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  Pierre de Format, Rene’ Descartes, Christian Huygens, and Isaac Barrow are given credit for finding partial solutions  Isaac Newton( ) is credited for finding the general solution to the tangent line problem

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

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 = slope of the secant =Recall

 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!!! this sounds like a limit!!!

Soo…  If f is 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))

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

Example 2 Tangent 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

Example 3 Finding 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 or Find the derivative of

Example 4 Differentiablilty  Derivatives have certain rules on certain rules on when they exist when they existContinuityDifeferentiability Vertical tangent

Example 5 Applications  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. 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

Example 6 Application 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 and Deriv. from left 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

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

The End