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MATH 237 MIDTERM By: Niall W. MacGillivray

MATH 237 MIDTERM By: Niall W. MacGillivray. Outline. 1.1-1.2 Scalar Functions 2.1-2.2 Limit Theorems 2.3-2.4 Proving a Limit Does (Not) Exist 3.1-3.2 Continuity Theorems 3.3 Limits Revisited 4.1-4.2 Partial Derivatives 4.3-4.4 Linear Approximations 5.1 Differentiability.

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MATH 237 MIDTERM By: Niall W. MacGillivray

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  1. MATH 237 MIDTERM By: Niall W. MacGillivray

  2. Outline 1.1-1.2 Scalar Functions 2.1-2.2 Limit Theorems 2.3-2.4 Proving a Limit Does (Not) Exist 3.1-3.2 Continuity Theorems 3.3 Limits Revisited 4.1-4.2 Partial Derivatives 4.3-4.4 Linear Approximations 5.1 Differentiability

  3. Outline cont’d 5.2-5.3 Partial Derivatives, Continuous Functions, and Differentiable Functions 5.4 Linear Approximation Revisited 6.1 Basic Chain Rule 6.2 Extensions of the Chain Rule 7.1 Directional Derivatives 7.2-7.3 Gradient Vector Revisited 8.1 Taylor Polynomials 8.2 Taylor’s Theorem

  4. 1.1-1.2 Scalar Functions • Function: a “well-defined” rule • Domain: subset for which function is defined • Range: subset which is attained by function • Scalar Function: Domain is subset of n-space and range is subset of 1-space

  5. 1.1-1.2 Scalar Functions

  6. 1.1-1.2 Scalar Functions • Level curves: curves given by setting the function equal to a constant

  7. 1.1-1.2 Scalar Functions

  8. 1.1-1.2 Scalar Functions

  9. 1.1-1.2 Scalar Functions • Cross sections: curves given by setting one of the variables equal to a constant

  10. 1.1-1.2 Scalar Functions

  11. 2.1-2.2 Limit Theorems • Euclidean distance: “usual” distance in n-space

  12. 2.1-2.2 Limit Theorems

  13. 2.1-2.2 Limit Theorems

  14. 2.3-2.4 Proving a Limit Does (Not) Exist

  15. 2.3-2.4 Proving a Limit Does (Not) Exist

  16. 3.1-3.2 Continuity Theorems • Limit exists • f defined at a • The equality holds

  17. 3.1-3.2 Continuity Theorems

  18. 3.1-3.2 Continuity Theorems

  19. 3.1-3.2 Continuity Theorems

  20. 3.1-3.2 Continuity Theorems

  21. 3.1-3.2 Continuity Theorems

  22. 3.1-3.2 Continuity Theorems

  23. 3.1-3.2 Continuity Theorems

  24. 3.1-3.2 Continuity Theorems

  25. 3.3 Limits Revisited

  26. 3.3 Limits Revisited

  27. 4.1-4.2 Partial Derivatives

  28. 4.1-4.2 Partial Derivatives

  29. 4.1-4.2 Partial Derivatives

  30. 4.1-4.2 Partial Derivatives • Careful about the order of subscripts

  31. 4.1-4.2 Partial Derivatives

  32. 4.1-4.2 Partial Derivatives

  33. 4.1-4.2 Partial Derivatives

  34. 4.1-4.2 Partial Derivatives

  35. 4.3-4.4 Linear Approximations • Recall in 1-D: The tangent line • Recall in 1-D: The linear approximation

  36. 4.3-4.4 Linear Approximations

  37. 4.3-4.4 Linear Approximations

  38. 4.3-4.4 Linear Approximations • Hint: Don’t expand terms

  39. 4.3-4.4 Linear Approximations • Calculator value: -1.034, error of 0.004

  40. 4.3-4.4 Linear Approximations

  41. 4.3-4.4 Linear Approximations

  42. 5.1 Differentiability • How good is this approximation?

  43. 5.1 Differentiability

  44. 5.1 Differentiability

  45. 5.1 Differentiability

  46. 5.1 Differentiability

  47. 5.1 Differentiability

  48. 5.2-5.3 Partials, Cts Fncs, and Diff Fncs • Recall in 1-D: Differentiability implies Continuity

  49. 5.2-5.3 Partials, Cts Fncs, and Diff Fncs

  50. 5.2-5.3 Partials, Cts Fncs, and Diff Fncs

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