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: F091

7. Sec 3.10: Linear approximation and Differentials
Example: F091

8. Sec 3.10: Linear approximation and Differentials
Example: F121

9. Sec 3.10: Linear approximation and Differentials

10. Sec 3.10: Linear approximation and Differentials
Example: 081

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

12. 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,

13. Sec 3.10
Differentials

14. Definition: Sec 3.10: Linear approximation and Differentials
If , where is a differentiable function, then the differential
is an independent variable; that is, can be given the value of any real number. The differential is then defined in terms of by the equation

15. Example: 092 Sec 3.10: Linear approximation and Differentials 0,014
0.001
0.01
0.021
0.045

16. Sec 3.10: Linear approximation and Differentials

17. Sec 3.10: Linear approximation and Differentials
Exams problems
can be approximated by
= 14*0.01 = 0.14
relative change in x
relative change in y
relative change in y
can be approximated by
= 0.035 =
percentage change in x
percentage change in x
percentage change in x
can be approximated by

18. Sec 3.10: Linear approximation and Differentials

19. Sec 3.10: Linear approximation and Differentials

20. Sec 3.10: Linear approximation and Differentials
FINAL-151

21. Sec 3.10: Linear approximation and Differentials
If , where is a differentiable function, then the differential
is an independent variable; that is, can be given the value of any real number. The differential is then defined in terms of by the equation