2.1. Tangent Lines and Rates of Change http://www.ck12.org
2.1 Tangent Lines and Rates of Change
Learning Objectives
A student will be able to:
- Demonstrate an understanding of the slope of the tangent line to the graph.
- Demonstrate an understanding of the instantaneous rate of change.
A car speeding down the street, the inflation of currency, the number of bacteria in a culture, and the AC voltage
of an electric signal are all examples of quantities that change with time. In this section, we will study the rate of
change of a quantity and how is it related to the tangent lines on a curve.
The Tangent Line
If two pointsP(x 0 ,y 0 )andQ(x 1 ,y 1 )are two different points of the curvey=f(x), then the slope of the secant line
connecting the two points is given by
msec=yx^1 −y^0
1 −x 0=f(xx^1 )−f(x^0 )
1 −x 0( 1 )
Now if we let the pointx 1 approachx 0 ,Qwill approachPalong the graphf. Thus the slope of the secant line will
gradually approach the slope of the tangent line asx 1 approachesx 0. Therefore (1) becomes
mtan=x 1 lim→x 0 f(x^1 x 1 )−−xf 0 (x^0 ). ( 2 )If we leth=x 1 −x 0 ,thenx 1 =x 0 +handh→0 becomes equivalent tox 1 →x 0 , so( 2 )becomes
mtan=limh→ 0 f(x^0 +hh)−f(x^0 ).If the pointP(x 0 ,y 0 )is on the curvef, then the tangent line atPhas a slope that is given by
mtan=hlim→ 0 f(x^0 +hh)−f(x^0 )provided that the limit exists.
Recall from algebra that thepoint-slopeform for the tangent line is given by