1. Calculus Chapter 3 Derivatives

2. 3.1 Informal definition of derivative

3.  A derivative is a formula for the rate at which a function changes.

4. Formal Definition of the Derivative of a function

5.  You’ll need to “snow” this

6. Formal Definition of 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 curve see diagram

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

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

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. 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 1. Derivative of a constant

20. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives

21. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple

22. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules

23. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules 5. Higher order derivatives

24. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules 5. Higher order derivatives 6. Product rule

25. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules 5. Higher order derivatives 6. Product rule 7. Quotient rule

26. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules 5. Higher order derivatives 6. Product rule 7. Quotient rule 8. Negative integer power rule

27. 3.3 Differentiation Rules 1. Derivative of a constant 2. Power Rule for derivatives 3. Derivative of a constant multiple 4. Sum and difference rules 5. Higher order derivatives 6. Product rule 7. Quotient rule 8. Negative integer power rule 9. 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