2 1 THE DERIVATIVE AND THE TANGENT LINE

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2. 1 THE DERIVATIVE AND THE TANGENT LINE PROBLEM We are learning: • The

2. 1 THE DERIVATIVE AND THE TANGENT LINE PROBLEM We are learning: • The Tangent Line Problem • The Derivative of a Function • Differentiability and Continuity

THE TANGENT LINE PROBLEM: Possible definitions: How would we define what a tangent line

THE TANGENT LINE PROBLEM: Possible definitions: How would we define what a tangent line is?

LET US EXPLORE TO HELP US THINK OF A DEFINITION Let’s work together to

LET US EXPLORE TO HELP US THINK OF A DEFINITION Let’s work together to see what the secant lines’ slopes are approaching! Instead of doing the chart method of finding the limit we can use our analytical methods

ANALYTICAL METHOD:

ANALYTICAL METHOD:

EX: TANGENT LINES TO THE GRAPH OF A NONLINEAR FUNCTION

EX: TANGENT LINES TO THE GRAPH OF A NONLINEAR FUNCTION

THE DERIVATIVE OF A FUNCTION!!! The limit used to define the slope of a

THE DERIVATIVE OF A FUNCTION!!! The limit used to define the slope of a tangent line is also:

EXAMPLE 2: FINDING THE DERIVATIVE BY THE LIMIT PROCESS

EXAMPLE 2: FINDING THE DERIVATIVE BY THE LIMIT PROCESS

EXAMPLE 3: USING THE DERIVATIVE TO FIND THE SLOPE AT A POINT

EXAMPLE 3: USING THE DERIVATIVE TO FIND THE SLOPE AT A POINT

STOP FOR SOME CLASS WORK: 2. 1 P. 99: 1, 4, 5, 7, 9,

STOP FOR SOME CLASS WORK: 2. 1 P. 99: 1, 4, 5, 7, 9, 13, 17, 23, 27 -32, 33, 35

DIFFERENTIABILITY AND CONTINUITY

DIFFERENTIABILITY AND CONTINUITY

DIFFERENTIABILITY AND CONTINUITY Continuity does not imply differentiability as shown by the next examples!

DIFFERENTIABILITY AND CONTINUITY Continuity does not imply differentiability as shown by the next examples!

EXAMPLE 6: A GRAPH WITH A SHARP TURN Conclusion: At a sharp turn the

EXAMPLE 6: A GRAPH WITH A SHARP TURN Conclusion: At a sharp turn the derivative does not exist since it could be on two tangent lines!

EXAMPLE 7: A GRAPH WITH A VERTICAL TANGENT LINE Conclusion: If a portion of

EXAMPLE 7: A GRAPH WITH A VERTICAL TANGENT LINE Conclusion: If a portion of the graph gets to steep and has undefined slope then the derivative will also be undefined! Cool proof in textbook! Last 2. 1 Hmwr: 45, 48, 50, 51 -63 odd, 69 -71.

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