Linear Approximation and Differentials

1. Linear Approximation and Differentials Lesson 4.8

2. Tangent Line Approximation • Consider a tangent to a function at a point x = a • Close to the point, the tangent line is an approximation for f(x) y=f(x) • The equation of the tangent line:y = f(a) + f ‘(a)(x – a) f(a) • a

3. Tangent Line Approximation • We claim that • This is called linearization of the function at the point a. • Recall that when we zoom in on an interval of a function far enough, it looks like a line

4. New Look at • dy = rise of tangent relative to x = dx • y = change in y that occurs relative to x = dx y dy • • • x + x x x = dx

5. New Look at • We know that • then • Recall that dy/dx is NOT a quotient • it is the notation for the derivative • However … sometimes it is useful to use dy and dx as actual quantities

6. The Differential of y • Consider • Then we can say • this is called the differential of y • the notation is d(f(x)) = f ’(x) * dx • it is an approximation of the actual change of y for a small change of x

7. Animated Graphical View • Note how the "del y" and the dy in the figure get closer and closer

8. Try It Out • Note the rules for differentialsPage 274 • Find the differential of3 – 5x2x e-2x

9. Differentials for Approximations • Consider • Use • Then with x = 25, dx = .3 obtain approximation

10. 2x x Propagated Error • Consider a rectangular box with a square base • Height is 2 times length of sides of base • Given that x = 3.5 • You are able to measure with 3% accuracy • What is the error propagated for the volume? x

11. Propagated Error • We know that • Then dy = 6x2 dx = 6 * 3.52 * 0.105 = 7.7175This is the approximate propagated error for the volume

12. Propagated Error • The propagated error is the dy • sometimes called the df • The relative error is • The percentage of error • relative error * 100%

13. Assignment • Lesson 4.8 • Page 276 • Exercises 1 – 45 odd