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2WCB0 - Summary Calculus C
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2WCB0 - Calculus CA Summary
Fe Fan Li
August 10, 2020
, Contents
Preface 3
1. Limits and Continuity 4
1.1 Examples of Velocity, Growth Rate, and Area . . . . . . . . . . . 4
1.2 Limits and Functions . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Limits at Infinity and Infinite Limits . . . . . . . . . . . . . . . . 6
1.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 The Formal Definition of Limit (!) . . . . . . . . . . . . . . . . . 9
2. Differentiation 11
2.1 Tangent Lines and Their Slopes . . . . . . . . . . . . . . . . . . . 11
2.2 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . 13
2.6 Higher-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Using Differentials and Derivatives . . . . . . . . . . . . . . . . . 14
2.8 The Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 16
3. Transcendental Functions 17
3.1 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Exponential and Logarithmic Functions . . . . . . . . . . . . . . 18
3.3 The Natural Logarithm and Exponential and Exponential Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 The Inverse Trigonometric Functions . . . . . . . . . . . . . . . . 19
4. More Applications of Differentiation 20
4.1 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Finding Roots of Equations . . . . . . . . . . . . . . . . . . . . . 20
4.3 Indeterminate Forms . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Concavity and Inflections . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Sketching the Graph of a Function . . . . . . . . . . . . . . . . . 24
1
, 2
4.7 Graphing with Computer . . . . . . . . . . . . . . . . . . . . . . 24
4.8 Extreme-Value Problems . . . . . . . . . . . . . . . . . . . . . . . 24
4.9 Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . 24
4.10 Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5. Integration 27
5.1 Sums and Sigma Notation . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Areas of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.4 Properties of the Definite Integral . . . . . . . . . . . . . . . . . . 27
5.5 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . 27
6. Techniques of Integration 29
6.1 The Method of Substitution . . . . . . . . . . . . . . . . . . . . . 29
6.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 Integrals of Rational Functions . . . . . . . . . . . . . . . . . . . 30
6.4 Inverse Substitution . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.5 Other Methods for Evaluating Integrals . . . . . . . . . . . . . . 33
6.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.9 First-Order Differential Equations 35
,Preface
This document is a summary of the course 2WCB0, based on the course given
in the year 2018-2019 and the book Calculus a Complete Course. Use this
summary at your own risk; I am not responsible for your exam and
sections in this summary may contain errors, redundant information
or absence of important information. My recommendation is that
you also study the book and exercises as well as exam questions thor-
oughly. This summary will not be a relisting of standard limits and other
expressions, instead useful expressions can be found in these chapters your-
selves:
1. Chapter 2.5: derivatives of trigonometric functions
2. Chapter 3.2: Exponential and logarithmic functions: Limits and calcula-
tion properties
3. Chapter 3.3: Natural Logarithm: Limit and properties
4. Chapter 3.5 Inverse trigonometric functions: definition,derivatives,integrals,properties
5. Chapter 4.10: Maclaurin series for some widely used functions
6. Chapter 5.6: Elementery integrals,integrals of trigonometric functions
7. Chapter 6.1: One useful integral involving the secant
8. Chapter 6.2: Integrals involving the natural logarithm and arctangent
9. Chapter 6.3: Integrals involving trigonometric functions: inverse substi-
tutions
Also: this doc is probably unfinished and some parts may not be relevant any-
more but here you have it anyway
3
,1. Limits and Continuity
1.1 Examples of Velocity, Growth Rate, and Area
Skip.
1.2 Limits and Functions
Definition 1:
An informal definition of limit
If f (x) is defined for all x near a, except possibly at a itself, and if we can ensure
that f (x) is as close as we want to L by taking x close enough to a, but not
equal to a, we say that the function f approaches the limit L as x approaches
a, and we write
lim f (x) = L (1)
x→a
Definition 2:
Informal definition of left and right limits
If f (x) is well defined on some interval (b, a) extending to the left of x = a, and
if we can ensure that f (x) is as close as we want to L by taking x to the left of
a close enough to a, then we say f (x) has left limit L at x = a, and we write
lim f (x) = L. (2)
x→a−
If f (x) is defined on some interval (a, b) extending to the right of x = a, and if
we can ensure that f (x) is as close as we want to L b taking x to the right of
a and close enough to a, then we say f (x) has right limit L at x = a, and we
write
lim+ f (x) = L (3)
x→a
4
, 5
Note the use of suffix + to denote approach from the right (the positive side)
and the suffix - to denote approach from the left side (the negative side).
Theorem 1:
Relationship between one-sided and two-sided limits
A function f (x) has limit L at x = a if and only if it has both left and right
limits there and these one-sided limits are both equal to L:
lim f (x) = L ⇐⇒ lim = lim f (x) = L. (4)
x→a x→a− x→a+
Theorem 2: Limit Rules
Look for these rules yourself in the book at page 69.
Theorem 3:
Limits of Polynomials and Rational Functions
1. if P (x) is a polynomial and a is any real number, then
lim P (X) = P (a) (5)
x→a
2. If P (x) and Q(x) are polynomials and Q(a) 6= 0, then
P (x) P (a)
lim = . (6)
x→a Q(x) Q(a)
Theorem 4:
The Squeeze Theorem
Suppose that f (x) ≤ g(x) ≤ h(x) holds for all x in some open interval containing
a except possibly at x = a itself. Suppose also that
lim f (x) = lim h(x) = L. (7)
x→a x→a
Then limx→a g(x) = L also. Sometimes similar statements can hold for left and
right limits.
Example 10
Given that 3 − x2 ≤ u(x) ≤ 3 + x2 for all x 6= 0, find limx→0 u(x)
Since limx→0 u(x) = 3 and limx→0 (3 + x2 ) = 3, the Squeeze implies that
limx→0 u(x) = 3.
Examples of exam questions will be displayed for certain topics. It is recom-
mended to make these questions yourself first before looking at the answer.
These solutions of these examples are only meant as a recap.
, 6
Final exam - 29-01-2018 Question 9a
Determine the following limits ( give a good argumentation):
(a) limx→0 x sin( x12 )
Solution:
Apply the squeeze theorem: for all x ∈ R we have
0 ≤ | sin( x12 )| ≤ |x| → 0
So limx→0 x sin( x12 ) = 0
1.3 Limits at Infinity and Infinite Limits
Definition 3: Limits at infinite and negative infinity (informal defini-
tion)
If the function f is defined on some interval (a, ∞) and if we can ensure that
f (x) is as close as we want to the number L by taking x large enough, then we
say that f (x) approaches the limit L as x approaches infinity, and we
write
lim f (x) = L (8)
x→∞
If f is defined on an interval −∞, b and if we can ensure that the f (x) is as close
as we want to the number M by taking x negative and large enough in absolute
value, then we say that f (x) approaches the limit M as x approaches
negative infinity, and we write
lim f (x) = M (9)
x→−∞
Example 3:
(Numerator and denominator of the same degree) evaluate
2x2 −x+3
limx→±∞ 3x2 +5
solution: Divide the numerator and denominator by x2 , the highest power of
x appearing in the denominator:
2x2 −x+3 2−(1/x)+(3/x2 ) 2−0+0
limx→∞ 3x2 +5 = limx→∞ 3+(5/x2 ) = 3−0
Final exam - 3-11-2014
√ Question
√ 7
Evaluate limx→∞ 4x2 + 7x − 4x2 − 7x.
solution:
The limit can be solved using the so-called√
square root√
trick: √ √
√ √ (4x2 +7x− 4x2 −7x)∗( 4x2 +7x+ 4x2 −7x)
2 2
limx→∞ 4x + 7x− 4x − 7x = limx→∞ √ 2 √
2
( 4x +7x)− 4x −7x)
, 7
2 2
(4x +7x)−(4x −7x) 14x √ x 7√
= limx→∞ √ √ = limx→∞ √ 7 7
= √
|x| · ( 1+ 7 7
=
( 2 4x +7x)− 4x2 −7x) |2x|( 1+ 4x + 1− 4x ) 4x + 1+ 4x )
7
2.
Example 6: (A two-sided infinite limit) Describe the behaviour of the
function f (x) = x12 near x = 0.
solution
As x approaches 0 from either side, the values of f (x) are positive and grow
larger and larger, so the limit of f (x) as x approaches 0 does not exist. It is
nevertheless convenient to describe the behaviour of f near 0 by saying that
f (x) approaches ∞ as x approaches zero. We write
limx→0 (f (x) = limx→0 = ∞
Similar results are possible for one-sided limits.
1.4 Continuity
Most functions that we encounter have domains that are interval, or unions of
separate intervals. A point P in the domain of such a function is called an
interior point of the domain if it belongs to some open interval contained in
the domain. If it is not an interior point then P is called an endpoint of the
domain.
Definition 4:
Continuity at an interior point
We say that a function f is continuous at an interior point c of its domain if
lim f (x) = f (c) (10)
x→c
If either limx→c f (x) fails to exist or it exists but is not equal to f (c), then we
will say that f is discontinuous at c.
Definition 5:
Right and left continuity
We say that f is right continuous at c if limx→c+ f (x) = f (c).
We say that f is left continuous at c if limx→c− f (x) = f (c).
Theorem 5:
Function f is continuous at c if and only if it is both right continuous and left
continuous at c.
, 8
Figure 1: (a) f is continuous(b)limx→c f (x) 6= f (c) (c) limx→c f (x) does not
exist
Definition 6:
Continuity at an endpoint
We say that f is continuous at a left endpoint c of its domain if it is right
continuous there.
We say that f is continuous at a right endpoint c of its domain if it is left
continuous there.
Definition 7:
Continuity on an interval
We say that f is continuous on the interval I if it is continuous at each
point of I. In particular , we will say that f is a continuous function if f is
continuous at every point of its domain.
Theorem 8:
The Max-Min Theorem
If f (x) is continuous on the closed, finite interval [a, b], then there exist numbers
p and q in [a, b] such that for all x in [a, b],
f (p) ≤ f (x) ≤ f (q). (11)
Thus, f has the absolute minumum value m = f (p) taken on at the point p,
and the absolute maximum value M = f (q) taken on at the point q.
Theorem 9:
The Intermediate-Value Theorem
If f(x) is continuous on the interval [a, b] and if s is a number between f (a) and
f (b)m then there exists a number c in [a, b] such that f (c) = s.
In particular, a continuous function defined on a closed interval takes on all
values between its minimum value m and its maximum value M , so its range is
also a closed interval, [m, M ].
, 9
Figure 2: The continuous function f takes on the value s at some point c
between a and b
1.5 The Formal Definition of Limit (!)
Definition 8:
A formal definition of limit
We say that f (x) approaches the limit L as x approaches a, and we write
lim f (x) = L or lim = L, (12)
x→a x→a
if the following condition is satisfied: for every number > 0 there exists a
number δ > 0, possibly depending on , such that if 0 < |x − a| < δ, then x
belongs to the domain of f and
|f (x) − L| < (13)
Definition 9:
Right limits
We say that f (x) has right limit L at a, and we write
lim f (x) = L, (14)
x→a+
if the following condition is satisfied:
for every number > 0 there exists a number δ > 0, possibly depending on ,
such that if a < x < a + δ, then x belongs to the domain of f and
|f (x) − L| < (15)
The definition for left limit is formulated in a similar way.
Definition 10:
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