Posts Tagged ‘maximum’
Tuesday, November 10th, 2009
How to Make a Box-and-Whisker Plot
Description
A detailed tutorial on how to make a box-and-whisker plot. Step by step tutorial including several examples of how to make a box-and-whisker plot for reference.
Overview
A box-and-whisker plot is named for it’s resemblance to a cat’s face – the box is the face of the cat, and the lines extending out from either side are known as whiskers. Sometimes box-and-whisker plots are simply called box plots. They are used to graph sets of numbers according to five values: the highest value, known as the maximum, the second highest value, known as the upper quartile, the median, or the middle, the second lowest value, known as the lower quartile, and the lowest value, known as the minimum. The box centers around the median and the whiskers extend out to the other numbers.
Tags: algebra, box, box-and-whisker, boxplot, diagram, graph, highest, line, lower, lowest, maximum, median, middle, minimum, plot, quartile, upper, value, whisker
Posted in Algebra | No Comments »
Tuesday, October 20th, 2009
How to Graph the Cosecant Function
Description
A detailed tutorial on solving the graph of the cosecant function. Step by step tutorial including several examples of how to solve the graph of the cosecant function for reference.
Overview
The graph of cosecant is very closely related to the graph of secant. The graph appears to be many concave up and concave down curves placed in periods of 2pi. In reality, the local maximums and minimums on the graph of cosecant match up with the local maximums and minimums on the graph of sine, making it easy to line them up together. This is because sine and cosecant are the opposite of each other – sine is equal to one over cosecant.
Tags: amplitude, asymptote, cosecant, function, graph, intervals, maximum, minimum, period, pi, secant, sine, trigonometric, trigonometry, x, y
Posted in Trigonometry | No Comments »
Tuesday, October 20th, 2009
How to Graph the Secant Function
Description
A detailed tutorial on solving the graph of the secant function. Step by step tutorial including several examples of how to solve the graph of the secant function for reference.
Overview
The graph of secant is very closely related to the graph of cosecant. The graph appears to be many concave up and concave down curves placed in periods of 2pi. In reality, the local maximums and minimums on the graph of secant match up with the local maximums and minimums on the graph of cosine, making it easy to line them up together. This is because cosine and secant are the opposite of each other - cosine is equal to one over secant.
Tags: amplitude, asymptote, cosecant, cosine, function, graph, intervals, maximum, minimum, period, pi, secant, trigonometric, trigonometry, x, y
Posted in Trigonometry | No Comments »
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Boundedness Theorem Explained
Description
A detailed tutorial of the boundedness theorem. Step by step tutorial including an explanation of the boundedness theorem for reference. Knowledge of the boundedness theorem is required in calculus.
Overview
The boundedness theorem is a theorem that is very closely linked to the extreme value theorem. The boundedness theorem states that a continuous function f in the closed interval [a, b] is bounded on that interval. In mathematical terms, this means that there exist real numbers m and M such that ![m \le f(x) \le M\quad\text{for all }x \in [a,b].\,](http://s.wordpress.com/latex.php?latex=m%20%5Cle%20f%28x%29%20%5Cle%20M%5Cquad%5Ctext%7Bfor%20all%20%7Dx%20%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "m \le f(x) \le M\quad\text{for all }x \in [a,b].\,")
This translates to mean “m is less than or equal to f(x) which is less than or equal to M for all x belonging to [a, b]“.
Tags: a, b, bounded, boundedness theorem, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, m, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Extreme Value Theorem Explained
Description
A detailed tutorial of the extreme value theorem. Step by step tutorial including an explanation of the extreme value theorem for reference. Knowledge of the extreme value theorem is required in calculus.
Overview
The extreme value theorem states that if a real valued function f is continuous in the closed and bounded interval [a, b], then f must attain its maximum and minimum value at least once. In mathematical terms, this means that there exist numbers c and d in [a, b] such that ![f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,](http://s.wordpress.com/latex.php?latex=f%28c%29%20%5Cge%20f%28x%29%20%5Cge%20f%28d%29%5Cquad%5Ctext%7Bfor%20all%20%7Dx%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,")
The translation of that formula is “f(c) is greater than or equal to f(x) which is greater than or equal to f(d), for all x belonging to [a, b]“. In order for something to belong to an interval, it must be found in the interval.
Tags: a, b, bounded, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
