Definition of the Derivative LeTourneau Univesity拉托紐大學(xué)的導(dǎo)數(shù)的定義

1、definition of the derivativelesson 3.4tangent linerecall from geometrytangent is a line that touches the circle at only one pointlet us generalize the concept to functionsa tangent will just touch the line but not pass through itwhich of the above lines are tangent?2a secant linecrosses the curve tw

2、icethe slope of the secant will be3()( )()( )yf ahf af ahf axahahx = ahtangent linenow let h get smaller and smaller4x = ah0()( )limhf ahf ahthe slope of the tangent linetry this with the ti nspire againtangent linetry it out givendetermine the slope of the tangent line at x = 0evaluate52( )64f xxx0

3、(0)(0)limhfhfhtangent line use the limit command on your calculator to determine the slope offor x = 1 once you have the slope and the pointyou can determine the equation of the line611()yymxx2( )64f xxxthe derivativewe will define the derivative of f(x) asnote the derivative is the rate of change f

4、unction for f(x) the derivative is also a function of xthe limit must exist70()( )( )limhf ahf afxhcomparisondifference quotient slope of secantaverage rate of changeaverage velocityderivative slope of tangentinstantaneous rate of changeinstantaneous velocity8( )( )f af bab0()( )limhf ahf ahfinding

5、f(x) from definitionstrategy1.find f(x + h)2.find and simplify f(x + h) f(x)3.divide by h to get4.let h 09()( )f xhf xh0()( )( )limhf xhf xfxhtry it outuse the strategy to find the derivatives of these functions105( )f xx3( )42f xxx( )12w xxhint: rationalize the numeratorequation of the tangent line

6、we stated previously that once we determine the slope of the tangentwe can cut to the chase and state it as1111()yymxx11( ) ()yyfxxxwarningour definition of derivative includedthe phrase if the limit existsderivatives do not exist at corners or sharp points on the graphthe slope is different on each side of th