MATH 140 - Summary Notes
bh2smith@gmail.com
These notes are intended to provide a quick, concise reference to the course material, but should not be
considered as a sufficient replacement of the textbook and/or attendance in class. Each week of tutorial
will be separated by a horizontal line and I hope to arrange that the sections and subsections be numbered
according to those in your textbook.
Contents
1. Functions and Limits
1.1. Four ways to represent a function
1.2. A catalogue of essential functions
1.3. New functions from old
1.4. Graphing calculators and computers
1.5. Exponential functions
1.6. Inverse functions
1.7. The Limit of a Function
1.8. Limit Laws
1.9. Precise Definition of a Limit
1.10. Continuity
2. Derivatives
2.1. Derivatives and Rates of Change
2.2. The Derivatives as a Function
2.3. Differentiation Formulas
2.4. Derivatives of Trig functions
2.5. Chain Rule
2.6. Implicit Differentiation
2.7. Rates of Change in Natural and Social Science
2.8. Linear approximation and Differentials
3. Applications of Differentiation
3.1. Maximum and Minimum Values
3.2. Mean Value Theorem
3.3. How Derivatives Affect the Shape of a Graph
3.4. Limits at Infinity; Horizontal asymptotes
3.5. Summary of Curve Sketching
3.6. Optimization Problems
3.7. Newton’s Method
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1. Functions and Limits
1.1. Four ways to represent a function. Functions can be represented verbally, numerically (table of
values), visually (a graph) or algebraically (by formulae). They have a domain, a range and personality.
Some functions are defined piecewise like, for example,
(
x,
x≥0
|x| =
.
−x, x < 0
Some can admit symmetry (usually about the y-axis) and some functions are increasing, decreasing or both.
We use the vertical line test to determine if a given sketch is a function.
1.2. A catalogue of essential functions. Most functions you will see (and, in fact, these are all the
functions we use in mathematics) arise as sums, products, quotients, scalar multiples, roots and compositions
of the elementary functions:
1
Pn
(1) Polynomials; p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 = i=0 ai xi where n ≥ 0 is called the degree
of p and the ai ’s are called coefficients.
Domains of polynomials are all of R.
(2) Power functions: f (x) = xa with a ∈ R is constant.
• When a is a positive integer, these are the basic powers of x we know and love from grade 101.
• When a = 1/n, these are the compositional inverses of the integer powers of x (sketched by
reflection about the axis y = x).
Domains for x1/n is {x ≥ 0} when n is odd and all of R when n is even.
• When a = −1 this is the function f (x) = x−1 = x1 (the multiplicative inverse of y = x)
Domain of y = 1/x is R\{0} = (−∞, 0) ∪ (0, ∞) is everywhere except 0.
P (x)
(3) Rational functions: f (x) = Q(x)
a ratio of polynomials.
Domain of a rational function is R\Z(Q) (all of R except for the zeros of Q).
(4) Algebraic functions: Constructed using algebraic operations (sums, products, powers, quotients
or roots) of polynomials.
Domains here are not as easily stated, but we will see examples to get comfortable.
(5) Trigonometric: f (x) = cos(x) (or sin(x)) defined by the x (or y) coordinates of a particle travelling
(counter-clockwise) along the unit circle in R2 .
Domains are all of R and Ranges are [−1, 1].
(6) Exponential: f (x) = ax with a > 0 is constant.
Domain is R and range is (0, ∞).
(7) Logarithmic: f (x) = loga (x) is the compositional inverse of the exponential ax .
Domain is (0, ∞) and range is R.
Notice that the domains and ranges of inverse functions switch.
1.3. New functions from old. Functions can be vertically or horizontally shifted, stretched, scrunched or
reflected. Domains and ranges will change with these transformations respectively. One can also construct
new functions by composition of the elementary functions. One must keep track and be very careful of
domains and ranges when constructing new functions.
1.4. Graphing calculators and computers. Not covered
1.5. Exponential functions. Everyone should know the laws of exponents.
1.6. Inverse functions. A function is one-to-one if it passes the horizontal line test and (for the purpose
of this course) this is equivalent to being invertible. The domain of f −1 is the range of f and vice-versa.
The book provides (cf. page 61) a short algorithm for finding the inverse of a function. Note that f −1 is
NOT the same as 1/f . The graph of f −1 is given by the reflection in the line y = x of the graph of f .
Logarithms. The logarithm loga (x) is the inverse of ax . From this we obtain the laws of logarithms from the
laws of exponents. For example
loga (xy) = loga (x) + loga (y) comes from/corresponds to ax+y = ax · ay .
The natural logarithm appears most frequently because it is so natural. That is, ln(x) is the inverse of ex
where e is called Euler’s number.... I think. Remember that ln(e) = 1 or in words that
“Lonny is number one”
There is a change of base formula, loga (x) =
ln x
ln a .
Inverse trig. The trig functions are periodic and hence, do not pass the horizontal line test. This would
imply they are not invertible. However, restricting their domains to a domain on which they are one-to one
allows us to define inverses.
1See textbook page 29 for standard sketches of these curves
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1.7. The Limit of a Function. If f (x) is defined for x near some number a (i.e. in some open interval
around a, except possibly at a), write limx→a f (x) = L to say the limit of f (x) as x approaches a is L. One
can also talk about left and right handed limits and the result is that limx→a f (x) = L if and only if both
the left and right hand limits of f at a equal L (i.e. limx→a± f (x) = L).
An important limit is limx→0 sinx x = 1.(
The Heavy-side step function H(x) =
0, t < 0
1, t ≥ 0
is has that limx→0− H(x) = 0 and limx→0+ H(x) = 1
so that the limit at 0 does not exist.
A vertical asymptote of f at a is found when one has that limx→a f (x) = ±∞ or limx→a± f (x) = ±∞.
1.8. Limit Laws. If c is a constant and both limx→a f (x), limx→a g(x) exist then
(1)
(2)
(3)
(4)
limx→a [f (x) ± g(x)] = limx→a f (x) ± limx→a g(x)
limx→a c · f (x) = c · limx→a f (x)
limx→a f (x)g(x) = limx→a f (x) · limx→a g(x)
(x)
x→a f (x)
limx→a fg(x)
= lim
limx→a g(x) provided that limx→a g(x) 6= 0.
Theorem 1 (Squeeze Theorem). If f (x) ≤ g(x) ≤ h(x) for x near a and limx→a f (x) = limx→a h(x) = L
then limx→a g(x) = L.
1.9. Precise Definition of a Limit. The limit of f (x) as x approaches a is L if for all > 0, there exists
a δ > 0 such that |f (x) − L| < whenever 0 < |x − a| < δ. There are analogous definitions for left and right
handed limits.
1.10. Continuity. A function f is continuous at a if limx→a f (x) = f (a). Similarly, for any subset A ⊂ R
f is said to be continuous on A if it is continuous at every point in A. From the limit laws, we get the
following result
Lemma 1. If f and g are continuous at a and c is some constant, then f ± g, cf, f g and f /g are also
continuous at a (provided g(a) 6= 0).
Polynomials are continuous on all of R and any rational function (of polynomials) is continuous everywhere
that it is defined. Furthermore, root functions, trig functions, exponential and logarithmic functions are all
continuous on their domains.
Theorem 2. If f is continuous at b and limx→a g(x) = b, then
lim f (g(x)) = f ( lim g(x)) = f (b).
x→a
x→a
This theorem implies that if g is continuous at a and f is continuous at g(a) then their composition
f ◦ g(x) = f (g(x)) is continuous at a.
Theorem 3 (Intermediate Value Theorem). If f is continuous on the closed interval [a, b] and N is any
number between f (a) and f (b) (with f (a) 6= f (b)), then there exists some c ∈ (a, b) such that f (c) = N .
2. Derivatives
2.1. Derivatives and Rates of Change. The tangent line to a curve y = f (x) at the point p = (a, f (a))
(a)
is unique line passing through p with slope m = limx→a f (x)−f
(provided the limit exists). The derivative
x−a
0
of a function f at the number a, denoted by f (a), is defined as
f 0 (a) = lim
x→a
f (x) − f (a)
f (a + h) − f (a)
= lim
.
h→0
x−a
h
We write ∆x = x2 − x1 to denote small differences in x. The average rate of change of a function f over
(x1 )
∆y
the interval [x1 , x2 ] is given as ∆x
= f (xx22)−f
. The instantaneous rate of change is found by taking the
−x1
limit as ∆x → 0 (this is just the derivative again).
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2.2. The Derivatives as a Function. Notice we can differentiate a function at any point (on the interior)
of its domain and the result is again a number (whose value is the slope of the tangent line to f at that
point). Hence the derivative of f is another (often different) function associated to, or derived from, f . This
function, denoted f 0 (x) is called the derivative of f .
Notation: For y = f (x) we can write
f 0 (x) = y 0 =
df
d
dy
=
=
f (x) = Df (x) = Dx f (x).
dx
dx
dx
dy
∆y
is called Leibnitz notation and was meant to represent the limit lim∆x→0 ∆x
. A function is called
The dx
0
differentiable at a if f (a) exists. For a subset A ⊂ R (say, for example, A = (a, b)), f is called differentiable
on A if it is differentiable at every point in A.
2.3. Differentiation Formulas.
Constants
Product Rule
d
c=0
dx
(f g)0 = f 0 g + g 0 f
Powers
d
d n
x = 1 and more generally
x = nxn−1
dx
dx
Scalar multiple
Quotient Rule
0
f 0 g − g0 f
f
=
g
g2
(cf )0 = cf 0
General Powers
Sums
dxa
= −nxa−1 for all a ∈ R
dx
(f ± g)0 = f 0 ± g 0
2.4. Derivatives of Trig functions. By the squeeze theorem along with the fact that limθ→0
one finds that
d sin x
d cos x
= cos x
and
= − sin x.
dx
dx
Then, by quotient rule,
d tan x
d sin x
=
= · · · = sec2 x.
dx
dx cos x
Similarly,
d sec x
d cot x
d csc x
= − csc x cot x,
= sec x tan x and
= − csc2 x
dx
dx
dx
sin(θ)
θ
=1
2.5. Chain Rule. For differentiable functions f, g, the derivative of their composition is
(f ◦ g)0 (x) = f 0 (g(x)) · g 0 (x)
In Leibnitz notation, if y = f (u) and u = g(x) then
dy
df du
=
·
.
dx
du dx
For example
d
[g(x)n ] = ng(x)n−1 · g 0 (x).
dx
2.6. Implicit Differentiation. By invoking the chain rule, we can now differentiate ‘functions’ (formally
referred to as relations) for which we cannot isolate y-variable2. Lets give it a whirl with the relation
tan(x/y) = x + y
2For example, the relation x2 + y 2 = 1 (describing a circle)
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where y is thought to be y = y(x) (a function of x). Then applying
dx + y
d tan(x/y)
=
dx
dx
d(x/y)
dy
2
sec (x/y) ·
=1+
dx
dx
dy
1 · y − x dx
dy
=1+
sec2 (x/y) ·
y2
dx
sec2 (x/y)y − y 2 = sec2 (x/y)x
and solving for
dy
dx
d
dx ;
we have
(Chain-rule)
(Quotient-rule)
dy
dy
+ y2
dx
dx
shows that
dy
sec2 (x/y)y − y 2
=
dx
sec2 (x/y)x + y 2
2.7. Rates of Change in Natural and Social Science. Slopes of secants are average rates of change
where slopes of tangents are instantaneous rates of change. In physics, when s = f (t) is the position of a
0
00
particle attire t, then its velocity is v(t) = ds
dt and its acceleration a(t) = v (t) = s (t). I don’t really feel like
doing this section.
2.8. Linear approximation and Differentials. The linear approximation of y = f (x) at some point a in
the domain of f is given by
Lf,a (x) = f (a) + f 0 (a)(x − a)
and this is really just the equation of the tangent line to f at a. The differential dy is given by
dy = f 0 (x)dx
and is related to the approximate change in f at x if one is to move over by ∆x.
3. Applications of Differentiation
3.1. Maximum and Minimum Values. An absolute maximum of a function y = f (x) is a value of f at,
say c, for which f (c) ≥ f (x) for all x in the domain. A local maximum is similarly defined as f (c) ≥ f (x)
for all x in some neighbourhood of c.
Theorem 4 (Extreme value theorem). If f is continuous on a closed interval [a, b] then f attains an absolute
maximum and minimum at some points c, d ∈ [a, b] respectively.
Theorem 5. If f has a local minimum or maximum at c, and if f 0 (c) exists, then f 0 (c) = 0.
A critical number of f is a number c in the domain of f such that either f 0 (c) = 0 or does not exist.
To find absolute and/or local minimums and/or maximums of f on a closed interval, it suffices to evaluate
f 0 (x), find the critical points, check the values of f at endpoints of the interval as well as at critical points.
3.2. Mean Value Theorem.
Theorem 6 (Rolle’s Theorem). Let f be a function such that
(1) f is continuous on [a, b],
(2) f is differentiable on (a, b), and
(3) f (a) = f (b).
There there is a number c ∈ (a, b) such that f 0 (c) = 0.
Theorem 7 (Mean Value Theorem). Let f be a function such that
(1) f ic continuous on [a, b], and
(2) f is differentiable on (a, b).
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There there is a number c ∈ (a, b) such that
f 0 (c) =
f (b) − f (a)
b−a
Corollary 1. If f 0 (x) = 0 for all x ∈ (a, b) then f is constant on (a, b).
Corollary 2. If f 0 (x) = g 0 (x) on (a, b) then f (x) − g(x) is constant on (a, b) and hence f (x) = g(x) + c for
some constant c.
3.3. How Derivatives Affect the Shape of a Graph. If f 0 (x) > 0 then f is increasing at x, if f 0 (x) < 0
then f is decreasing.
Proposition 1 (First derivative test). Suppose c is a critical number of a continuous function f .
a) If f 0 changes from positive to negative at c then f has a local maximum at c,
b) If f 0 changes from negative to positive at c then f has a local mminimum at c,
c) If f 0 does not change sign at c, then f has no local min or max at c.
If the graph of f lies entirely above all of its tangent lines then f is concave up. In the opposite setting f
is concave down.
Proposition 2 (Concavity test).
a) If f 00 (x) > 0 on an interval I , then f is concave up on I
00
b) If f (x) < 0 on an interval I , then f is concave down on I
A point p on a curve y = f (x) is an inflection point if f is continuous there and the curve changes from
concave up to concave down or vice-versa.
Proposition 3 (Second derivative test). Suppose f 00 is continuous near c.
a) If f 0 (c) = 0 and f 00 (c) > 0 then f has a local maximum at c, and
b) If f 0 (c) = 0 and f 00 (c) < 0 then f has a local minimum at c.
3.4. Limits at Infinity; Horizontal asymptotes. Let f be a function defined on some interval (a, ∞).
Then limx→∞ f (x) = L means that f has a horizontal asymptote y = L. This limit can be approaching L
from above, below or in a damped harmonic (sinusoidal) fashion. Same ideas, hold for the limit as x goes to
−∞.
Proposition 4. If r > 0 is a rational number, then
1
= 0.
xr
3.5. Summary of Curve Sketching. The following recipe will always yield successful calculus cupcakes!
Let y = f (x) be some function.
(1) Information from f
(a) Domain
(b) Intercepts (x and y)
(c) Symmetry (is f , even? odd? periodic?)
(d) Asymptotes
(i) Horizontal; check limx→±∞
(ii) Vertical; check limx→a± when a is an isolated point missing from the domain or an
endpoint of an open interval.
(iii) Oblique; f (x) is said to be asymptotic to g(x) if limx→∞ (f (x) − g(x)) = 0.
Many examples take the form f (x) = p(x)
q(x) where p, q are polynomials and the degree of
p is greater than the degree of q. Another common example is f (x) = g(x) + h(x) where
h(x) has a horizontal asymptote of 0 (e.g. h(x) = 1/x, 1/x2 , . . . or h(x) = e−x ), because
then certainly limx→∞ f (x) − g(x) = limx→∞ h(x) = 0.
(2) Information from f 0 (i.e. first derivative test)
(a) intervals of increasing/decreasing (i.e. find x such that f 0 (x) > 0/f 0 (x) < 0).
(b) Critical points; points where f 0 = 0 and/or when f 0 tends to infinity. These are used to find
local and global maxima and minima
lim
x→±∞
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(3) Information from f 00 (i.e. second derivative test)
(a) Concavity and inflection points.
3.6. Optimization Problems. Here, I suppose, we will neglect to summarize the theory, but solve a few
interesting/complicated problems.
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