This course covers the standard topics from an Algebra course. We start by covering exponents, radicals and factoring so we have the basics down. We will learn how to solve linear, quadratic equations and inequalities as well as some applications of these techniques. In addition, we will introduce function notation, domains and ranges of functions, and finding roots of functions. We also examine graphs of lines, circles, parabolas, ellipses, hyperbolas, and transformations. Finally, we will look at exponential functions, logarithmic functions, solutions to exponential and logarithmic functions, and solving linear systems of equations with two or three variables.
In this course we will look at various techniques for finding solutions to differential equations. We will spend most of our time solving homogeneous and nonhomogeneous first and second order differential equations and learn how to extend these solution methods to higher order differential equations. We will learn how to apply Laplace transforms to solve differential equations and we will learn how to solve systems of linear differential equations. We finish off the course with some topics that are typically not covered in a differential equations course but are included for your knowledge. These are series solutions, boundary value problems, Fourier series and some introductory topics in solving partial differential equations.
This is a full set of material for all the topics that are traditionally covered in the Calculus I, Calculus II, and Calculus III courses. Limits, derivatives, and integrals for single variable functions and multi-variable functions are all covered and examples are worked and the solution process examined in detail. Applications of limits, derivatives and integrals are all examined and examples worked. Major theorems such as Intermediate Value Theorem, Mean Value Theorem, Fundamental Theorem of Calculus, Green’s Theorem, Stoke’s Theorem, and Divergence Theorem are covered and discussed. Practice problems with full solutions are included to reinforce a students understanding of the concepts and topics covered.
Calculus I covers limits, derivatives and basic integration. Calculus II covers advanced integration techniques, parametric equations, polar coordinates, series, vectors and 3D space. Calculus III covers 3D space, partial derivatives, double and triple integrals, line integrals, and surface integrals. Of course, applications are also presented where applicable.
Select the book to see a full listing of topics covered.
Below are summaries of the options available. Select an option on the left to access it.
Algebra Trig Review: This is a series of problems, with solutions, in selected topics in Algebra and Trigonometry that are vital for being successful in a Calculus course. The material presented here is intended to be a refresher for these topics and so does not go into great detail in the solutions outside of what is needed to remind you of the topics.
Common Math Errors: This is a series of errors and misconceptions that students in a math course will often run into. In most of the sections, the errors/misconceptions are presented in the incorrect form and the correct from. There will also be discussion about the error or misconception in most cases and what to watch out for so you don’t make the error/misconception. Most of the sections are accessible to anyone with some basic Algebra background. There are, however, some errors involving Trigonometry and Calculus.
Complex Number Primer: Many students never really see complex numbers and then are, all of a sudden, expected to know them. This quick primer is intended to introduce you to the basics of complex numbers as well as arithmetic involving complex numbers and some of the more common operations involving complex numbers. Hopefully, if you are in the situation of all of a sudden needing to know about complex numbers, this primer will give you all the information you need.
How To Study Math: This is a list of general tips on how to study and be successful in a math course. They have been broken down into distinct topics (taking notes, homework, exams, etc.). Every person is different so you may not find all of them useful but hopefully you can use this as a guide to determine what works for you in being successful in your math course.
Cheat Sheets and Tables: A series of cheat sheets and tables for quick reference. Topics include Algebra, Trigonometry, and Calculus. There are also tables for common derivatives and integrals as well as a table of Laplace transforms.
This is a series of problems, with solutions, in selected topics in Algebra and Trigonometry that are vital for being successful in a Calculus course. The material presented here is intended to be a refresher for these topics and so does not go into great detail in the solutions outside of what is needed to remind you of the topics.
Find two positive numbers whose sum of six times one of them and the second is 250 and whose product is a maximum.
Find two positive numbers whose sum of twice the first and seven times the second is 600 and whose product is a maximum.
Let \(x\) and \(y\) be two positive numbers whose sum is 175 and \(\left( {x + 3} \right)\left( {y + 4} \right)\) is a maximum. Determine \(x\) and \(y\).
Find two positive numbers such that the sum of the one and the square of the other is 200 and whose product is a maximum.
Find two positive numbers whose product is 400 and such that the sum of twice the first and three times the second is a minimum.
Find two positive numbers whose product is 250 and such that the sum of the first and four times the second is a minimum.
Let \(x\) and \(y\) be two positive numbers such that \(y\left( {x + 2} \right) = 100\) and whose sum is a minimum. Determine \(x\) and \(y\).
Find a positive number such that the sum of the number and its reciprocal is a minimum.
We are going to fence in a rectangular field and have 200 feet of material to construct the fence. Determine the dimensions of the field that will enclose the maximum area.
We are going to fence in a rectangular field. Starting at the bottom of the field and moving around the field in a counter clockwise manner the cost of material for each side is $6/ft, $9/ft, $12/ft and $14/ft respectively. If we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area.
We are going to fence in a rectangular field that encloses 75 ft2. Determine the dimensions of the field that will require the least amount of fencing material to be used.
We are going to fence in a rectangular field that encloses 200 m2. If the cost of the material for of one pair of parallel sides is $3/m and cost of the material for the other pair of parallel sides is $8/m determine the dimensions of the field that will minimize the cost to build the fence around the field.
Show that a rectangle with a fixed area and minimum perimeter is a square.
Show that a rectangle with a fixed perimeter and a maximum area is a square.
We have 350 m2 of material to build a box whose base width is four times the base length. Determine the dimensions of the box that will maximize the enclosed volume.
We have $1000 to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides cost $10/cm2 and the material for the bottom cost $15/cm2 determine the dimensions of the box that will maximize the enclosed volume.
We want to build a box whose base length is twice the base width and the box will enclose 80 ft3. The cost of the material of the sides is $0.5/ft2 and the cost of the top/bottom is $3/ft2. Determine the dimensions of the box that will minimize the cost.
We want to build a box whose base is a square, has no top and will enclose 100 m3. Determine the dimensions of the box that will minimize the amount of material needed to construct the box.
We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 in3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can.
We want to construct a cylindrical can whose volume is 105 mm3. The material for the wall of the can costs $3/mm2, the material for the bottom of the can costs $7/mm2 and the material for the top of the can costs $2/mm2. Determine the dimensions of the can that will minimize the cost of the materials needed to construct the can.
We have a piece of cardboard that is 30 cm by 16 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
We have a piece of cardboard that is 5 in by 20 in and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
A printer needs to make a poster that will have a total of 500 cm2 that will have 3 cm margins on the sides and 2 cm margins on the top and bottom. What dimensions of the poster will give the largest printed area?
A printer needs to make a poster that will have a total of 125 in2 that will have ½ inch margin on the bottom, 1 inch margin on the right, 2 inch margin on the left and 4 inch margin on the top. What dimensions of the poster will give the largest printed area?
