1 / 15

Lesson 2.1 The Tangent Line Problem

Lesson 2.1 The Tangent Line Problem. By Darren Drake 05/16/06. History. Calculus grew from 4 major problems Velocity/acceleration problem Max and min value problem Area problem THE TANGENT LNE PROBLEM. History.

gage-gaines
Download Presentation

Lesson 2.1 The Tangent Line Problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lesson 2.1The Tangent Line Problem By Darren Drake 05/16/06

  2. History • Calculus grew from 4 major problems • Velocity/acceleration problem • Max and min value problem • Area problem • THE TANGENT LNE PROBLEM

  3. History • Pierre de Format, Rene’ Descartes, Christian Huygens, and Isaac Barrow are given credit for finding partial solutions • Isaac Newton(1642-1727) is credited for finding the general solution to the tangent line problem

  4. What is a Tangent Line? • The Tangent line touches the curve at one point • But this doesn’t work for all curves

  5. How do you find the tangent line? • To find the tangent line at point c, you must first find the slope of the tangent line at point c • You can approximate this slope by of the secant line containing the points and slope of the secant = Recall

  6. The closer is to 0, the closer is to c • The more accurate the tangent line approximation will be • Hmmmm…. this sounds like a limit!!!

  7. Soo… • If fis defined on an openinterval containing c, and if the limit • Exists, then the line passing through the point (c, f(c)) with the slope m is the line tangent to the graph f(x) at the point (c, f(c))

  8. Example 1The slope of the graph of a linear function Given f(x) = 2x-3, find the slope at (2,1)

  9. Example 2Tangent Lines to the graph of a nonlinear function • Find the slope at (0,1) of the tangent line to the graph of and write an equation for the tangent line at this point

  10. Example 3Finding the derivative by the limit process The limit used to define the slope of the tangent line is also used to define differentiation. The derivative of f at x is given by or Find the derivative of

  11. Example 4Differentiablilty • Derivatives have certain rules on when they exist Continuity Difeferentiability Vertical tangent

  12. Example 5Applications • There is a hill whose cross section forms the equation . Your car just died so you have to push your car up the hill. But the hill is too steep at first. So You make a ramp to make it up the hill but you don’t know how steep it is. You won’t be able to push your car up the ramp if it is steeper than 1/3 When the ramp is at your feet, it touches the hill at one point 4 feet from the start of the hill. How steep is the ramp? Will you be able to push you car up it? Use the definition of the tangent line to find your answer

  13. Example 6Application A see saw sits on the pivot formed by the equation What is the slope of the of the see saw at x=0? Use the definition of the Derivative to find your answer Recall Deriv. from left and Deriv. from right The derivative from the left and right do not equal eachother; thereofre, f is not differentiable at x = 0 so we don’t know what the slope of the see saw is at x = 0

  14. Trick Question!!!!!! The slope of the see saw is indeterminate, we don’t what it is!!

  15. The End

More Related