We call the equation of the tangent the linearization of the function.

1. We call the equation of the tangent the linearization of the function. For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

3. is the standard linear approximation of f at a. Start with the point/slope equation: linearization of f at a The linearization is the equation of the tangent line, and you can use the old formulas if you like.

4. Linearization

7. Example Finding a Linearization

8. Important linearizations for x near zero: This formula also leads to non-linear approximations:

9. Differentials: When we first started to talk about derivatives, we said that becomes when the change in x and change in y become very small. dy can be considered a very small change in y. dx can be considered a very small change in x.

10. Estimating Change with Differentials

11. Let be a differentiable function. The differential is an independent variable. The differential is:

12. Example Finding the Differential dy

13. Examples Find dy if • y = x5 + 37x Ans: dy = (5x4 + 37) dx • y = sin 3x Ans: dy = (3 cos 3x) dx

14. Differential Estimate of Change Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is df = f ‘(a) dx.

15. Example The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in circle’s area A. Compare this to the true change ΔA.

16. Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in r very small change in A (approximate change in area)

17. (approximate change in area) Compare to actual change: New area: Old area: