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MAT 1234 Calculus I

MAT 1234 Calculus I. Section 2.9 Linear Approximations and Differentials. http://myhome.spu.edu/lauw. Next. WebAssign 2.9 Quiz (Tuesday)– 2.7, 2.9 (No 2.8? Why?). Preview. The need for approximations : Formulas can be simplified. Very popular method used in physical sciences. Preview.

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MAT 1234 Calculus I

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  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

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