1. Warm-Up- Test in the box…
Find the equation of the line tangent to
at (1,1).
(1,1)

2. 3-9: Differentials Objectives: Understand the concept of differentials
Use differentials to estimate change
©2003 Roy L. Gover (

3. Analysis x
0.9
1.0
1.1
f(x) y
(1,1)
Complete the table

4. Important Idea
The equation of the line tangent to f(x) at c can be used to approximate values of f(x) near f(c). Let y represent the change in f(x) that corresponds to a small change in x called x…

5. Definition and are traditionally known as dy and dx…
The differential of x (dx) is any nonzero real number. The differential of y is:

6. Analysis dx

7. Analysis
approximates x
as x c.

8. Important Idea
The differential of y (dy) can be used as an approximation for the change in y when dx is small:

9. Analysis
As values of x get close to c, y becomes a better approxi-mation of values close to f(c) x y x x y

10. Important Idea
The definition of differentials allows for dy and dx in the symbol for a derivative:
to be treated as separate quantities.

11. Example
Find the differential dy of the function:

12. Try This
Find the differential dy of the function:

13. Example
Use differentials to approximate
Let , x=25 & dx=.3 then…

14. Definition
The percentage error is the ratio of the error to the original amount.
Example: If dy= .1 sq. in. is the error in the area of a circle of radius 2 in., then the percentage error is:

15. Lesson Close
The equation dy=f’(x)dx tells how sensitive f(x) is to changes in x.

16. Assignment
240/11-15 all (No calculator),18,21,23

17. Example
Use differentials to find the error and percentage error in the volume of a ball bearing whose radius is measured to be 0.7 inch correct to within 0.01 inch.
Ex 3,p233

18. Try This
The profit p for a company is Use differentials to approximate the change and percentage change in profit as production changes from x=95 to x=100 units. Hint: Let x=95 & dx=5.

19. Solution
Profit dx
Units