1. SECTION 8-L Linearization Linear Approximations Tangent line approximations

2. Let’s start with the function at x = 4 because we know what is Suddenly your calculator batteries are dead and you need to find out the square root of 5. The best you are going to be able to do now is approximate it. But how? Lets find the tangent line at x = 4? Now graph both the function and its tangent line…

3. Notice that for numbers close to 4, the tangent line is very close to the curve itself… So lets try plugging 5 into the tangent line equation and see what we get… Use the tangent line to the functionat x = 4 to approximate

4. So let’s plug 5 into the tangent line to get… since    (4, 2) (5, 2.25) Notice how close the point on the line is to the curve when we zoom in close to 4 So in this case, our approximation will be a very good one as long as we use a number close to 4.

5. For any function f (x), the tangent line is a close approximation of the function for some small distance from the point of tangency. We call the equation of the tangent line the linearization of the function at a and it is denoted by the function notation L(x).

6. Note: Linearization is just using the equation of the tangent line for an approximation. The linearization of f at a is the standard linear approximation of f at a. y – y 0 = m(x – x 0 ) We take the point-slope form… Make a few notational substitutions… And solve for L(x)

7. linearization near zero: The graph would look like this:.

8. Notice how close the graphs are to each other near x = 0

9. 2). If (2, -2) is a point on the graph of use the equation of a tangent line passing through the point to approximate the y-coordinate a) when the x-coordinate is 2.1 *Use the tangent line at (2, -2) for the linearization

10. 2). If (2, -2) is a point on the graph of use the equation of a tangent line passing through the point to approximate the y-coordinate when the x-coordinate is 2.1 b) When the x-coordinate is 1.9

11. 3) Use linearization to approximate using the function Now use the tangent line at x = 8 for the linearization

12. Homework: Worksheet 8-L