1. Calculus

2. 7.5 Indeterminant Forms

3. L’Hopital’s Rule
If f(a)=g(a)=0,

4. If f(a)=g(a)=0, f’(a), g’(a) exist, g’(a) = 0 NOT
L’Hopital’s Rule
If f(a)=g(a)=0,
f’(a), g’(a) exist, g’(a) = 0 NOT

5. f’(a), g’(a) exist, g’(a) = 0 NOT,
L’Hopital’s Rule
If f(a)=g(a)=0,
f’(a), g’(a) exist, g’(a) = 0 NOT,
then lim x a f(x) = f’(a)
g(x) g’(a)

6. Examples

7. Other indeterminant forms are

8. Examples

9. 7.6 Rates at which functions grow

11. f grows faster than g as x approaches infinity if

12. f and g grow at the same rate as x approaches infinity if

13. Show y=e^x grows faster than y= x^2 as x approaches infinity.
example
Show y=e^x grows faster than y= x^2
as x approaches infinity.

14. Show y= ln x grows more slowly than y=x as x approaches infinity.
example
Show y= ln x grows more slowly than y=x as x approaches infinity.

15. Compare the growth of y=2x and y=x as x approaches infinity.
example
Compare the growth of y=2x and y=x as x approaches infinity.

16. 7.7 trig review

17. This is a picnic !!!!!

18. 7.8 derivatives of inverse trig functions

19. 7.8 integrals of inverse trig functions

20. 7.9 Hyperbolic Functions

21. Def of hyperbolic functions
cosh x =

22. Def of hyperbolic functions
cosh x =
sinh x =

23. Def of hyperbolic functions
cosh x =
sinh x =
tanh x =

24. Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =

25. Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =
csch x =

26. Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =
csch x =
coth x =

27. Identities
cosh^2 – sinh^2 = 1

28. Identities
cosh^2 x– sinh^2 x= 1
cosh 2x = cosh^2 x + sinh^2 x

29. Identities cosh^2 x – sinh^2 x = 1 cosh 2x = cosh^2 x + sinh^2 x
sinh 2x = 2 sinh x cosh x

30. Identities cosh^2 x – sinh^2 x = 1 cosh 2x = cosh^2x + sinh^2x
sinh 2x = 2 sinh x cosh x
coth^2 x = 1 + csch^ 2 x

31. Identities cosh^2 x – sinh^2 x = 1 cosh 2x = cosh^2x + sinh^2x
sinh 2x = 2 sinh x cosh x
coth^2 x = 1 + csch^ 2 x
tanh^2 x = 1- sech^2 x

32. These are cool
cosh 4x + sinh 4x =

33. clearly
cosh 4x – sinh 4x =

34. therefore
sinh e^(nx) +
cosh e^(nx) = e^(nx)

35. (sinh x + cosh x ) = e^x

36. So ( sinh x + cosh x )^4 = (e^x)^4

37. So ( sinh x + cosh x )^4 = (e^x)^4 = e^(4x)

38. MORE
sinh (-x) = - sinh x

39. MORE
sinh (-x) = - sinh x
cosh (-x) = cosh x

40. Derivatives of hyperbolic functions

41. Integrals of hyperbolic functions

42. Can you guess what’s next?

43. Of course!

44. Inverse hyperbolic functions

45. Inverse hyperbolic functions
Derivatives

46. Inverse hyperbolic functions
Integrals

47. 7.5 – 7.9 Test