Difference between revisions of "Slope"
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| − | + | '''Slope''' is the steepness of a [[line]]. A positive slope rises; a negative slope falls. Another term for slope is gradient. | |
| − | Slope is the steepness of a line. A positive slope rises; a negative slope falls. Another term for slope is gradient. | ||
For a straight line, the slope (m) is constant and represented by the difference in the vertical direction (y) divided by the difference in the horizontal direction (x): | For a straight line, the slope (m) is constant and represented by the difference in the vertical direction (y) divided by the difference in the horizontal direction (x): | ||
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:: <math>m = \frac{\Delta y}{\Delta x}</math> | :: <math>m = \frac{\Delta y}{\Delta x}</math> | ||
| − | The delta, Δ, represents the difference in values between any two points in the x or y direction for straight line. For a curve, the delta, Δ, represents the difference in values for two points in very close proximity to each other. | + | The delta, Δ, represents the difference in values between any two points in the x or y direction for straight line. For a [[curve]], the delta, Δ, represents the difference in values for two points in very close proximity to each other. |
For a straight line, another way of representing the slope (m) is as follows: | For a straight line, another way of representing the slope (m) is as follows: | ||
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where the two points are located at (x<sub>1</sub>, y<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>). | where the two points are located at (x<sub>1</sub>, y<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>). | ||
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| + | ==Introduction to derivative== | ||
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| + | If a function has value <math>f(x)</math> at <math>x</math> and <math>f(x+h)</math> at <math>x+h</math> with <math>h>0</math> than the slope of the line joining <math>(x,f(x))</math> to <math>(x+h,f(x+h))</math> is, | ||
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| + | :<math>\frac{f(x+h)-f(x)}{h}</math>. | ||
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| + | The slope of the line that meets (is tangential to) <math>f(x)</math> at <math>x</math> is the limit as <math>h</math> tends to zero, or, | ||
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| + | :<math>\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h},</math> | ||
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| + | which is denoted <math>f'(x)</math>, which is called the [[Derivative (calculus)|derivative]] of <math>f(x)</math>. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Latest revision as of 21:08, September 8, 2020
Slope is the steepness of a line. A positive slope rises; a negative slope falls. Another term for slope is gradient.
For a straight line, the slope (m) is constant and represented by the difference in the vertical direction (y) divided by the difference in the horizontal direction (x):
The delta, Δ, represents the difference in values between any two points in the x or y direction for straight line. For a curve, the delta, Δ, represents the difference in values for two points in very close proximity to each other.
For a straight line, another way of representing the slope (m) is as follows:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}}
where the two points are located at (x1, y1) and (x2, y2).
Introduction to derivative
If a function has value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} at and at with than the slope of the line joining to is,
- .
The slope of the line that meets (is tangential to) at is the limit as tends to zero, or,
which is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)} , which is called the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} .