We havenot yet developedany computationalmachineryfor computing derivatives and integrals so
we will just stateone popularapplicationof integralcalculusthat relatesthe derivativeand integralsof a
function.
Example2:
Thereare quite a few applicationsof calculusin business.One of theseis the cost function of producing
itemsof a product.It can be shownthat the derivativeof the cost functionthat givesthe slopeof the
tangentline is anotherfunctionthat that givesthe cost to producean additionalunit of the product.This is
calledthemarginalcostand is a very importantpieceof informationfor managementto have.Conversely,
if one knowsthe marginalcost as a functionof then findingthe area underthe curveof the functionwill
give backthe cost function
LessonSummary
- We usedlinearapproximationsto studythe limit process.
- We computedapproximationsfor the slopeof tangentlines to a graph.
- We analyzedapplicationsof differentialcalculus.
- We analyzedapplicationsof integralcalculus.
ReviewQuestions
- For the function approximatethe slopeof the tangentline to the graphat the point
a. Use the followingset of -valuesto generatethe sequenceof secantline slopes:
b. Whatvaluedoesthe sequenceof slopesapproach?
- Considerthe function
a. For whatvaluesof wouldyou expectthe slopeof the tangentline to be negative?
b. For whatvalueof wouldyou expectthe tangentline to haveslope?
c. Give an exampleof a functionthat has two differenthorizontaltangentlines?
- Considerthe function Generatethe graphof usingyour calculator.
a. Approximatethe slopeof the tangentline to the graphat the point Use the followingset of
-valuesto generatethe sequenceof secantline slopes.
b. For whatvaluesof do the tangentlinesappearto haveslopeof? (Hint:Use the calculatefunction
in your calculatorto approximatethe -values.)
c. For whatvaluesof do the tangentlines appearto havepositiveslope?