Substitution 2

Graphs a function f(x) and dynamically explores and the modifications made to areas associated with f (x) made by a substitution u=g(x)
How to use   ||   Examples   ||   Other Notes

How to use
• Enter the function f (x) in the text input field marked "f (x)=" (Example: f (x)=2x/(x2+1)).
• Click the "Graph" button (this button refreshes both graphs) -- at this point, the left graph will show the graph of f (x), and the right graph will remain empty.
• Enter the substitution function g(x) in the text input field marked "u=g(x)=", and its derivative g'(x) in the text input field marked "u'=g'(x)=". Then click the "Graph" button to refresh. Now the right graph will show the function transformed by the substitution. (Example: u=x2+1, u'=2x).
• Select a value for a -- this can be done either of two ways:
• Click and drag the red point in the left graph. The point moves along the x-axis, and its horizontal position gives the value for a.
• Enter a value directly into the text input field marked "a=".
(Example: a=1)
• Enter a value for the base length of the rectangle shown in the left graph in the text input field marked "base=". (Example: b=1)
• To erase the graph and all input fields, setting a and the base width to default values, click the "Clear" button
The text input fields for f (x), g(x), and g'(x) can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. The text input fields for a and the base length can accept real numbers in decimal notation.

For assistance computing the derivative g'(x), try the Derivative Calculator.

Examples

Other Notes
The graphs show a function f (x) and the modifications made to f (x) and areas by a substitution u=g(x). The graph on the left shows f (x), and the graph on the right shows the effect making the substitution u=g(x), so plotting the curve given by points of the form (g(x), f (x)/g'(x)).

The graph on the left also shows a green or orange rectangle with a red point at one corner. The height of the rectangle is the value of f (a), where a is the horizontal position of the red point. This value of f (a) is given in a label under the graph. The base width of the rectangle is given in the text input field marked "base=". If the area of the rectangle is positive, it is shaded in green; if negative, orange. The area of the rectangle is also shown in a label under the graph.

A corresponding green or orange rectangle and red point are drawn in the graph on the right. The horizontal position of the red point is given by g(a). The height of the rectangle is given by f (a)/g'(a) (the point on the left graph over the red point). The base width of the rectangle is b.g'(a), where b is the base width from the left graph. If the area of the rectangle is positive, it is shaded in green; if negative, orange. Labels under the right graph show g(a), the base width and height of the rectangle, and the area of the rectangle.

The areas of the rectangles in the two graphs are equal. With the two graphs giving different heights, and the base width in the left graph fixed by the text input field, the critical dimension is the base width in the right graph -- this is the effect of the substitution u=g(x).