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Rules of differentiation

Rules of differentiation. REVIEW:. The Chain Rule. Taylor series. Approximating the derivative. Monday Sept 14th: Univariate Calculus 2. Integrals ODEs Exponential functions. Antiderivative (indefinite integral). Antiderivative (indefinite integral).

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Rules of differentiation

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  1. Rules of differentiation REVIEW:

  2. The Chain Rule

  3. Taylor series

  4. Approximating the derivative

  5. Monday Sept 14th: Univariate Calculus 2 Integrals ODEs Exponential functions

  6. Antiderivative (indefinite integral)

  7. Antiderivative (indefinite integral)

  8. Area under a curve = definite integral

  9. Integrating data: the trapezoidal rule Very similar!

  10. Example: integrating a linear function

  11. Another angle: the upper limit as an argument

  12. Another angle: the upper limit as an argument

  13. Another angle: the upper limit as an argument

  14. Differential equations Algebraicequation: involves functions; solutions are numbers. Differential equation: involves derivatives; solutions are functions. INITIAL CONDITION

  15. e.g. dead reckoning

  16. Example

  17. Classification of ODEs Linearity: Homogeneity: Order:

  18. Superposition(linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.

  19. Superposition(linear, inhomogeneous equations)

  20. Superposition(linear, inhomogeneous equations)

  21. Superposition(nonlinear equations)

  22. ORDINARY differential equation (ODE): solutions are univariate functions PARTIAL differential equation (PDE): solutions are multivariate functions

  23. Exponential functions: start with ODE Qualitative solution: slope=1 1

  24. Exponential functions: start with ODE Analytical solution

  25. Exponential functions: start with ODE Analytical solution

  26. Rules for addition, multiplication, exponentiation

  27. Rules for addition, multiplication, exponentiation

  28. Differentiation, integration (chain rule)

  29. Properties of the exponential function Taylor series: Sum rule: Power rule: Derivative Indefinite integral

  30. Cooling

  31. Sinking

  32. Homework: • Do exercises for section 2.6, 2.8 and 2.9. Omit 2.9, #1. • This will include: • Exercise with antiderivatives and classifying ODEs. • Carbon dating (for Thursday field trip) • Derive further well-known functions from f’’=-f

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