Activities/Turtle Art/Tutorials/Measuring Slopes of tangents - Revision history

Rdrsadhu: Migrate to GitHub

2018-07-30T10:54:32Z

Migrate to GitHub

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-{{stub}}
+== Turtle Art/Tutorials/Measuring Slopes of tangents ==
-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/help-activity/blob/master/source/turtleart_tutorials/measuring_slopes_of_tangents.rst GitHub]

FGrose: moved Activities/TurtleArt/Tutorials/Measuring Slopes of tangents to Activities/Turtle Art/Tutorials/Measuring Slopes of tangents: deCamelCase

2012-07-23T15:37:26Z

moved Activities/TurtleArt/Tutorials/Measuring Slopes of tangents to Activities/Turtle Art/Tutorials/Measuring Slopes of tangents: deCamelCase

← Older revision	Revision as of 11:37, 23 July 2012
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Mokurai: /* Tangent Function */ Fix link

2011-08-07T23:12:54Z

Tangent Function: Fix link

← Older revision		Revision as of 19:12, 7 August 2011
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	This function of the angle is called the tangent function.		This function of the angle is called the tangent function.

	[[File:TATangentFunctionGraph		[[File:TATangentFunctionGraph]]

	==Slope==		==Slope==

Mokurai: /* Tangent Function */ Fix link

2011-08-07T23:12:30Z

Tangent Function: Fix link

← Older revision		Revision as of 19:12, 7 August 2011
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	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.		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		[[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.		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.

Mokurai: New page stub

2011-08-07T23:08:06Z

New page stub

New page

{{stub}}
==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.

[[File:TATangentAngle.png]]

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, single-valued functions, which cannot turn backwards, there is another measure that leads us to many useful consequences.

* 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 single-valued function intersects any vertical line in only one point.

[[File:SingleValuedFunction.png]]
[[File:MultipleValuedFunction.png]]

A continuous, multiple-valued function necessarily turns backward in some interval.

The tangent to a continuous, single-valued 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.

[[File:TATangentofAngle.png]]

This function of the angle is called the tangent function.

[[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.

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.