A Preview of CalculusLesson 2.1

What Do You Think?What things could be considered the greatest achievements of the human mind?

It's the Greatest!Consider that all these things emerged because of technological advancesThose advances relied on CALCULUS !Calculus has made it possible to:Build giant bridgesTravel to the moonPredict patterns of population change

True or False?

The Genius of Calculus is SimpleIt relies on only two ideasThe DerivativeThe IntegralBoth come from a common sense analysis of motionMotion is change in position over timeAll you have to do is drop your pencil to see it happen

What Is CalculusIt is the mathematics of changeIt is the mathematics of tangent linesslopesareasvolumesIt enables us to model real life situationsIt is dynamicIn contrast to algebra/precalc which is static

What Is CalculusOne answer is to say it is a "limit machine"Involves three stagesPrecalculus/algebra mathematics processBuilding blocks to produce calculus techniquesLimit processThe stepping stone to calculusCalculusDerivatives, integrals

Contrasting Algebra & Calculus

Use f(x) to find the height of the curve at x = c

Find the limit of f(x) as x approaches c

Contrasting Algebra & Calculus

Find the average rate of change between t = a and t = b

Find the instantaneous rate of change at t = c

Contrasting Algebra & Calculus

Area of a rectangle

Area between two curves

Tangent Line ProblemApproximate slope of tangent to a lineStart with slope of secant line

Tangent Line ProblemNow allow the x to get smaller

The Area ProblemWe seek the area under a curve, the graph f(x)

We approximate that area with a number of rectangles Sum = 31.9Actual = 33.33

The Area ProblemThe approximation is improved by increasing the number of rectangles

Number of rectangles = 10Sum = 32.92Actual = 33.33

The Area ProblemThe approximation is improved by increasing the number of rectangles

Number of rectangles = 25Sum = 33.19Actual = 33.33View Applet Example

AssignmentLesson 2.1Page 67Exercises 1 10 all