Table of Contents
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Precalculus: Algebra & Geometry (1-16)
View the graphs of 7 basic functions.
Learn to shift the graph of a function left or right and up or down.
Understand the reflection of a function through the x-axis, the y-axis and the origin.
Find the distance between two points in the plane.
Find the slope of the line through two points in the plane.
Practice solving equations involving the absolute value of a linear function.
Practice factoring quadratic polynomials.
Determine how the shape of the graph of a quadratic polynomial depends on the parameters when the square has been completed.
Determine how the shape of the graph of a quadratic polynomial depends on its coefficients (without completing the square).
Find the maximum or minimum of a quadratic polynomial by completing the square.
Find the vertical asymptotes of a rational function.
Identify the domain and range of a function.
Learn the graphs of exponential, logarithmic and trigonomic functions by shifting them left or right and up or down.
Calculate the exponential decay of a radioactive element.
Calculate the exponential growth of a bacteria culture.
Practice solving logarithmic equations and checking for extraneous roots.
Precalculus: Trigonometry & 2D Vectors (17-24)
Practice computing the third side of a right triangle using the Pythagorean Theorem.
Practice computing sin, cos, tan, cot, sec, and csc using the triangle definitions.
Practice computing sin, cos, tan, cot, sec, and csc using the circle definitions.
Practice identifying the graphs of shifted trigonometric functions.
Identify the mean value, amplitude, period, and phase shift of the graph of a sine or cosine curve.
Practice finding the sides and angles of a triangle using the Law of Sines for Angle-Side-Angle triangles.
Practice finding the sides and angles of a triangle using the Laws of Sines and Cosines for Side-Angle-Side triangles.
Practice the algebra of two-dimensional vectors, including addition, subtraction, scalar multiplication, linear combination, length and dot product.
Limits & Continuity (25-35)
Learn to identify left and right limits and the function value and their relation to two-sided limits, continuity from the left and right and continuity, given the graph of the function.
Learn to identify left and right limits and the function value and their relation to two-sided limits, continuity from the left and right and continuity, given a piecewise formula for the function.
Learn to identify left and right limits and the function value and their relation to two-sided limits, continuity from the left and right and continuity, given the numerical values of the function.
Visualize the precise definition of continuity by finding the delta for a given epsilon.
Practice proving that the limit of a linear function exists by using the precise definition of a limit.
Find the constant which makes a piecewise defined function continuous.
Use the Intermediate Value Theorem to find an interval on which a continuous function has a prescribed value.
Use the Bracketing Method to solve a continuous equation given the numerical values of the function.
Use the Bisection Method to solve a continuous equation given the numerical values of the function.
Use the Bracketing Method to solve a continuous equation given the formula for the function.
Use the Bisection Method to solve a continuous equation given the formula for the function.
Derivatives: Foundations (36-43)
Calculate the slopes of secant lines and use them to find the slope of the tangent line.
Use the slopes of secant lines to find the slope of the tangent line algebraically. Try some numeric points and then some symbolic points.
Find the equation of the line tangent to the graph of a function. Try some numeric points and then some symbolic points.
Practice computing derivatives with the chain rule.
Practice computing derivatives.
Find the derivative of an implicitly defined function.
Practice computing derivatives using logarithmic differentiation.
Learn how the graph of an inverse function is related to the graph of the function. Compute the derivatives of the function and its inverse function at corresponding points and then view the respective tangent lines.
Derivatives: Applications (44-62)
Practice finding the linear approximation to a function and using it to estimate the value of the function at a nearby point.
When two variables are related by an equation their derivatives are also related.
Solve a related rates problem involving a ladder sliding down a wall.
Given the position of an object, find its velocity and acceleration.
Practice using l'Hospital's rule for computing limits.
Find the parametric formula for the line tangent to a parametric curve.
Practice identifying the graphs of the first and second derivatives of a function given the formula and/or the plot of the function.
Learn the terminology to describe the properties of the graph of a function.
Learn to identify the properties of a function, its derivative, and its second derivative from the graph of the function.
Learn to identify the properties of a function, its derivative, and its second derivative from the graph of the first derivative.
Learn to identify the properties of a function, its derivative, and its second derivative from the graph of the second derivative.
Find the critical points of a function and classify them using the Second Derivative Test.
Minimize the surface area of a tin can with fixed volume.
Maximize the area of a field enclosed in a fence with several pens.
Minimize the surface area of a box with a square base and fixed volume.
Maximize the area of a rectangle inscribed in an ellipse or part of an ellipse.
Minimize the distance from a point to a line.
Minimize the time needed to row across a river and run downstream.
Practice finding the linear and quadratic approximations to a function and using them to estimate the value of the function at a nearby point.
Anti-Derivatives (63-67)
Practice finding anti-derivatives of simple functions.
Practice finding the anti-derivative that satisfies an initial condition.
Given the acceleration of an object and initial conditions, find its velocity and position.
Derive the Fundamental Theorem of Calculus by computing the derivative of the area under the graph of a function.
Using the fact that the area under the graph of a function is an anti-derivative of the function, set up an initial value problem for the area under a curve and solve it.
Integrals: Foundations (68-87)
Practice computing integrals by the method of substitution.
Practice computing integrals by integration by parts.
Practice computing integrals by integration by parts using tabular integration.
Practice computing integrals of products of powers of trigonometric functions.
Practice computing integrals by making trigonometric substitutions.
Practice finding general partial fraction decompositions.
Practice finding the coefficients in a partial fraction decomposition.
Practice evaluating the integrals that arise from a partial fraction decomposition.
Determine whether an improper integral converges or diverges; then compute its value.
Practice computing indefinite integrals.
Practice computing definite integrals.
Practice approximating definite integrals using left Riemann sums.
Practice approximating definite integrals using right Riemann sums.
Practice approximating definite integrals using the trapezoid rule.
Practice approximating definite integrals using Simpson's rule.
Find a bound on the error when approximating a definite integral using left Riemann sums.
Find a bound on the error when approximating a definite integral using right Riemann sums.
Find a bound on the error when approximating a definite integral using the trapezoid rule.
Find a bound on the error when approximating a definite integral using the midpoint Riemann sums.
Find a bound on the error when approximating a definite integral using Simpson's rule.
Integrals: Applications (88-100)
Visualize the slicing of a region in the plane and compute its area.
Find the average value of a function approximately and exactly.
Visualize the slicing of a solid and compute its volume.
Visualize a solid of revolution and compute its volume using disks, washers, or cylinders.
Visualize and compute the arc length of a curve in the plane.
Visualize a surface of revolution and compute the surface area.
Compute the center of mass of a bar with non-uniform density.
Compute the centroid of an region in the plane.
Compute the force and work required to stretch a spring.
Compute the force and work due to gravity.
Compute the force and work needed to lift a weight using a massive rope.
Compute the work needed to pump water out of a tank.
Compute the force on a plate submerged in a bucket of water.
Differential Equations (101-104)
Practice solving separable differential equations without and with initial conditions.
Practice solving linear differential equations without and with initial conditions.
Identify the direction field of a differential equation.
Set up and solve mixing problems using differential equations and inital conditions.
Sequences and Series (105-110)
Determine the convergence and limit of a sequence of numbers.
Practice computing the sum of a geometric series.
Practice computing the sum of a telescoping series with one or two term telescoping.
Identify and apply convergence and divergence tests for series.
Use the Maclaurin series of a function to compute its limit.
Use the Maclaurin polynomial approximation for the integrand to approximate an integral.
Geometry & 3D Vectors (111-116)
Use vectors and dot products to compute an angle of a triangle in three dimensional space.
Use vectors and cross products to compute the area of a triangle in three dimensional space.
Use vectors, dot products, and cross products to compute the angles and area of a triangle in three dimensional space.
Practice identifying a quadratic surface from its graph. Optional: Get out your 3D stereo glasses.
View the graphs of 14 basic polar curves.
Practice identifying a polar curve from its graph.
Curves & Line Integrals (117-118)
Find the arc length of a parametric curve in three dimensional space.
Find the work done by a force along a parametric curve.
Partial Derivatives (119-121)
Compute the first and second partial derivatives of a function of two variables.
Find the local maxima, local minima, and saddle points of a function of two variables.
Find the tangent plane to a surface defined as the level surface of a function.
Multiple Integrals (122-123)
Compute the center of mass of a solid given in rectangular coordinates.
Find the limits of integration on a triple integral in rectangular, cylindrical, or spherical coordinates.
Surfaces and Surface Integrals (124-124)
Compute the surface area of cones and paraboloids described parametrically.
Games (125-129)
Solve a cryptogram by replacing each letter of the alphabet by another letter to find the hidden quotation.
Guess points in the plane until you find the hidden zero. Once you discover the rule you should be able to find the zero with only four guesses.
Play a game of classic Tic Tac Toe against the computer or against a friend.
Find your way through a virtual reality maze by remembering your sequence of turns.
Play Four in a Row against the computer or against a friend. The winner is the first person to get four disks in a row -- horizontally, vertically, or diagonally.
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