Difference between revisions of "Slope"
m (new link) |
|||
| (9 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
| − | 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): | ||
| Line 5: | Line 5: | ||
:: <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: | ||
| Line 12: | Line 12: | ||
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>). | ||
| + | |||
| + | ==Introduction to derivative== | ||
| + | |||
| + | 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, | ||
| + | |||
| + | :<math>\frac{f(x+h)-f(x)}{h}</math>. | ||
| + | |||
| + | 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, | ||
| + | |||
| + | :<math>\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h},</math> | ||
| + | |||
| + | which is denoted <math>f'(x)</math>, which is called the [[Derivative (calculus)|derivative]] of <math>f(x)</math>. | ||
| + | |||
| + | [[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):
- <math>m = \frac{\Delta y}{\Delta x}</math>
The delta, Î, represents the difference in values between any two points in the x or y direction for straight line. For a curve, the delta, Î, 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:
- <math>m = \frac{y_2 - y_1}{x_2 - x_1}</math>
where the two points are located at (x1, y1) and (x2, y2).
Introduction to derivative
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,
- <math>\frac{f(x+h)-f(x)}{h}</math>.
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,
- <math>\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h},</math>
which is denoted <math>f'(x)</math>, which is called the derivative of <math>f(x)</math>.