2-1: The Derivative Objectives: Explore the tangent line problem

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Derivative and the Tangent Line Problem

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Presentation transcript:

2-1: The Derivative Objectives: Explore the tangent line problem Define the derivative Discuss the relationship between differentiability and continuity

Important Idea The tangent line problem is: in general, how do you find the slope of the tangent line at a point?

Analysis

Analysis As , the slope of the secant linethe slope of the tangent line at x

Analysis As , the slope of the secant linethe slope of the tangent line at x

Analysis As , the slope of the secant linethe slope of the tangent line at x

Analysis As , the slope of the secant linethe slope of the tangent line at x is very, very close to 0

Analysis Slope of secant line: AP Exam: instead of

Analysis The slope of the secant line becomes the slope of the tangent line:

Example Find the slope of the line tangent to the following graph at the point (2,5):

Try This Find the slope of the line tangent to the following graph at the point (3,17): m=12

Try This Find the slope of the line tangent to the following graph at the point (1,1): Hint: rationalize the numerator

Try This Estimate the slope of the line tangent to the following graph at the indicated point:

Try This Estimate the slope of the line tangent to the following graph at the point (0,0)

Definition The slope of the tangent line, if it exists, is: sometimes:

Definition The slope of the tangent line at a point is called the derivative of f(x) and the derivative is: if the limit exists.

Important Idea Differentiation at a point implies continuity at the point, however,… continuity at a point does not imply differentiability at a point

f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1) Example f(x) f(x) is differentiable at (3,1) implies f(x) is continuous at (3,1) (3,1)

Example f(x) continuous at (3,1) does not imply f(x) is differentiable at (3,1) f(x) (3,1)

Definition An alternative form of the derivative at a point c is: providing the limit exists

Example Use the alternative form to find f’(x) at x=2:

Example Use your knowledge of derivatives and algebra to find the equation of the line tangent to at (2,8).

Try This Use your knowledge of derivatives and algebra to find the equation of the line tangent to at (-3,4).

Lesson Close A derivative is…