1. Sec 3.10: Linear approximation and Differentials
The equation of the tangent line at x = 4
by zooming in toward the point (4,2) on the graph of the function, we noticed that the graph looks more and more like its tangent line L(x) .
we use the tangent line L(x) as an approximation to the curve when x is near 4.

2. The tangent line is considered as an approximation of the curve y=f(x)
Sec 3.10: Linear approximation and Differentials
y=L(x) is the tangent line
If we are very close to the point a
The tangent line is considered as an approximation of the curve y=f(x)

3. Sec 3.10: Linear approximation and Differentials
Why do we need the approximation of f (we have f)
Example:
Compute:
Smart Way:
Find the tangent line at x=1

4. Sec 3.10: Linear approximation and Differentials
The equation of the tangent line at x = 4
Example:
Approximate:
we use the tangent line L(x) as an approximation to the curve when x is near 4.

5. The tangent line is considered as an approximation of the curve y=f(x)
Sec 3.10: Linear approximation and Differentials
The tangent line is considered as an approximation of the curve y=f(x)
is called the linear approximation or tangent line approximation
is called the linearization of f at a.
standard linear approximation

6. Sec 3.10: Linear approximation and Differentials
Example

7. Sec 3.10: Linear approximation and Differentials
Example

8. Sec 3.10: Linear approximation and Differentials
An important linear approximation for roots and powers
Examples:
x sufficiently close to zero,
Examples:
By calculator

9. Sec 3.10: Linear approximation and Differentials
APPLICATIONS TO PHYSICS
Linear approximations are often used in physics. In analyzing the consequences of an equation, a physicist sometimes needs to simplify a function by replacing it with its linear approximation.
x sufficiently close to zero,

10. Sec 3.10
Differentials

11. Sec 3.10: Linear approximation and Differentials

12. Sec 3.10: Linear approximation and Differentials

13. Example Sec 3.10: Linear approximation and Differentials 0,014 0.001
0.01
0.021
0.045

14. Sec 3.10: Linear approximation and Differentials
relative error
relative error
percentage errors
percentage errors

15. Sec 3.10: Linear approximation and Differentials

16. Sec 3.10: Linear approximation and Differentials

17. Sec 3.10: Linear approximation and Differentials
Estimated
True
Change in x
The change
Relative change
Percentage change

18. Sec 3.10: Linear approximation and Differentials

19. Sec 3.10: Linear approximation and Differentials

20. Sec 3.10: Linear approximation and Differentials