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Line 1: − :This article deals with the simplified concept of gradient of a straight line+ − :For the advanced vector field concept See: [[Gradient]] − − {{math-m}} − − In early [[mathematics]], a '''gradient''' or '''slope''' is the increase of a straight line joining two points. − − In schooling, usually early high school, students are taught that the gradient is equal to "the rise over run" or more formally, − − :<math>\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}</math>, − − the rise being defined as the difference between the highest point and the lowest point, negative if the highest is on the left and positive if the highest point in on the right. The run is defined as difference of the right <math>x</math>-value and the left <math>x</math>-value (see [[Cartesian coordinates]]). − − If the Cartesian coordinates of two points <math>(x_{1},y_{1})</math> and <math>(x_{2},y_{2})</math>, with <math>x_{2}>x_{1}</math> then the gradient of the line joining them <math>m</math> is, − − :<math>m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}</math>. − − This concept is usually first introduced with the introduction of [[linear equation]]s. The equation of a straight line in Cartesian coordinates is given by, − − :<math>y=mx+c</math> − − where <math>m</math> is the gradient of the line and <math>c</math> is the value of the <math>y</math> coordinate when <math>x=0</math>, this is called the [[y-intercept]], e.g, where the line intercepts the [[y-axis]]. − − ===Introduction to derivative=== − − {{math-h}} − 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 gradient 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>. − − Therefor the gradient of the line that meet (is tangential to) <math>f(x)</math> at <math>x</math> is the limit as <math>h</math> tend 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>. − − ==References== − # [http://www.teacherschoice.com.au/Maths_Library/Gradient/gradient_-_two_fixed_points.htm Theachers' choice - gradient] − − [[Category:mathematics]]
Redirecting to Gradient (two points)
#REDIRECT [[Gradient (two points)]]