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

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

2-1: The Derivative Objectives: Explore the tangent line problem Define the derivative Discuss the relationship between differentiability and continuity ©2002 Roy L. Gover (www.mrgover.com)

Example Average speed is 50 mph measured over 10 hours 500 distance in miles 10 time in hours

Example Average speed is 35.7 mph measured over 7 hours distance in miles 250 7 time in hours

Important Idea The average speed over a time period is the slope of the line connecting the beginning and end of the time period.

Example Average speed is 24 mph measured over 5 hours distance in miles 120 5 time in hours

Try This Describe in words how you could find the speed at exactly the 5th hour. distance in miles 5 time in hours

Solution The instantaneous velocity at exactly the 5th hour is the slope of the line tangent to the velocity function at t=5.

Example What is the slope of the tangent line at t=5? 5 distance in miles 5 time in hours

Solution 42.9 mph distance in miles 300 mi. 7 hrs. 5

Try This What is the instantaneous velocity at 8 hours? 70 mph Distance in miles 70 mph

Important Idea The instantaneous velocity at a point, or any other rate of change, is the slope of the tangent line at the point

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 (2,1): Why did we not use the point (2,1) in the solution?

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 point (3,1): (3,1)

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 right end point

Try This Estimate the slope of the line tangent to the following graph at the left end 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).

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

Warm-Up at (-3,4).

Lesson Close A derivative is…

Assignment Page 103 Problems 3, 4, 5- 17 odd, 71, 73