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General Information
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For Students
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For Instructors
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Miscellaneous
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Algebra
- Fundamental Manipulations
- Factoring
- Solving Equations
- Inequalities
Functions
- Notations
- Operations on Functions
- Applet 7 (Graphs of f+g, f-g, f*g, and f/g)
- Symmetry
- Applet 8 (Odd and Even Functions) : Test if a function is odd or even by looking at the graphs of f(-x) and -f(-x)
- New Functions from Old
- Applet 9 (Stretching and Compression of a Function) : Look at the graph of A*f(B*x) and observe the effects of changing A and B.
- Applet 10 (Translation of a Function) : Look at the graph of A+f(B+x) and observe the effects of changing A and B.
- Applet 11 (New Functions from Old) : Make new functions from old ones by reflecting, translating and stretching.
- Applet 12 (New Functions from Old Game) : For a function out of a list, there is a proposed target function that can be obtained by performing a sequence of transformations on the given function. In each example you have to find such a sequence of transformations that make the graphs coincide.
- Symmetry
- New Functions from Old
- Self Test 2 (Quiz on Transformation of Functions) : Determine the relationship between two graphs where one is f(x) and the other is A*f(x) or f(A*x), or f(x)+B, or f(x+B) etc. Comes with instant checking.
- Applet 13 (Function Composition)
- Trigonometric Functions
- Applet 14 (Defining sin(x) and cos(x) on a circle)
- Trigonometric Identities
- Applet 15 (Trigonometric Identities) : Choose a target function, and then choose a basic function that you believe can be transformed into the target through a series of transformations -- stretching, shifting, or reflections.
- Exponents
- Logarithms
- Applet 17 (Introduction to Log Functions) : A step by step introduction to log functions with interactive graphs.
- Applet 18 (Logarithms)
- Inverse of Functions
- Intersection of Two Functions
- Overview
- Least Squares Approximation
- Applet 23 (Least Squares Approximation) : This applet explores fitting a polynomial p(x) of degree n to a given set of data points. It computes the best least squares approximation to the data, "best" in the sense that SUM (p(xi) - yi)^2 is minimized.
- Word Problems
Limits
- Limit Laws
- Continuity
- Self Test 4 (Quiz I on Continuous Functions) : Use the definition of continuity to determine if a function is continuous or not. Comes with instant checking.
- Self Test 5 (Quiz on Properties of Continuous Functions)
Derivatives
- Definition of Derivatives
- Trigonometric Derivatives
- Velocity
- Applet 26 (Average Velocity) : Galileo experimented with falling objects. By meas-uring the distance fallen as a function of time he was able to conclude that the velocity is a linear function and that the acceleration is constant. This applet mir-rors his experiments.
- Curve Sketching
- Applet 27 (Behaviour of f, f' and f'' at given points) : Viewing a graph, determine information about a function and its first and second derivative.
- Self Test 9 (Quiz I on the behaviour of f, f' and f'' at given points) : This is a quiz to test your ability to use concepts of differentiablity to infer information about the graph of a function. Comes with instant checking.
- Self Test 10 (Quiz II on the behaviour of f, f' and f'' at given points) : You are given the graphs of f, f' and f'' and you have to determine which is which. Comes with instant checking.
Applications of Derivatives
- Tangent and Secant Lines
- Mean Value Theorem
- Linear Approximations
- Applet 30 (Linear Approximation using the First Derivative)
- Newtons Method
Integration
- Riemann Sums
- The Definite Integral
- Area Between Curves
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Inproper Integrals
Applications of Integration
- Area Between Curves
- Applet 37 (Graph and Calculation of the Area between two curves)
- Working Example 2 (Area Between 2 Curves with Visualization) : Integrals are used to find the area between two curves. An integral formula is developed and its applicability is discussed in a variety of examples.
- Applet 38 (Area Between two Curves)
- Arc Length
- Working Example 3 (Visualization of the Arclength Integral)
- Applet 39 (Numerical Approximation + Visualization) : Another illustration of the Riemann Sum modeling method is given, this time to compute the length of a curve in the plane. An integral formula is developed to compute the arc length. It is pointed out that the formula often leads to integrals that must be approximated by numerical methods.
- Applet 40 (Arc Length)
- Volumes of Solids of Revolution
- Applet 41 (A Riemann Sum for a Solid of Revolution) : Integrals find application in many modeling situations involving continuous variables such as area. They allow us to model physical entities that can be described through a process of adding up, or accumulating, smaller infinitesimal parts. In this section, the Riemann Sum approach is used to develop an integral formula for the volume of a solid of revolution.
- Surfaces of Revolution
- Numerical Approximations
- Applet 43 (Trapezoid Rule Visualization) : Many applications of calculus involve definite integrals. If it is possible to find an antiderivative for the integrand, then the integral can be evaluated using the Fundamental Theorem. When an antiderivative is not apparent, numerical (approximate) methods are invoked. The numerical method that is discussed in this section is called the Trapezoid Rule.
- Work
- Expected Value
Sequences and Series
- Sequences
- Convergence Criteria
- Alternating Series
- Power Series
- Taylor Series
Parametric Equations
- Parametric Equations
- Calculus with Parametric Curves
- Area in Polar Coordinates
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