The tutorials are divided into 2 separate
sections. The first set of tutorials is based on general mathematical
concepts. These tutorials will seem familiar to students since most of
the content is a review of calculus-related concepts from high schools
courses. A short description of each tutorial from the General
Mathematics section is listed below. To visit the tutorial, click the
link below each description or the link from the menu on the left-hand
side of the page.
The number sets tutorial defines and gives
examples of the different types of numbers that students may encounter
in the calculus course.
| NUMBER SETS Tutorial
Absolute Value & Inequalities
The absolute value and inequalities tutorial
illustrates how to solve inequalities and expressions that involve
absolute value symbols. The examples also show how to graph
> | ABSOLUTE VALUE &
Sets & Intervals
The sets and intervals tutorial contains
definitions, such as the union or intersection of sets, from Discrete
Mathematics. The rules for proper set notation are defined. The
tutorial also contains information about the different types of
intervals and how these intervals are expressed in set builder notation
and as segments of the real number line.
The fractions tutorial shows students how to
evaluate algebraic expressions that involve operations of fractions.
The tutorial also illustrates how to convert fractions to decimal form
and how to convert between improper fractions and mixed numbers. An
added feature is a greatest common denominator / least common multiple
calculator that will help students to reduce fractions.
| FRACTIONS Tutorial
The polynomials tutorial begins with a few
definitions of different terms. It also contains a section on
evaluating rational expressions. The most important part of this
tutorial is the section on factoring polynomial expressions. Students
will be required to factor expressions in all sections of the calculus
| POLYNOMIALS Tutorial
The linear equations tutorial defines the
different types of linear equations and illustrates how to find the
equation of a line using several different formulas. This will become
useful when students reach the tangent line section of the calculus
course. The tutorial wraps up with a few line theorems and section on
graphing linear equations.
| LINEAR EQUATIONS Tutorial
The quadratic equations tutorial begins with a
description of the method of completing the square and a few examples
of factoring. The most important part of this tutorial, however, is the
section on the quadratic formula. This formula will allow students to
find the roots of any quadratic equation, if they exist. The section
concludes with a section on graphing quadratic equations (i.e.
| QUADRATIC EQUATIONS Tutorial
The geometry tutorial features several different
formulas for the area and perimeter and surface area and volume of
common 2- and 3-dimensional figures, as well as several other well
known formulas from geometry, such as the Pythagorean Theorem. The
tutorial also contains information on the equations of circles and
other conics. This tutorial will be particularly important to students
during the related rates and optimization problems section of the
course. Many of the formulas from the geometry tutorial will be
required to solve the application problems.
| GEOMETRY Tutorial
The finite series tutorial contains the rules for
series notation and describes
some of the different types of sequences and series. It also contains
the different properties of finite series, as well as some theorems
that make the calculation of series much easier.
> | FINITE SERIES Tutorial
The trigonometry tutorial is one of the largest
of the general mathematics tutorials because it covers so much
information. The different trigonometric functions and their respective
graphs are covered, along with the periodicity of these functions. The
tutorial also contains sections covering the sine and cosine laws and
arc length. The tutorial concludes with a section covering
trigonometric identities and formulas.
It's important that students are familiar with
all of the content of this tutorial because almost all calculus
sections will include concepts from trigonometry in one way or another.
> | TRIGONOMETRY Tutorial
This very short tutorial features information
about exponents and how to simplify exponential expressions using the
laws of exponents.
| EXPONENTS Tutorial
The logarithms tutorial introduces the concept of
a logarithm and how to simplify logarithmic expressions using the laws
of logarithms. It also contains sections on the natural logarithm and
the change of base formula. Students will realize how closely related
the concepts of natural logarithms and calculus are once they reach the
inverse functions section of the calculus course.
| LOGARITHMS Tutorial
The induction tutorial contains guidelines for proving statements using mathematical induction. This is a very powerful method of proof. The only drawback is that proof by mathematical induction can only be used for statements that have number sets that increase or decrease by integer values.
| MATHEMATICAL INDUCTION Tutorial
The second set of tutorials is based on the
mathematics covered in the MATH 1036/1037 courses. Students may be
familiar with some of this material from their high school calculus
course. However, the content of these tutorials and the calculus course
are much more in-depth than the high school level course. A short
description of each of the Calculus tutorials is listed below. To visit
the tutorial, click the link below each description or the link from
the menu on the left-hand side of the page.
The limits tutorial begins with a definition of
limit using tangent and secant lines. It continues with sections on
left- and right-handed limits and infinite limits and vertical
asymptotes. Examples show how to calculate limits using the properties
of limits. Some new material is covered in the squeeze theorem and the
intermediate value theorem. An important section deals with continuous
and discontinuous functions and illustrates the different type of
discontinuity through graphs. The tutorial ends by revisiting the
concepts of tangents and limits and providing a new definition of a
| LIMITS tutorial
The derivatives tutorial defines a derivative
using the definition of a limit from the previous tutorial. It touches
on the different notations for derivatives and the concept of the
derivative as a function of x. A theorem for differentiable functions
is accompanied by graphs illustrating cases where functions are not
differentiable. One of the most important parts of the tutorial is the
section on differentiation, where the different formulas for
differentiation are listed. These formulas will allow students to
easily take the derivative of simple functions.
The tutorial also shows students how to take the
derivatives of trigonometric functions. Knowledge of trigonometry will
be especially helpful in this section. There are a few short, but
important sections that describe the method of implicit differentiation
and higher derivatives. Students will need to use these concepts in the
later parts of the course. The tutorial wraps up with a short section
on linear approximation.
| DERIVATIVES tutorial
Related Rates and Optimization
This tutorial lets students put their knowledge
of the previous calculus concepts to good use with practical
application problems. It begins with related rates and a few examples
of the different types of problems. Before tackling optimization
problems, the tutorial covers maximum and minimum values. This concept
is necessary for optimization problems, as well as drawing the graphs
of functions in the curve sketching tutorial.
> | RELATED
RATES & OPTIMIZATION tutorial
The curve sketching tutorial illustrates all of
the steps that students must take in order to sketch a given function.
Some of these steps include finding the intervals of increase/decrease
and concavity and the applying the first and second derivative tests to
find maximum and minimum values. The tutorial also covers vertical and
horizontal asymptotes, as well as the more complicated slant
asymptotes. It concludes with a couple examples of sketching curves
given only the equation of the function.
| CURVE SKETCHING tutorial
The integration tutorial shows students how to
solve definite and indefinite integrals using special properties and
formulas for integration. The tutorial also covers the Fundamental
Theorem of Calculus, whose name implies its importance to the calculus
course. Another important concept is the substitution rule, which
allows students to solve integrals of more complicated functions.
| INTEGRALS tutorial
Area and Volume
The area and volume tutorial shows some of the
applications of integral calculus. Although these problems are not as
practical as other applications of calculus, they show how the concepts
from the course can be applied to other areas of mathematics. After
completing this tutorial, students should be able to find the area
between two given curves and determine the volume of a solid of
rotation using one of the methods outlined in the tutorial.
& VOLUME tutorial
The inverse functions tutorial combines the
concepts of inverse functions, trigonometry, logarithms and calculus.
It begins with a thorough definition of inverse functions. The next
section illustrates the relation between the e and the natural
logarithm and calculus. The formulas for differentiation and
integration of logarithmic and exponential functions are given with
several examples illustrating the correct use of each formula. The
tutorial also covers the method of logarithmic differentiation, which
will allow students to take the derivative of functions that were
previously too complicated to differentiate.
The last section of the tutorial covers inverse
trigonometric functions. The graphs and formulas for differentiation
and integration of the three main trigonometric functions are provided,
along with a few other formulas. Several examples are included so
students can fully grasp the concepts of this tutorial.
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