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−  {{stub}}
 +  == Turtle Art/Tutorials/Measuring Slopes of tangents == 
−  ==Tangents to curves==  
   
−  The most obvious measure of direction is an angle. This program can draw a line and a circle that are tangent, at any angle. You can see the angle, in degrees, in the program.
 +  Read at https://help.sugarlabs.org/turtleart_tutorials/measuring_slopes_of_tangents.html 
   
−  [[File:TATangentAngle.png]]
 +  The source file has been moved to [https://github.com/godiard/helpactivity/blob/master/source/turtleart_tutorials/measuring_slopes_of_tangents.rst GitHub] 
−   
−  For general curves that can go in any direction, such as a circle or any other loop, this is often the most appropriate measure.
 
−   
−  [[File:TATangentAngles.png]]
 
−   
−  ==Tangents to functions==
 
−   
−  But for continuous, singlevalued functions, which cannot turn backwards, there is another measure that leads us to many useful consequences.
 
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−  * A continuous function takes no jumps. That means that for small changes in the x direction, there cannot be a much larger change in the y direction.
 
−   
−  [[File:TAContinuousFunction.png]]
 
−  [[File:TADiscontinuousFunction.png]]
 
−   
−  There is no tangent at a point of discontinuity.
 
−   
−  * A singlevalued function intersects any vertical line in only one point.
 
−   
−  [[File:SingleValuedFunction.png]]
 
−  [[File:MultipleValuedFunction.png]]
 
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−  A continuous, multiplevalued function necessarily turns backward in some interval.
 
−   
−  The tangent to a continuous, singlevalued function has an angle between 90 degrees and 90 degrees.  
−   
−  ==Tangent Function==
 
−   
−  If we take any angle inside the range [90,90], and take the x and y coordinates of a point on one side, we can build a right triangle with that angle and with sides equal to the coordinates, thus.
 
−   
−  [[File:TACoordinateTriangle.png]]
 
−   
−  If we make the lower side of a fixed length, and take that length as the unit of measure, the length of the other side is a function of the angle. In the following diagram, we use a radius of a triangle as the fixed side. The other side in this triangle is tangent to the circle.
 
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−  [[File:TATangentofAngle.png]]
 
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−  This function of the angle is called the tangent function.
 
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−  [[File:TATangentFunctionGraph]]
 
−   
−  ==Slope==
 
−   
−  The tangent function gives us a convenient measure of the direction of a tangent line, one that will lead us into calculus. This is often called the slope of the function. We have seen the tangent function defined using a fixed radius, but it works for any x value. Going back to our diagram of an angle, a point and its coordinates, and the triangle built from those coordinates, the slope, or the tangent of the angle, is the y coordinate divided by the x coordinate, y/x. Since y is how much the line rises when we run from 0 to x, the tangent can be described as rise/run (rise over run). Some students find this a helpful mnemonic, while others find it completely confusing.
 
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−  The next steps in the use of slopes are the derivative function, the definite integral, and the Fundamental Theorem of the Calculus, which will be the subjects of further tutorials.
 