1. Section 4.1 Linear Approximations and Applications
MAT 1234 Calculus I
Section 4.1
Linear Approximations and Applications

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WebAssign 4.1

3. Preview The need for approximations: Formulas can be simplified.
Very popular method used in physical sciences.

4. Preview
Introduce a simple approximation method (linear approximation) by using the first derivative of the function.
It is a fundamental idea of how computing technology works.
Formula  Idea+Evidence  Applications
Introduce the concept of differentials

5. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by
Why?

6. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ?
Easy to find

7. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ?
Easy to find

8. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ?
Easy to find

9. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ?
Easy to find

10. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ?
Easy to find

11. Linear Approximations
When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by
Why?

12. 𝑦 𝑎 𝑥

13. 𝑦
𝑓(𝑥)
𝑓(𝑎) 𝑎 𝑥

14. 𝑦 𝑎 𝑥
𝑓(𝑥)
𝑓(𝑎)

15. Example 1 Estimate the value of 9.036 is near 9
Let us consider the function
when x is near 9

16. Step 1: Define the function and the near by point
Estimate the value of
9.036 is near 9

17. Step 2: Find

18. Step 3: Find the linear approximation

19. Step 4: Substitute x=9.036 into the approximation in Step 3

20. Compare this with your calculator!
Example 1
Estimate the value of
Compare this with your calculator!

21. Example 1 Remarks
Pay attention to the usage of the approximate and equal signs.
Correct
Incorrect

22. Example 1 Remarks
Pay attention to the usage of the approximate and equal signs.

23. Expectations You are expected show all 4 steps.
Historically, a few students each year skip step 3.
In the quiz and exam, I will specially ask “Find the linear approximation of 𝑓 𝑥 .”
A few students also ignore 𝑓 in

24. Example 2
Estimate the value of

25. Step 1: Define the function and the near by point
Estimate the value of

26. Step 2: Find

27. Step 3: Find the linear approximation

28. Step 4: Substitute x=2.001 into the approximation in Step 3

29. Compare this with your calculator!
Example 2
Estimate the value of
Compare this with your calculator!

30. Better Approximations
Taylor Polynomials

31. Differentials

32. Differentials 𝑦 𝑎 𝑥
𝑓(𝑥)
𝑓(𝑎)

33. Differentials 𝑦 𝑎
𝑎+𝑑𝑥

34. Differentials Suppose 𝑦=𝑓(𝑥) Let 𝑑𝑥 be an independent variable
We define a new dependent variable 𝑑𝑦 as
There are 2 dependent variables and 2 independent variables

35. Differentials Suppose 𝑦=𝑓(𝑥) Let 𝑑𝑥 be an independent variable
We define a new dependent variable 𝑑𝑦 as
There are 2 dependent variables and 2 independent variables

36. Differentials 𝑦 depends on 𝑥 dy depends on x and dx
dx and dy are called differentials
f’(x)=dy/dx (This explains the notation )
Use differentials to find anti-derivatives

37. Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥
dx and dy are called differentials
f’(x)=dy/dx (This explains the notation )
Use differentials to find anti-derivatives

38. Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥
𝑑𝑥 and 𝑑𝑦 are called differentials
f’(x)=dy/dx (This explains the notation )
Use differentials to find anti-derivatives

39. Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥
𝑑𝑥 and 𝑑𝑦 are called differentials
𝑓’(𝑥)=𝑑𝑦/𝑑𝑥 (This explains the notation 𝑑𝑦 𝑑𝑥 )

40. Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥
𝑑𝑥 and 𝑑𝑦 are called differentials
𝑓’(𝑥)=𝑑𝑦/𝑑𝑥 (This explains the notation 𝑑𝑦 𝑑𝑥 )
Use differentials to find anti-derivatives

41. Example 3

42. Example 4