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Understanding Derivatives: Informal Definition and Calculation Techniques

This section covers the informal definition of derivatives, illustrating how they represent the rate at which a function changes. Key concepts include the formal definition, various notations including f'(x) and dy/dx, and differentiation rules. We explore the slope of secant and tangent lines, average and instantaneous rates of change, as well as the roles of continuity and existence of derivatives. The chapter includes practical examples, discussions on functions with horizontal tangents, and rules for finding derivatives, including power, product, quotient, and trigonometric functions.

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Understanding Derivatives: Informal Definition and Calculation Techniques

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  1. Calculus Chapter 3 Derivatives

  2. 3.1 Informal definition of derivative

  3. 3.1 Informal definition of derivative • A derivative is a formula for the rate at which a function changes.

  4. Formal Definitionof the Derivative of a function

  5. You’ll need to “snow” this

  6. Formal Definitionof the Derivative of a function • f’(x)= lim f(x+h) – f(x) • h->0 h

  7. Notation for derivative • y’ • dy/dx • df/dx • d/dx (f) • f’(x) • D (f)

  8. Rate of change and slope Slope of a secant line See diagram

  9. The slope of the secant line gives the change between 2 distinct points on a curve. i.e. average rate of change

  10. Rate of change and slope-slope of the tangent line to a curvesee diagram

  11. The slope of the tangent line gives the rate of change at that one point i.e. the instantaneous change.

  12. Slope= y-y x-x Slope of secant line m= f ’(x) Slope of tangent line compare

  13. Time for examples • Finding the derivative using the formal definition • This is music to my ears!

  14. A function has a derivative at a point

  15. A function has a derivative at a point iff the function’s right-hand and left-hand derivatives exist and are equal.

  16. Theorem If f (x) has a derivative at x=c,

  17. Theorem If f (x) has a derivative at x=c, then f(x) is continuous at x=c.

  18. Finding points where horizontal tangents to a curve occur

  19. 3.3 Differentiation Rules • Derivative of a constant

  20. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives

  21. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple

  22. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules

  23. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules • Higher order derivatives

  24. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules • Higher order derivatives • Product rule

  25. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules • Higher order derivatives • Product rule • Quotient rule

  26. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules • Higher order derivatives • Product rule • Quotient rule • Negative integer power rule

  27. 3.3 Differentiation Rules • Derivative of a constant • Power Rule for derivatives • Derivative of a constant multiple • Sum and difference rules • Higher order derivatives • Product rule • Quotient rule • Negative integer power rule • Rational power rule

  28. 3.4 Definition Average velocity of a “body” moving along a line

  29. Defintion Instantaneous Velocity is the derivative of the position function

  30. Def. speed

  31. Definition Speed The absolute value of velocity

  32. Definition Acceleration

  33. acceleration • Don’t drop the ball on this one.

  34. Definition Acceleration The derivative of velocity,

  35. Definition Acceleration The derivative of velocity, Also ,the second derivative of position

  36. 3.5 Derivatives of trig functions • Y= sin x

  37. 3.5 Derivatives of trig functions • Y= sin x • Y= cos x

  38. 3.5 Derivatives of trig functions • Y= sin x • Y= cos x • Y= tan x

  39. 3.5 Derivatives of trig functions • Y= sin x • Y= cos x • Y= tan x • Y= csc x

  40. 3.5 Derivatives of trig functions • Y= sin x • Y= cos x • Y= tan x • Y= csc x • Y= sec x

  41. 3.5 Derivatives of trig functions • Y= sin x • Y= cos x • Y= tan x • Y= csc x • Y= sec x • Y= cot x

  42. TEST 3.1-3.5 • Formal def derivative • Rules for derivatives • Notation for derivatives • Increasing/decreasing • Eq of tangent line • Position, vel, acc • Graph of fct and der • Anything else mentioned, assigned or results of these

  43. Whereas The slope of the secant line gives the change between 2 distinct points on a curve. i.e. average rate of change

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