Archive for the ‘Calculus’ Category
Friday, October 9th, 2009
Explanation of Leibniz Notation
Description
A detailed tutorial on Leibniz notation. Step by step tutorial including several examples of Leibniz notation for reference. Knowledge of Leibniz notation is mandatory for calculus.
Overview
Leibniz notation is a common notation in calculus that helps to identify derivaties. In Leibniz notation, the terms dx and dy are used for derivatives of x and y. This can be used with any variable. Typically this will be expressed in a fraction form, as dy / dx. This form says that you take the derivative of x in respect to y. This notation can be used for integrals as well as derivatives, although it was first developed for use with derivatives.
Tags: anti-derivative, Calculus, change, derivative, dx, dy, function, Gottfried Wilhelm Leibniz, infinitely small, integral, Leibniz, Math, notation, with respect to
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Thursday, October 8th, 2009
Introduction to Inflection Points
Description
A detailed tutorial on inflection points. Step by step tutorial including several examples of inflection points and how to locate inflection points for reference.
Overview
An inflection point, sometimes also known as a point of inflection, is a point on the graph of a function at which the function changes sign. This means that a concave up curve will become a concave down curve, or a concave down curve will become a concave up curve. Inflection points are also points of local maxima and local minima of a function. There are two ways to categorize inflection points. There are stationary points of inflection, and non-stationary points of inflection. Stationary points are formed when the function is zero, and non-stationary points are when the function is not zero.
Tags: Calculus, concave, curve, down, function, inflection, inflexion, local, Math, maxima, minima, non-stationary, point, saddle-point, sign, stationary, up
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Thursday, October 8th, 2009
How to Use the Second Derivative Test
Description
A detailed tutorial on how to use the second derivative test. Step by step tutorial including several examples of how to use the second derivative test for reference.
Overview
The second derivative test is more well-known than the first derivative test, and is often thought to be more accurate. The second derivative test states that if the second derivative of a function is less than zero, then there is a local maximum at x. If the second derivative of a function is greater than zero, then there is a local minimum at x. However, if the second derivative of a function is equal to zero, then the local maximum or minimum cannot be determined. Then you must use the first derivative test to figure it out. The second derivative test can also be used to figure out the concavity of a function – that is, if a curve is pointing up or down. This is normally used to help create the image of the function on a graph.
Tags: Calculus, chart, concavity, critical points, curve, derivative, equals, extrema, extremum, first derivative test, function, graph, Math, maxima, maximum, minima, minimum, negative, positive, second derivative test
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Thursday, October 8th, 2009
How to Use the First Derivative Test
Description
A detailed tutorial on how to use the first derivative test. Step by step tutorial including several examples of how to use the first derivative test for reference.
Overview
The first derivative test involves taking the derivative of a function that you would like to find the local maximum or minimum of. Once you have the derivative, you must determine if the function is increasing or decreasing. If the derivative is positive, the function is increasing, and when the derivative is negative, the function is decreasing. If the derivative cannot be determined as positive or negative, then the test fails.
Tags: Calculus, chart, critical points, decreasing, derivative, extrema, extremum, first derivative test, function, graph, increasing, Math, maxima, maximum, minima, minimum, negative, positive, second derivative test
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Thursday, October 8th, 2009
The Local Maximum and Minimum of a Function
Description
A detailed tutorial on finding the local maximum and minimum of a function. Step by step tutorial including several examples of finding the local maximum and minumum of a function for reference.
Overview
The local maximum of a function is the largest value that a function can be. The local minimum of a function is the smallest value that a function can be. When given a graph, it is easy to point out local maxima or minima – what is the highest point on the graph you see? What is the lowest? Functions can have more than one local maximum or minimum. The local maxima and minima can be found by using the first or the second derivative test, if they are to be found locally. If they are to be found globally, a method of optimization must be used.
Tags: Calculus, critical points, extrema, extreme value theorem, extremum, Fermat's theorem, first derivative test, function, globalm local, graph, Math, maxima, maximum, minima, minimum, optimization, second derivative test
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Tuesday, October 6th, 2009
How to Test for Convergence Using the Alternating Series Test
Description
A detailed tutorial on testing for convergence using the alternating series test. Step by step tutorial including several examples of testing for convergence using the alternating series test for reference.
Overview
The alternating series test, like all convergence and divergence tests, is fairly easy. The hardest part is figuring out if you should use the AST, or a different test. An easy way to tell is, is the equation negative? What would happen if you pulled a negative one out? Or maybe, there is already a negative one outside of the equation. If you see any fraction, function, or any equation at all with a -1 to an odd power at the front (or at the front of the numerator, in a fraction) then you should use the alternating series test for it. If the series is decreasing over time, and the limit is approaching zero, then the series is convergent. The alternating series test is normally used in conjunction with another test for convergence.
Tags: -1, alternating, AST, Calculus, converge, convergence, decreasing, diverge, divergence, fraction, function, limit, Math, negative, one, series, test, zero
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Tuesday, October 6th, 2009
How to Test for Convergence Using the Geometric Series Test
Description
A detailed tutorial on how to test for convergence using the geometric series test. Step by step tutorial including several examples of testing for convergence using the geometric series test for reference.
Overview
A geometric series is a series that maintains a constant ratio between a set of terms. This series is an addition series, and would be expressed as 1/a + 1/2a + 1/4a, extending as far as you wish in either direction. If a series does not have that constant ratio, then it is not a geometric series. The series should converge at one, because as all the numbers are added they get closer and closer to one. The first term of a geometric series is given by a, and the ratio of a geometric series is given by r. If the ratio is less than one, then the geometric series converges to a / (1 – r). If the ratio is greater than or equal to one, then the series diverges. Usually the series will converge, which is why this is considered a test for convergence and not for divergence.
Tags: a, addition, Calculus, converge, convergence, diverge, divergence, equal to, first term, geometric, greater than, less than, Math, notation, r, ratio, series, summation, test
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Tuesday, October 6th, 2009
Plotting Points in the Polar Coordinate System
Description
A detailed tutorial on plotting points in the polar coordinate system. Step by step tutorial including several examples of how to plot points on the polar coordinate system for reference.
Overview
By this point, everyone should know how to plot points on a normal graph. But what about a circular graph? This circular graph is called the polar coordinate system or the polar plane. Instead of using the points (x, y), the polar coordinate system uses the points (r, theta). Theta is a greek letter that looks like a zero with a horizontal line drawn through the center. Most of the points you will be finding for the polar coordinate system will be used with trigonometric functions – sine, cosine, and tangent. Graphing occurs in about the same way as it would on a normal graph – just match up the points, even if they are on a circle.
Tags: Calculus, circle, coordinate, cosine, function, functions, graph, Math, points, polar, r, sine, system, tangent, theta, trig, trigonometric, x, y
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Friday, October 2nd, 2009
How to Find the Reference Angle
Description
A detailed tutorial on finding the reference angle. Step by step tutorial with several examples of how to find the reference angle for reference.
Overview
The reference angle is something you run into in precalculus and calculus. The reference angle is only used when working with radian measure, which while being more precise than degree notation, can sometimes be difficult to figure out and out into something you can use when solving an equation. The reference angle uses the unit circle, which has four points of 0, pi/2, pi, 3pi/2, and 2pi. When calculating an angle that is not exact, you place it on your unti circle and find the closest of those points. Subtract them. This is your reference angle.
Tags: Calculus, degrees, Math, pi, radians, reference, reference angle, subtract, unit circle
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Friday, October 2nd, 2009
Introduction to Simple Harmonic Motion
Description
A detailed tutorial on simple harmonic motion. Step by step tutorial including several examples of simple harmonic motion for reference.
Overview
Simple harmonic motion is related to Hooke’s law – as what Hooke’s law does is measure harmonic motion. Simple harmonic motion is motion that is neither driven nor damped. It is one movement, one force. This motion can also be periodic. Simple harmonic motion is expressed by the equation:
Tags: amplitude, Calculus, constant, force, frequency, harmonic, Hooke's law, Math, motion, oscillator, period, periodic, phase, simple
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