What is Calculus?Three Basic Concepts

Lesson 2.1

Limit

• A mathematical tool • It studies the tendency of a

function – as its variable approaches some value

Derivative:

• Defined as a limit

• Computes variety of values – rates of change

– slopes of tangent lines of curves

• Known as "differential calculus"

Integral

• Found by taking a special limit

• Often a limit of a sum of terms• Computes things such as

– area

– volume

– arc length

• Known as "integral calculus"

Limit

• Consider the sequence:

• As n gets very large, what value does the fraction approximate?

• We say the limit of the fraction approaches 1 as n gets very large

1 2 3, , , ...

2 3 4 1

n

n

lim 11n

nL

n

The Derivative

• Consider a function   f(x) = y

• We seek the slope of the tangent line at point x = a P = (a,f(a))

• An approximation of the tangent is the secant PQ

• Slope of the secant is ( ) ( )f x h f xm

h

The Derivative

• Now allow h to get very small • The secant becomes a very close

approximation of the tangent• Then the slope of the tangent line at x =

a is

0

( ) ( )limh

f x h f x

h

The Integral

• Consider the function f(x) = y

• We seek the area under the curve between points x0 and x3

• An approximation is the sum of the areas of the three boxes

The Integral

• We can get a closer approximation by increasing the number of partitions at x0, x1, x2, ... xn where n is very large

• The limit of the sum as n -> infinity is the actual area under the curve

1

lim ( )n

nn

k

Area box

Mathematical ModelingSteps• Make assumptions about the real world• View the real world problem

• variables• formulas• relationships

• This abstraction becomes the model• Simplify the math and derive mathematical

facts from the model• Use the resulting facts to make predictions

about the real world• compare predictions to real world events• fine tune the model

Example of Mathematical Modeling

• Gather data

• Plot on graph

Example of Mathematical Modeling

• Observe and compare to known functions

• Which is it??– linear

– quadratic

– exponential – logarithmic

– trigonometric

Assignment

• Lesson 2.1

• Page 81

• Exercises: 3, 5, 9, 13, 15, 19, 21, 29, 33, 37, 39