Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 21
Page 1
MA110 Week 21 Trig Integrals; Trig Substitution
Concepts
Trigonometric Identities
The focus of the lab assignment this week will be to use known techniques of integration and apply
them in a particular manner to integrate expressions
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 17
Page 1
MA110 Week 17 Antiderivatives; Areas and the Denite
Integral
Antiderivatives
A function F is called an antiderivative of the continuous function f on interval I if F 0 (x) = f (x)
for all x 2 I.
For example, the general antiderivative
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 18
Page 1
MA110 Week 18 Denite and Indenite Integrals
Concepts
Antiderivatives
A function F is called an antiderivative of the continuous function f on interval I if F 0 (x) = f (x)
for all x 2 I.
For example, the general antiderivative of f (x
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 19
Page 1
MA110 Week 19 The Substitution Rule
Concepts
Antiderivatives
A function F is called an antiderivative of the continuous function f on interval I if F 0 (x) = f (x)
for all x 2 I.
For example, the general antiderivative of f (x) = 2x i
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 16
Page 1
MA110 Week 16 Optimization; Curve Sketching
Background Concepts
Properties of Functions
Domain: If the domain is not explicitly defined for a given function, it is assumed that the domain
consists of all real numbers for which the fun
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 13
Page 1
MA110 Week 13 cfw_ Approximation Techniques; Extrema
Di erentials
For y = f (x), let
x be a small nonzero change in x and
y the corresponding change in y
along the curve. Thus y = f (x + x) f (x). Let y 0 = f 0 (x) be the slope of th
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 14
Page 1
MA110 Week 14 cfw_ Existence Theorems; Shapes of Curves
Existence Theorems
Back in the Week 5 lab, we worked with our first existence theorem, the Intermediate Value Theorem.
Recall that the theorem gives us, as long as required condi
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 15
Page 1
MA110 Week 15 L
Hospital Rule
s
Concepts
Indeterminate Forms and L
Hospitals Rule
0
1
f (x)
and
, arising from lim
. L
Hospital Rule
s
x!a g (x)
0
1
f (x)
f 0 (x)
states that if a limit expression takes one of these two basic forms, t
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 12 cfw_ Derivatives of Logarithmic Functions;
Related Rates
Concepts
Calculus of Logarithmic Functions
1
d
[loga x] =
dx
x ln a
The di erentiation rule for logarithmic functions is as follows:
d
g 0 (x)
[loga g (x)] =
dx
g (x) ln a
d
1
d
g 0 (x
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 10 cfw_ Di erentiation Continued
Concepts
Limits Involving Trigonometric Functions
When determining limits of trig expressions, the following facts are very useful:
lim
sin
=1
!0
and
lim
cos
1
!0
=0
.
Note that consequently we also have:
lim
!0
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 11 cfw_ Di erentiation Continued
Concepts
Derivatives of Exponential Functions
d x
[e ] = ex
dx
d f (x)
e
= ef (x) f 0 (x)
dx
d x
[a ] = ax ln a
dx
d
af (x) = af (x) ln a f 0 (x) (Chain Rule)
dx
Derivatives of Inverse Trigonometric Functions
d
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 9 cfw_ Derivatives
Derivatives and Di erentiability
A secant is a line which joins two points along a curve. Thus, the slope of the secant joining the
points (a; f (a) and (a + h; f (a + h) along the curve defined by y = f (x) is given by
mseca
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 8 cfw_ Limits Continued
Continuity
A function f is said to be continuous at a number x = a if and only if lim f (x) = f (a). If
x!a
lim f (x) = f (a) [ lim+ f (x) = f (a) ], the function is said to be continuous from the left [ right ]
x!a
x!a
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 7
Page 1
MA110 Week 7 cfw_ Evaluating Limits
Background Concepts
Limits at a Number (Text: 2.2, 2.3)
To determine lim f (x) we are asking the question:
x!a
As x gets closer and closer to the finite
number a, which single value, if any, does f (
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 4 cfw_ Exponential and Trigonometric Functions
Background Concepts
Exponential Functions (Text: 1.5)
An exponential function is a function of the form f (x) = ax = expa (x), where a > 0, a 6= 1
is constant and is called the base, and x is a var
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 5 cfw_ Inverse Functions
Inverse Functions (Text: 1.6)
A function f is onetoone if, for all x1 , x2 in the domain of f , x1 6= x2 implies f (x1 ) 6= f (x2 ) ;
that is, no two di erent domain values can produce the same function value. A onet
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 20
Page 1
MA110 Week 20 Integration Techniques Continued
The Substitution Rule
The substitution rule allows some integrals that are not in a form yielding a known antiderivative to
R
R
be expressed in such a form so that they may be evaluated:
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 22
Page 1
MA110 Week 22 Partial Fraction Expansion
Concepts
Division of Polynomials
A rational expression for which the degree of the numerator is greater than or equal to the degree
of the denominator can be thought of as an improper fraction
Introduction to Differential and Integral Calculus
MA 110

Fall 2013
MA110 Week 24
Page 1
MA110 Week 24 Higher Order Partial Derivatives; Chain
Rule
Concepts
Partial Derivatives
Given a function of two variables, z = f (x; y), we find the partial derivative of f with respect to
@f
@z
x, denoted
=
= fx , by holding y consta
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
a Fall Tenn, 2014
6 O + o %$ p
Lecture Section: _ Lab Section:
Name:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 110* — Introduction to Differential and
Integral Calculus
PreCalculus Test — Version 1 — October 23rd, 2014
(ﬁlmsL l/erSM‘O‘I/t
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab Report 4  Inverse Functions
Name:
Student Number:
Lab
Fall 2014
1. [4 marks] Evaluate the following without using your calculator. Show all work and leave results as exact values.
1
1
2
(a) log5
= log5 1 log5 125 = 0 3 = 3
(b) e2 ln 5 = eln 5 =
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab Report 5  Limits and Limit Applications
Name:
Student Number:
Lab
Fall 2014
1. [7 marks] Consider the graph of f (x) given below.
(a) Evaluate each of the following. Each answer should be
a finite number, , , or d.n.e. (does not exist).
f (5) =
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA101/103/110  Maple Tutorial
Name:
Lab:
Fall 2014
This worksheet is designed to familiarize you with Maple and some of its common commands. These commands, as
well as many others (which will be shown to you in upcoming labs), will be used throughout the
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab Report 3  Transformation and Composition of Functions; Exponential
Functions
Name:
Student Number:
Lab
Fall 2014
1. [5 marks] Recall, from Question 1 on last weeks lab, the following functions and their domains:
1
f (x) = x2 16
g(x) =
5x
domf :
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab 1  PreCalculus Topics
Lab Objectives
The main aim of the first two labs is to review some basic precalculus material: functions and their properties,
simplifying expressions and solving equations, and working with absolute value and inequalit
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab 2  Functions; Trigonometry
Background Concepts
Interval Notation
When interval notation is used, we assume that the real numbers are being considered. A square [closed] bracket
indicates that the endpoint value is included whereas a round (open
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab Report 6  Limit Applications; Derivatives
Background Concepts
Limits at Infinity and Horizontal Asymptotes (Text: 2.6)
We can consider limits at infinity of a function f , denoted by
lim f (x) and
x
lim f (x).
x
If, as x becomes
arbitrarily lar
Introduction to Differential and Integral Calculus
MA 110

Fall 2014
MA110 Lab Report 3  Properties of Functions continued; Exponential
Functions
Background Concepts
Function Properties
1. Domain: If the domain is not explicitly defined for a given function, it is assumed that the domain consists
of all real numbers for w