MTH 233—Calculus I
Tan, Single Variable Calculus: Early Transcendentals, 1st ed.
Part 1: Introduction, Limits and Continuity
0
Preliminaries
It is assumed that students are comfortable with the material in Chapter
0: Preliminaries. Selected (very important) topics will get a brief review,
beginning with section 0.2; other sections will be reviewed when needed.
0.2
Functions and Their Graphs
Informal Definition:
A function is a rule for which each input is assigned exactly one
output. The collection of all valid inputs for a given function is
called the domain of the function, and the set of images of the
domain elements is called the range of the function.
Finding the domain of a function: For most functions, keep the following
in mind:
• the denominator of a fractional expression should be nonzero
• the argument of an even root should be nonnegative
• the argument of a logarithms should be strictly positive
• most trigonometric functions have periodic values for which the function is not defined
• all inverse trigonometric functions have special domains
• functions used to model real-life situations may have additional (nonmathematical) limitations on their domains
Representations of functions: Functions have several useful representations:
• Algebraic: f (x) = 2x + 3
• Rhetorical: The function ‘f ’ doubles the input and adds three to produce the output
• Ordered pairs/tables: {(x, y) : y = 2x + 3}
• Graphs
Definition:
Two functions are equal if and only if they have a common domain
and give the same output for every input in that domain.
Section 0.2: 1-25 every other odd, 27, 28, 31-57 odd
1
1.1
Limits
An Intuitive Introduction to Limits
In this section, we start with an intuitive notion of limits and discuss two
strategies to estimate limits.
Informal Definitions:
Given a function f defined in some neighborhood of a point a,
we say that
lim f (x) = L
x→a
provided the output can be made arbitrarily close to L for all
inputs sufficiently close to a.
In addition, we can restrict to ‘one-sided limits’:
lim f (x) = L
x→a±
by taking all inputs sufficiently close to a, but larger (for a+ ) or
smaller (for a− ).
Key points:
• the existence/value of a limit is independent of the behavior of the
function at a
• limits may not exist for several reasons
– different behaviors of the output on the different sides of a; that
is, lim+ f (x) 6= lim− f (x)
x→a
x→a
– unbounded behavior near a (sometimes indicated using ∞ or −∞)
– unrestricted oscillation near a
– others
Estimating limits: There are two basic ways to estimate limits:
• Graphically: graphing a function on a domain that is ‘near’ the desired
value can frequently yield an estimate of the limit. Possible problems:
– finding an appropriate domain to graph on
– computers frequently leave artifacts that are not actually part of
the graph
• Numerically: creating a table of values that get nearer the desired
value (from both sides) and finding the outputs of those values sometimes leads to a pattern that can be used to deduce the limit. Possible
problems:
– finding appropriate inputs–getting ‘near’ enough
– the patterns found can sometimes be misleading
Section 1.1: Concepts 1–4; Exercises 1-21 odd, 29, 31, 43–46
1.2
Techniques for Finding Limits
In this section, we discuss techniques for evaluating (rather than estimating)
limits.
The Limit Laws: The following theorems can be proven (using the formal
definition):
• Constant rule: x→a
lim k = k
• Identity rule: lim x = a
x→a
• If the limits exist individually, then
– (Sum rule) the limit of a sum is the sum of the individual limits
– (Difference rule) the limit of a difference is the difference of the
individual limits
– (Product rule) the limit of a product is the product of the individual limits
– (Quotient rule) the limit of a quotient is the quotient of the individual limits (if the limit of the denominator is nonzero)
– (Power rule) the limit of a function to a rational power is the limit
of the individual function raised to the rational power (if it makes
sense)
– (Basic trigonometric functions) the limit of a trigonometric function at any point in its domain is the value of that function evaluated at that point
With these limit laws we can prove that polynomials, rational functions,
most trigonometric expressions, and expressions involving powers/roots can
be evaluated by substitution where defined.
When substitution fails: Substitution usually fails because the desired
value is not in the domain of the given function; however, since we only
care what is happening ‘near’ the desired value, if substitution gives us the
indeterminate form “0/0,” we can use some algebraic tricks to get forms that
are equivalent near the desired value:
• (generally useful for rational expressions) factor the numerator and
denominator, remove common factors, use substitution
• (generally useful for rational expressions involving roots) rationalize
the numerator/denominator, factor, remove common factors, use substitution
cos θ − 1
sin θ
= 1, lim
= 0 as long as
• Special trigonometric limits: lim
θ→0
θ→0 θ
θ
the inputs are in radians
The Squeeze Theorem: Another technique that can be used to evaluate
some limits appears below; before we discuss it, we will look at a somewhat
similar and simpler result that is used in the proof of the technique.
Lemma: (The Monotonicity Lemma)
If one function dominates another in a neighborhood of a desired
value (except perhaps at the desired value itself), then the limit
of the larger is greater than or equal to the limit of the smaller
at the desired value (if the limits exist).
This dominance can be used to create a ‘squeeze’ or ‘sandwich’: we can frequently find the limit of the ‘meat’ (f , below) if we can find the appropriate
‘bread’ (g and h, below):
Theorem: (The Squeeze Theorem)
Given a function f defined in some neighborhood of a point x = a,
if there exist functions g and h each defined on the same set and
such that
1. g(x) ≤ f (x) ≤ h(x) for all x in the given set and
2. x→a
lim g(x) = L = x→a
lim h(x)
then x→a
lim f (x) = L also.
Section 1.2: Concepts 1–4; Exercises 1-37 odd, 38, 41–73 every other odd,
75, 77, 85, 87, 93, 99–102
1.3
The Precise Definition of the Limit
(We will either skip this section or do a related lab exercise, but feel free to
read about it.)
3.5
Limits Involving Infinity; Asymptotes
To give the (almost) full picture for limits, we will skip ahead to a later
section and discuss the oft-misunderstood symbol “∞.” In the context of
limits, this symbol indicates unbounded behavior, either in the output:
lim f (x) = ∞
x→a
or in the input
lim f (x) = L
x→∞
or both
lim f (x) = ∞
x→∞
“Limits to infinity”: When the outputs are unbounded, we have:
Informal Definition:
Given a function f defined in some neighborhood of a point a,
we say that
lim f (x) = ±∞
x→a
provided the output can be made arbitrarily large/negative for
all inputs sufficiently close to a.
Note carefully: this is a special case of a limit not existing because of
unbounded behavior; the limit laws do not apply!
One-sided unbounded limits can be defined similarly with an analogous
slight change to the statement. Unbounded behavior of this type near a
particular input value is associated with vertical asymptotes:
Definition:
A function f has a vertical asymptote at x = a provided one
or more of the following hold:
lim f (x) = ±∞
x→a±
(where any combination of the plus/minus signs may hold.)
In general, vertical asymptotes will attain when the denominator of an expression is zero and the numerator is nonzero; the plus/minus behavior may
usually be determined via a sign diagram or testing a point ‘near enough.’
“Limits at infinity”: When the inputs are unbounded, we have:
Informal Definition:
Given a function f defined on some unbounded interval in the
positive/negative direction, we say that
lim f (x) = L
x→±∞
provided the output can be made arbitrarily close to L for all
inputs sufficiently large/negative.
Here the limits exist (since we can give them the finite value L), and
the limit laws apply normally as do both the Monotonicity Lemma and the
Squeeze Theorem. In addition these are already special cases of one-sided limits. Should the outputs become unbounded when the inputs are unbounded,
we can use our unbounded behavior symbol “∞” in both places. In any
event, since we are discussing behavior as the inputs become unbounded,
these limits describe the end behavior of the function. If the end behavior is
finite, we get horizontal asymptotes:
Definition:
The line y = L is a horizontal asymptote for the function f
provided at least one of the following hold:
lim f (x) = L
x→±∞
Note two things:
• A graph may have up to two horizontal asymptotes (think the graph
of the Arctangent function.)
• Graphs may cross horizontal asymptotes.
The following theorem is not difficult to prove:
Theorem:
If r > 0 is a rational number, then
lim
x→±∞
1
=0
xr
This fact can be useful in determining the horizontal asymptotes of many
functions.
Section 3.5: Concepts 1–4; Exercises 1–37 every other odd, 39, 45–55 odd,
89–94
1.4
Continuous Functions
Definitions:
• A function f is continuous at x = a provided x→a
lim f (x) = f (a).
• A function f is continuous on a set A provided f is continuous at every point in A.
• Continuity from the right/left can be defined using the appropriate one-sided limits.
Most of the following theorems follow from the limit laws:
• Constant functions are continuous
• The identity function is continuous
• Sums/differences/products of continuous functions are continuous (so
polynomials are continuous)
• Quotients of continuous functions are continuous wherever the denominator is nonzero (so rational functions are continuous on their domains)
• The trigonometric functions are continuous on their domains
• Exponential functions are continuous
• Inverses of continuous functions are continuous (so logarithmic functions and the inverse trigonometric functions are continuous on their
domains)
• Compositions of continuous functions are continuous
Thus most of the functions we know are continuous wherever they are
defined (though it can be a bit tricky to prove some of them)—this is exactly
why/when substitution works to evaluate a limit.
Discontinuities: Functions can be discontinuous in two basic ways:
Definition:
A function f has a removable discontinuity at x = a provided f is not continuous at a, but x→a
lim f (x) exists (removable
discontinuities are often called ‘skip discontinuities’). Any other
discontinuity is a non-removable discontinuity. Examples of
non-removable discontinuities include vertical asymptotes, jump
discontinuities, and oscillating discontinuities.
Note that a function with a removable discontinuity at x = a can be redefined at that point to produce what is called the continuous extension of that
sin x
sin x
function. For example,
can be redefined piecewise so that f (x) =
x
x
for x 6= 0 and f (0) = 1 to produce a new function f that is continuous on
all real numbers. (Some advanced classes ignore removable discontinuities
altogether and always assume the continuous extension, where necessary.)
The Intermediate Value Theorem: One of the most important consequences of continuity is the following:
Theorem: (The Intermediate Value Theorem)
A function which is continuous on a closed interval must take on
all outputs intermediate to the outputs at the endpoints somewhere on the closed interval.
This theorem has many important applications; for example, any continuous function that changes sign on a closed interval must have a zero on the
interval.
Section 1.4: Concepts 1–4; Exercises 1-65 every other odd, 71, 96–100
1.5
Tangent Lines and Rates of Change
Definition:
Given a function f defined in some neighborhood of x = a, the
tangent line at the point P (a, f (a)) on the graph of f is the line
passing through P and having slope
f (a + h) − f (a)
h→0
h
provided the limit exists. The argument of the limit is called the
difference quotient.
mtan = lim
The idea behind this is the following:
1. The difference quotient represents the slope of the secant line through
the point P (a, f (a)) and the point Qh (a + h, f (a + h)); using the standard ‘rise over run’ formula for slope:
y2 − y1
f (a + h) − f (a)
f (a + h) − f (a)
=
=
x2 − x1
(a + h) − a
h
2. As h → 0, the ‘run’ for the secant lines goes to 0 and Qh → P
3. This tendency in the slopes of the secant lines can be taken as the slope
of the tangent line—essentially, this is the tendency when the ‘run’ goes
to zero
Interpretations:
• Slope of the line tangent to the curve at the given point
• Instantaneous rate of change (with the difference quotient as the average rate of change, let the time increment tend to 0 and take the
tendency in the average rate of change)
Section 1.5: Exercises 1–31 odd, 41–44
Part 2: Derivatives and Calculation
2
The Derivative
2.1
The Derivative
Definition:
The derivative of a function f with respect to x is given by
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
The domain of f 0 is the set of inputs for which the limit exists.
Interpretation: with the above definition, the derivative f 0 is the function
that takes an input x and returns with the slope of the graph of f at the
point (x, f (x)) (or the instantaneous rate of change in f at the input x.)
Notation: Other notations for the derivative include
d
df
=
(f (x))
dx
dx
and
Dx f = Dx f (x)
Definition:
A function f is differentiable on an interval if it has a derivative
at each point on the interval.
Note that ‘one-sided derivatives’ can be defined analogously to one-sided
limits (e.g., the limit of the secant lines on the right/left side of the point,
rather than both sides at the same time.)
There are three basic places that a derivative does not exist:
• sharp corners: the secant lines on the right and left sides of the point
do not have the same tendency (the one-sided derivatives do not agree)
• vertical tangent lines: the slope of a vertical line is undefined, therefore if either of the right/left tendencies in the secant lines increase or
decrease without bound near the point, the derivative will not exist at
the point
• discontinuities: if the function is not defined at the point, clearly the
derivative can not exist; if there is a jump or skip in the graph, at least
one of the sides will have a vertical tangent line.
Theorem:
A function is continuous wherever it is differentiable.
Section 2.1: Concepts 1,2; Exercises 1–59 every other odd, 65–70
2.2
Basic Rules of Differentiation
The theorems that establish the basic rules of differentiation are:
• Constant rule:
d
(c) = 0
dx
• Power rule: Given a positive integer n,
• Constant multiple rule:
• Sum/difference rule:
d n
(x ) = nxn−1
dx
d
(cf (x)) = cf 0 (x)
dx
d
(f (x) ± g(x)) = f 0 (x) ± g 0 (x)
dx
• (Natural) Exponential functions:
d x
(e ) = ex
dx
Section 2.2: Concepts 1–4; Exercises 1–51 every other odd, 59–63 odd, 72,
75–78
2.3
The Product and Quotient Rules
The theorems dealing with products and quotients of functions are:
• Product rule:
d
(f (x) · g(x)) = g(x) · f 0 (x) + f (x) · g 0 (x)
dx
d
• Quotient rule:
dx
f (x)
g(x)
!
=
g(x) · f 0 (x) − f (x) · g 0 (x)
[g(x)]2
Notation: further derivatives frequently exist; the second derivative is
given by
d
d
(f 0 (x)) = f 00 (x) =
dx
dx
d
d2
(f (x)) = 2 (f (x)) = Dx2 (f )
dx
dx
!
the third derivative is the derivative of f 00 , and so on. The third derivative
can be given by f 000 or f (3) , or by changing the 2’s in the above forms to 3’s.
For the fourth derivative and beyond, the ‘primes’ are usually dropped and
the f (4) notation favored.
Section 2.3: Concepts 1, 2; Exercises 1–45 every other odd, 47, 55-63 odd,
71–76
2.4
The Role of the Derivative in the Real World
This section clarifies what is meant by “instantaneous rate of change” with
some applications. In general, for a function y = f (x), the variable x is
called the independent (or controlled) variable, and the y or f is called the
dependent (or responding) variable. The derivative, then, can be interpreted
as the instantaneous rate of change in the dependent variable with respect
to the independent variable.
For example, the independent variable is frequently time, and the dependent variable position. In this case, x = f (t), and the derivative f 0 (t)
represents the instantaneous change in position with respect to time: the
velocity. The slope of a position vs. time curve is then the velocity: the
faster the object moves, the more steep the graph. In addition, f 00 (t) would
be the instantaneous rate of change in the velocity with respect to time: the
acceleration.
Other combinations are possible: in general, df /dx can be interpreted as
the instantaneous rate of change in (whatever f means) with respect to the
change in (whatever x means).
(This section is usually part of a lab, but those who use this material
outside of mathematics, especially science majors, are strongly advised to
read this section.)
2.5
Derivatives of the Trigonometric Functions
Review: Section 0.3. Suggested exercises: 1–41 every other odd
We continue the basic differentiation rules for the trigonometric functions
(again, each of these are theorems.) In each of the following cases, it is
assumed that the angles are measured in radians:
•
d
(sin x) = cos x
dx
•
d
(cos x) = − sin x
dx
•
d
(tan x) = sec2 x
dx
•
d
(cot x) = − csc2 x
dx
•
d
(sec x) = sec x tan x
dx
•
d
(csc x) = − csc x cot x
dx
Section 2.5: Exercises 1–43 odd
2.6
The Chain Rule
Review: Section 0.4. Suggested exercises: 1-25 odd
Recall the notion of function composition: given functions f and g with
domains Df and Dg , respectively, take a ∈ Dg and evaluate g(a); then, if
g(a) ∈ Df , then plug this output of g into f to get f (g(a)). The general
notation for this new function is (f ◦ g)(x). The new function f ◦ g can
not have a domain larger than Dg (since the original input needs to make
it through g first); additionally, one must omit from the domain of the new
function any values of x ∈ Dg that result in g(x) 6∈ Df .
Differentiating a composition of functions requires the Chain Rule:
Theorem: (The Chain Rule)
Given functions f and g, for all x in the domain of g, if g is
differentiable at x and f is differentiable at g(x), then
d
d
((f ◦ g)(x)) =
(f (g(x))) = f 0 (g(x)) · g 0 (x).
dx
dx
Alternately, if we let u = g(x), then this can be written as
df
df du
=
· .
dx
du dx
This allows all of the previous theorems on differentiation to be generalized. In addition, the rule for exponential functions becomes
d x
(a ) = ax ln a
dx
Section 2.6: Concepts 1–3; Exercises 1-79 odd
2.7
Implicit Differentiation
Occasionally, the slopes of tangent lines are interesting or important, but the
relation between the variables can not be expressed as a function (circles, for
instance.) For this, we have the technique of implicit differentiation.
Effectively, assume that one of the variables can be expressed as a function
locally—that is, at most points we can narrow to a small disc around that
point in such a way that, on that small disc, one variable can be viewed as
an input and the other will have only one value for each value of the input.
This is referred to as representing one variable implicitly as a function of the
other; generally, this is necessary when the algebraic form can not be solved
for one of the variables in terms of the other (the equation of a circle is a
good example of this—when solving for either variable, both square roots1
are necessary to get the whole picture.)
To find a derivative of an implicit representation, we view one of the
variables as the independent variable—when we take derivatives of the independent variable, everything works normally. The other variable2 is viewed
as a dependent variable, and treated as if it is a function of the independent
1
2
the positive and negative square roots
or variables
variable—when we take derivatives of expressions involving the dependent
variable, we must use the chain rule.
This technique allows extension of the Power Rule for derivatives to rational powers (except 0) and allows us to develop derivatives for inverse
functions.
Review: Section 0.7. Suggested practice: Concepts 1–5; Exercises 1–15 odd,
21–45 every other odd, 49–53 odd, 61–66
Recall that a function is invertible provided it is one-to-one: each input
has a distinct output (graphically: the graph of the original function passes
the horizontal line test.) The inverse of such a function is the function
that can be composed with the original to get back the original input—
that is, if f is one-to-one and f −1 is its inverse, then (f −1 ◦ f ) (x) = x and
(f ◦ f −1 ) (x) = x. The graph of the inverse function is the reflection of
the graph of the original across the line y = x—this switches order of the
input/output pairs: if (a, b) is on the graph of f , the (b, a) is on the graph
of f −1 .
Differentiating (f ◦ f −1 )(x) = x using the chain rule and rearranging
yields
0
1
f −1 (x) = 0 −1
f (f (x))
This gives the derivative of an inverse function in terms of that function and
the original. (A procedure can also be developed using implicit differentiation
directly.)
This can be used to find the derivatives of the inverse trigonometric functions (again, each of these is a theorem—each is only valid, at most, on the
domain of the original inverse trigonometric function):
•
d −1 1
sin x = √
dx
1 − x2
•
d −1 −1
cos x = √
dx
1 − x2
•
d −1 1
tan x = 2
dx
x +1
•
d −1 −1
cot x = 2
dx
x +1
•
1
d −1 sec x = √ 2
dx
|x| x − 1
•
d −1 −1
csc x = √ 2
dx
|x| x − 1
Section 2.7: Concepts 1b, 2, 3; Exercises 1–37 odd, 49–77 every other odd,
102–104
2.8
Derivatives of Logarithmic Functions
Review: Section 0.8. Suggested practice: Concepts 1–6; Exercises 1–31 odd
Since the natural logarithm is the inverse of the (natural) exponential
function,
eln x = x
and
ln(ex ) = x
so
1
d
(ln x) =
dx
x
by the above rule.
Also, using the change of base formula for logarithms, we can show that
for a > 0,
d
1
(loga x) =
dx
x ln a
Note that this means that other bases work similarly to the ‘natural’ base, but
an additional constant is required (recall that the derivative of exponential
expressions with base other than e works similarly.)
The Power Rule can be extended to all real powers (except 0) via the
inverse nature of the natural exponential and logarithmic functions.
In addition, complicated rational expressions and expressions with variables in the base and exponent can be differentiated using logarithmic differentiation, a technique where logarithms are applied to both sides of an
equation, the results simplified, and followed by implicit differentiation.
Section 2.8: Concepts 1–4; Exercises 1–43 odd, 52–56
Part 3: Applications of Derivatives
2.9
Related Rates
This section is about a particular type of application of implicit differentiation; a related rates problem essentially consists of several steps:
• Set up: draw picture, choose variables, identify the values of the variables/rates of change of those variables, where given
• Find a (physical) relationship between the variables and view each as
a function of time
• Implicitly differentiate with respect to time
• Fill in the values from the set up phase and solve for the unknown
Section 2.9: Concepts 1, 2; Exercises 1-41 every other odd
2.10
Linearization and Differentials
(We generally skip this section, though some in the sciences may be interested
in some of the shortcuts in this section, especially in estimating the error in a
value calculated using a measurement. For example, if you know the precision
of the measure of a diameter, the techniques in this section would allow you
to conveniently and accurately determine the expected error in the volume
of the sphere.)
3
3.1
Applications of the Derivative
Extrema of Functions
Definition:
A function f defined on a domain D has an absolute maximum at c provided f (c) ≥ f (x) for all x in D. (Absolute minimum is defined similarly.) The absolute maximum and absolute
minimum values of f on D are called the extreme values or
extrema of f on D.
It is possible for a function to an absolute maximum/minimum at multiple
points in its domain, or even for the function to have no absolute maximum
or minimum (or any combination of the two.) Values that are larger/smaller
than the values at the points ‘near’ them are called relative (or local) extrema:
Definition:
A function f has a relative maximum at an interior point c of
its domain provided f (c) ≥ f (x) for all x in some neighborhood
of c. (Local minimum is defined similarly.)
Local extremes occur when a particular input has an output larger/smaller
than his ‘neighbors’ (as opposed to an absolute extreme where the input has
an output larger/smaller than every other input in the whole domain.) It
should be evident that an absolute extreme will always be a local extreme,
but that the reverse need not hold.
The following theorem helps classify (and identify) local/absolute extrema:
Theorem: Fermat’s Theorem
If f has a relative extremum at c, then either f 0 (c) = 0 or f 0 (c)
does not exist.
Note carefully that the converse need not hold—that is, if f 0 (c) = 0
or f 0 (c) does not exist, it does not necessarily follow that f has a relative
extremum at c: consider, for example, f (x) = x3 which has f 0 (0) = 0 but
no relative extrema. The following definition can be used to identify possible
relative extrema:
Definition:
A number c is a critical number of a function f provided f (c)
exists and f 0 (c) is either zero or undefined.
Now that we know where extrema may occur, the conditions that guarantee existence of absolute extrema are found in the following theorem:
Theorem: (Extreme Value Theorem)
A function that is continuous on a closed, bounded interval has at
least one absolute maximum and at least one absolute minimum
somewhere on the interval.
Optimization: Since the critical numbers are the places where local extrema
(and hence absolute extrema) may occur, we can devise a nice procedure for
finding absolute extrema:
• Check the Extreme Value Theorem to make sure absolute extrema
exist.
• Find the critical numbers on the given interval.
• Plug in the critical numbers and the endpoints to find the maximum/minimum
output.
(This section is frequently paired with Section 3.7: Optimization Problems—
essentially, the problems in section 3.7 are the word-problem versions of the
similar problems in this section.)
Section 3.1: Concepts 1–3; Exercises 1–77 every other odd, 99–102
3.7
Optimization Problems
The problems in this section are applications of the optimization procedure
discussed in section 3.1. The only real difference is that we are not restricted
to bounded intervals, so the Extreme Value Theorem does not always apply.
On unbounded intervals, the ‘endpoints’ become the end behavior of the
function: for example, on [2, ∞), we would find the critical numbers on the
interval and check them as usual, and we would check the endpoint 2 as
usual, but need to find the “limit at infinity” for the other ‘endpoint’ of the
interval.
Generally, optimization problems identify themselves with a phrase like
“find the largest/smallest. . . ” The primary strategy on such problems is to
use a variable for the unknown and find an interval of possible values, then
find a function involving that variable that can be optimized on that interval.
Section 3.7: Concepts 1, 2; Exercises 1-41 every other odd
3.2
The Mean Value Theorem
This section contains two theorems; the first is primarily used to prove the
second:
Theorem: (Rolle’s Theorem)
If a function is continuous on a given closed interval, differentiable
on the open interior, and the endpoints both have the same output, then there must be a point on the interval where the derivative is zero.
Theorem: (The Mean Value Theorem)
If a function is continuous on a given closed interval and differentiable on the open interior, then there is some point on the
interval where the value of the derivative is the same as the slope
of the secant line through the endpoints.
The Mean Value Theorem is frequently referred to as the most important
theorem in Calculus (for reasons that will be evident later.) As an immediate
application, we can prove that functions that have a zero derivative on an
interval must be constant on that interval, and thence that two functions
with the same derivative on an interval can only differ by a constant on that
interval.
Section 3.2: Concepts 1–3; Exercises 1–23 odd, 47–52
3.3
Monotonic Functions and the First Derivative Test
In this section, we interpret the effect of the behavior of the first derivative
on the behavior of the graph of a function.
Definitions:
Given a function f defined on an interval I, we say that f is
• increasing if, for every pair of inputs in I, the outputs
increase when the inputs increase;
• decreasing if, for every pair of inputs in I, the outputs
decrease when the inputs increase;
• monotonic on I if it is increasing or decreasing on I.
The Mean Value Theorem gives the following characterization:
Theorem:
If the derivative of a function is positive/negative on an interval,
then the function is increasing/decreasing on that interval.
This leads to
Theorem: (The First Derivative Test)
If a function f has a critical number at x = c and is differentiable
in some neighborhood of c (except, perhaps, c itself), then, as the
inputs increase,
• if the sign of f 0 changes from positive to negative at x = c,
then there is a local maximum at x = c;
• if the sign of f 0 changes from negative to positive at x = c,
then there is a local minimum at x = c;
• if there is no sign change at x = c, then there is not a local
extreme at x = c.
Thus a sign chart for the first derivative will tell where the function is
increasing/decreasing, and help locate and identify any local extrema.
Section 3.3: Concepts 1–3; Exercises 1–43 odd, 73–78
3.4
Concavity and Inflection Points
In the previous section, we discussed what the first derivative tells about the
graph of a function; in this section we explore the second derivative and its
effect on the graph of the original function.
We say that a function f is concave up at a point x = c provided f (c)
is below the ‘local secant lines’ (that is, for every neighborhood about c, any
secant line through points on opposite sides of c is above f (c) at x = c);
similarly, the function is concave down at a point x = c provided f (c) is
above the ‘local secant lines.’ Effectively, this means that a differentiable
function is concave up when it ’stays above its tangent lines’ (similarly, concave down when ‘below.’) The text defines these terms only for differentiable
functions in terms of the derivative (f is concave up/down where f 0 is increasing/decreasing.) As such,
Theorem:
If the second derivative of a function is positive/negative on an
interval, then the function is concave up/down on that interval.
A sign chart for the second derivative will tell where the function is concave up/down, and locate inflection points: points where the original function is defined and the concavity changes. Inflection points are thus critical
points of the first derivative which are additionally accompanied by a sign
change.
Also, we can (sometimes) identify local extrema:
Theorem: (The Second Derivative Test)
Given a function that has a continuous second derivative in some
neighborhood of a point x = c where f 0 (c) = 0,
• if f 00 (c) > 0, then f has a local minimum at x = c;
• if f 00 (c) < 0, then f has a local maximum at x = c;
• if f 00 (c) = 0, then the test fails—no conclusion can be made.
Section 3.4: Concepts 1–3; Exercises 1–45 every other odd, 51–54, 83–86;
also, Section 3.5 Exercises 57–60
3.6
Curve Sketching
The material we have covered so far can be used to create a nice procedure
for sketching curves:
1. Analyze f : find its domain, intercepts, end behavior, and locate any
skips/vertical asymptotes.
2. Analyze f 0 : make a sign chart and use it to identify where the graph is
increasing/decreasing; locate any critical numbers and find their types.
3. Analyze f 00 : make a sign chart and use it to identify where the graph
is concave up/down; locate any inflection points.
4. If any regions seem ambiguous or need more precision, plot a few points.
Slant Asymptotes: We can also find slant (or oblique) asymptotes: these
are tendency lines that are not horizontal. For rational functions, slant
asymptotes occur when the numerator is one degree larger than the denominator. These asymptotes can be found by polynomial long division: the
quotient will be linear, and the remainder portion will be a proper rational
function whose end behavior is zero.
In general, however, slant asymptotes occur when
f (x)
=m
x
and will be of the form y = mx + b where m is determined above and b is
given by
lim f (x) − mx = b
lim
x→±∞
x→±∞
Section 3.6: Concepts 1, 2; Exercises 1–43 odd
3.8
Indeterminate Forms and L’Hôpital’s Rule
With limits, quotients usually cause trouble: when the expressions in the
numerator and denominator both look to be going to zero, we say that the
current form of the function is indeterminate and call this situation the ‘0/0’
indeterminate form. When the expressions in the numerator and denominator both exhibit unbounded behavior, the current form is also indeterminate
and we call this situation the ‘∞/∞’ indeterminate form (regardless of the
signs involved). The problem with these forms is that they are unpredictable;
examples for each indeterminate form can be found where the limit is unbounded, finite but nonzero, and zero, so no ‘arithmetic’ can be done on or
with these indeterminate forms.
All is not lost, however. We have already discussed rearrangements of the
functional forms that clear up the indeterminates (factoring and canceling,
for instance), but we have a more powerful tool—if both numerator and
denominator seem to be behaving the same way, the rates of growth become
important. The following theorem is very useful (though easy to mis-apply)
to resolve these indeterminate cases:
Theorem: (L’Hôpital’s Rule)
Given two functions f and g defined and differentiable in a neighborhood of a point x = a, if
lim
x→a
f (x)
g(x)
is an indeterminate expression of the form 0/0 or ∞/∞, then
f (x)
f 0 (x)
= lim 0
x→a g(x)
x→a g (x)
lim
provided the limit on the right side exists or is infinite.
L’Hôpital’s Rule can also be used with other indeterminate forms, but
some algebra will be necessary to get the expression into one of the two
forms above:
• ‘0·∞’: in general, one of the expressions can be reciprocated and moved
to the denominator to get ‘0/0’ or ‘∞/∞’ and L’Hôpital’s Rule can be
applied.
• ‘∞−∞’: a common denominator usually gets these into ‘0/0’ or ‘∞/∞’
and L’Hôpital’s Rule can be applied.
• Exponential forms (1∞ , 00 , and ∞0 ): since the exponential function is
continuous,
lim f (x) = x→a
lim eln f (x) = elimx→a ln f (x)
x→a
and the laws of logarithms can be used to bring the power in front of
the logarithm. The resulting expression can be manipulated to ‘0/0’ or
‘∞/∞’ and L’Hôpital’s Rule can be applied; the solution will then be
the exponential of the limit found via L’Hôpital’s Rule.
Section 3.8: Concepts 1–9; Exercises 1–59 odd, 73, 74
3.9
Newton’s Method
This section details the Newton-Raphson Method for approximating roots of
functions. It is generally covered in the lab, but reading this section before
the lab would be extremely helpful.
Part 4: Introduction to Integration
4
4.1
Integration
Indefinite Integrals
In the preceding material, we were given a function and used the derivative
to find the rate of change in the function; it stands to reason, then, that given
the rate of change of a function, we would reverse the process of differentiation
to get back the original function. This reversal results in an antiderivative
of the given function:
Definition:
A function F is an antiderivative of a function f on an interval
I provided F 0 (x) = f (x) for all x in I.
As one might guess from the indefinite article in the definition, antiderivatives are not unique for a given function—clearly they may differ by a constant. One of the consequences of the Mean Value Theorem in Section ??
was that two functions with the same derivative on an interval can differ by
at most a constant. (We frequently say this more compactly by stating that
antiderivatives are unique up to an additive constant.)
The process of finding an antiderivative is called integration; for notational purposes, we use the indefinite integral:
Z
f (x) dx = F (x) + C
Here f (x) is the integrand, dx the differential (which, for now, just indicates the variable of integration), F (x) the antiderivative of f , and C the
arbitrary constant of integration. The basic rules for integration are just
the reverses of the rules for derivatives (summarized nicely in the text). Indefinite integrals are particularly nice because they can always be checked:
the derivative of the solution should be the integrand.
Finally, we begin the discussion of differential equations: a differential
equation is simply an equation that involves derivatives. A regular equation
just involves variables and the solutions are numbers which make the equation
true: x2 − 4 = 0 has solutions x = ±2. A differential equation involves
functions; for example:
d
(f (x)) = x
dx
has general solution f (x) = x2 /2 + C since this function makes the equation
true.
To find a particular solution, we need additional information: for example, if we know that f (0) = 3, then the solution of the differential equation
particular to this situation would be f (x) = x2 /2 + 3—the additional information helps us to determine the particular constant for this situation.
Differential equations with enough information to find the constant(s) are
called Initial Value Problems.
Section 4.1: Concepts 1–4; Exercises 1–29 odd, 35–49 odd, 81–87
4.2
Integration by Substitution
Integration is usually more difficult than differentiation, so there are several
techniques of integration that have evolved to help. The first effectively
undoes the chain rule. If F is the antiderivative of f , then
Z
f (g(x))g 0 (x) dx = F (g(x)) + C
There are several different notational methods to do this: the most common
is to use u = g(x) and check that du = g 0 (x) dx, then make the substitution
so that the integral turns into
Z
f (u) du = F (u) + C = F (g(x)) + C
Section 4.2: Concepts 1, 2; Exercises 1–75 odd, 89, 90
4.3
Area
In this section, we develop a strategy for estimating the area between a curve
and the x-axis above a closed, bounded interval using rectangular regions on
a uniform partition. The development works like this:
• Start with a continuous function f on a closed interval [a, b]
• Partition [a, b] into n subintervals: the width of each interval will be
∆x =
total width
b−a
=
n
number of subintervals
Define xk = a + k · ∆x and note that x0 = a, xn = b, and the other
points are equally spaced between: these xk ’s become the endpoints of
the subintervals. The k th -subinterval will then be [xk−1 , xk ].
• For each subinterval, choose a point ck in [xk−1 , xk ] and use f (ck ) as
the height of the rectangle for that interval.
• Each subinterval is then associated with a rectangle of width ∆x and
height f (ck ): the area of this rectangle is f (ck )∆x
• Sum these areas over the number of subintervals to get an approximation to the area under the curve on this interval:
n
X
f (ck )∆x
k=1
• For better approximations, use more rectangles.
Section 4.3: Concepts 1,2
4.4
The Definite Integral
This section combines the previous area approximation strategy with the
notion of limits to develop the definite integral:
Informal Definition:
Given a continuous function f defined on an interval [a, b], define
each symbol used below as in the material for the previous section.
Then the definite integral of f over [a, b] is the number given
by
Z b
a
f (x) dx = lim
n→∞
n
X
f (ck )∆x
k=1
(This is an informal definition because it glosses over many important
theoretical points—it captures the basic idea, but we will refine it a bit
later.)
With this definition, we can begin studying the properties and interpretations of this new object. Though it was designed to give the area under
the curve, it only actually does so on intervals where the function is nonnegative. If the function dips below the x-axis, the ‘heights’ of the rectangles
f (ck ) may actually be negative, so the ‘areas’ of those rectangles will be
negative—a function that stays below the x-axis on an interval will actually
have a negative definite integral on that interval. As such, we tend to interpret the definite integral geometrically as the signed area between the curve
and the x-axis.
There are several applications/interpretations of the definite integral: in
an applied setting, the area of the rectangle associated with the subinterval
becomes the product of the approximate rate of change with the width of
the subinterval—this approximates the change in the original function over
this subinterval. Adding these little changes up over the whole interval then
taking the limit results in the total change over the interval. For example,
if the integrand represents velocity (in, say, meters per second) at a given
time (measured in seconds), then the product on each subinterval in time is
the approximate change in position with respect to time multiplied by the
difference in times on that interval: meters per second times seconds equals
meters traveled. Sum these approximations and you get the total change
in position from time a to time b. The signs, then, start to make sense: a
negative velocity on a subinterval means the particle is moving backwards
on that interval, and so will not travel as far in the positive direction.
Several properties of the definite integral can be explained with the above
informal definition and interpretations:
•
Ra
•
Rb
•
Rb
a
a
a
f (x) dx = 0: ‘going nowhere’ results in no change
f (x) dx = −
Ra
b
f (x) dx: ‘going backwards’ reverses the sign
c dx = c(b − a): constant functions result in rectangles
• Sums/difference and constant multiples work the way one would expect
•
Rb
a
f (x) dx =
Rc
a
f (x) dx +
Rb
c
f (x) dx: draw the picture(s) here
• Definite integrals of functions which are nonnegative over the integral
of integration will be nonnegative
• The
definite integral is monotone: if f (x) ≥ g(x) on [a, b], then
Rb
a g(x) dx
• Integral bounds: if m ≤ f (x) ≤ M on [a, b], then m(b−a) ≤
M (b − a)
Rb
a
Rb
a
f (x) dx ≥
f (x) dx ≤
Finally, we can generalize the definition above to a (possibly non-uniform)
partition P consisting of n subintervals determined by (n + 1) points:
a = x0 < x1 < x2 < . . . < xn = b
the widths of the subintervals are then given by
(∆x)k = xk − xk−1
(notice we need subscripts for the widths since they are no longer all the
same). Again, for each subinterval, choose ck in [xk−1 , xk ]. Then the approximate area under this scheme will be given by
n
X
f (ck )(∆x)k
k=1
To get the limit in is a bit trickier; since our partition is no longer uniform,
we can add more points to the partition without actually making all of the
rectangles get skinnier. To get around this, we let kP k denote the maximum
subinterval width on P and force kP k → 0. In this case, the definite integral
can be given by
Z b
a
f (x) dx = lim
kP k→0
n
X
f (ck )(∆x)k
k=1
provided the limit is independent of the choices of the ck ’s.
Section 4.4: Concepts 1, 2; Exercises 1–35 odd, 63–70
4.5
The Fundamental Theorem of Calculus
This section brings everything together. It begins with the Mean Value
Theorem for Integrals:
Theorem: (Mean Value Theorem for Integrals)
If f is continuous on [a, b], then there exists a c in [a, b] such that
f (c) =
1 Zb
f (x) dx
b−a a
The right side of the equation above is the definition of the average (or
mean) value for a function on an interval. Refer to the integral bounds property in the previous section; dividing the second chain of inequalities through
by (b − a) results in the average value sandwiched between the minimum and
maximum value of the function on the given interval—since the function is
continuous, the Intermediate Value Theorem says that the function must
take on all values intermediate, so such a point c must exist on the interval.
The author then works through The Fundamental Theorem of Calculus,
Part I—the Integral-differential Relationship. Basically, this says that the
derivative of the antiderivative is the original function (though it is a bit
more complicated than that. . . ) Then comes the Fundamental Theorem of
Calculus, Part II—the Evaluation of Definite Integrals, which says that if an
antiderivative can be found, the value of the definite integral is the difference
between the values of the antiderivative evaluated at the endpoints of the
interval of integration.
Finally, we discuss the (mild) changes in the procedure for evaluation of
definite integrals via substitution.
Section 4.5: Concepts 1–5; Exercises 1–61 every other odd, 77–83 odd,
124–128