MAT 1234 Calculus I

1. MAT 1234Calculus I Section 2.9 Linear Approximations and Differentials http://myhome.spu.edu/lauw

2. Next • WebAssign 2.9 • Quiz (Tuesday)– 2.7, 2.9 • (No 2.8? Why?)

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. • 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 f(x) f(a) a x

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. Example 1 • Estimate the value of Compare with calculator!

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

22. Expectations • You are expected to follow the 4 steps solution process. • Do not skip steps!

23. Example 2 • Estimate the value of

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

25. Step 2: Find

26. Step 3: Find the linear approximation

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

28. Example 2 • Estimate the value of Compare with calculator!

29. Expectations • You are expected to follow the 4 steps solution process.

30. Better Approximations • Taylor Polynomials (section 11.10)

31. Differentials

32. Differentials y f(x) f(a) a x

33. Differentials y x x+dx

34. Differentials Suppose y=f(x) Let dx be an independent variable We define a new dependent variabledy 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 variabledy 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 ) • Use differentials to find anti-derivatives

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 ) • Use differentials to find anti-derivatives

41. Example 3

42. Example 4