exploring the tangent line bob jantzen
Goal: Use the Explore command to plot the tangent line along the graph of a function over a specified interval. This is an example of a template file which only needs parameter editing to fit to the desired function and window. Right click menus can empower students to do some of these activities without knowing commands, but not the Explore command, which is more sophisticated.
We use numerical values for the horizontal window range to get a continous slider with the Explore command for the location of the tangent line, and we need to adjust the window and starting x value of the tangent line location to the function we are studying. Executing the worksheet then adjusts the example to the new data.
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We need to make a function of two variables for the variable tangent line function expression but we need to use unapply because of the derivative in the expression. Recall that the tangent line equation equates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== to the start value plus the slope times the increment in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== , namely 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 . The right hand side is the linear approximation to the function at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5 ., or the "tangent line function".
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SSJhRzYi #
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot([f(x), TL(x,a)],x = -3.14 .. 3.14,color = [black, red],view = -2 .. 2,gridlines = true), ':-Plot0', [':-a'], ':-sliders'=["Slidera"], ':-labels'=["Labela"],':-isplot'=true, ':-handlers'=[a=handlertab[a]]);
#
# End of autogenerated code.
#
We could easily extend this to a pair of movable tangent lines and then graphically determine where they intersect. Clearly we would have to adjust the plot window for intersections outside the current window. Or one could fix a target and see which tangent lines (nose gun of a jet trajectory in a screen game) hit the target. Clearly instead of testing how to write the equation of a single tangent line at a specific point (boring, and typical of calc 1 testing problems), by teaching how to represent the family of tangent lines, they can actually suggest interesting applications that the Explore command can help enliven. If a target were located at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtbkdGJDYlUSIzRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiMUYnRjRGN0Y0RjdGNEY3RjRGNw== , then these are the first two tangent lines in the first quadrant which hit the target (approximately 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 in the current configuration). A third tangent line from a point just after the second peak in the first quadrant would also to the job.
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SSJhRzYi #
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot([f(x), TL(x,a), TL(x,b)],x = -3.14 .. 3.14,color = [black, red, blue],view = -2 .. 2,gridlines = true), ':-Plot1', [':-a', ':-b'], ':-sliders'=["Slidera1", "Sliderb"], ':-labels'=["Labela1", "Labelb"],':-isplot'=true, ':-handlers'=[a=handlertab[a],b=handlertab[b]]);
#
# End of autogenerated code.
#
SSJiRzYi #
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot([f(x), TL(x,a), TL(x,b)],x = -3.14 .. 3.14,color = [black, red, blue],view = -2 .. 2,gridlines = true), ':-Plot1', [':-a', ':-b'], ':-sliders'=["Slidera1", "Sliderb"], ':-labels'=["Labela1", "Labelb"],':-isplot'=true, ':-handlers'=[a=handlertab[a],b=handlertab[b]]);
#
# End of autogenerated code.
#
One can easily use Maple to calculate (numerically in this case) where these tangent lines have to be located. For example, looking at the graph in its current configuration.
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