CK12 Calculus - Single Variable

2. Derivatives......................................................................................................................................

TangentLinesand Ratesof Change

LearningObjectives
A studentwill be able to:

• Demonstratean understandingof the slopeof the tangentline to the graph.


• Demonstratean understandingof the instantaneousrate of change.
  A car speedingdownthe street,the inflationof currency, the numberof bacteriain a culture,and the AC
  voltageof an electricsignalare all examplesof quantitiesthat changewith time.In this section,we will study
  the rate of changeof a quantityand how is it relatedto the tangentlines on a curve.

The TangentLine
If two pointsP(x 0 ,y 0 ) andQ(x 1 ,y 1 ) are two differentpointsof the curvey=f(x)(Figure1), then the slope
of the secantline connectingthe two pointsis givenby

Now if we let the the pointx 1 approachx 0 ,Qwill approachPalongthe graphf. Thusthe slopeof the secant
line will graduallyapproachthe slopeof the tangentline asx 1 approachesx 0. Therefore(1) becomes

If we let thenx 1 =x 0 +handh→ 0 becomesequivalenttox 1 →x 0 , so (2) becomes

If the pointP(x 0 ,y 0 ) is on the curvef, then the tangentline atPhas a slopethat

is givenby

providedthat the limit exists.

Recallfrom algebrathat thepoint-slopeform for the tangentline is givenby