.
of with the true value you got in the beginning of
36. Let A(r) be the area enclosed by a circle of radius
r, and let L(r) be the length of the circle. Show that
30. Simplify the algebraic expressions you get when you 0
A (r) =L(r). (Use the familiar formulas from geometry
compute y and y=x for the following functions
for the area and perimeter of a circle.)
(a) y =x 2x + 1
37. Let V (r) be the volume enclosed by a sphere of ra-
dius r, and let S(r) be the its surface area. Show that
(b) y =
0 4 3
V (r) = S(r). (Use the formulas V (r) = r and
(c) y = 2 2
S(r) = 4r .)