Calculus
7.5-7.9
7.5 Indeterminant Forms
L’Hopital’s Rule
If f(a)=g(a)=0,
L’Hopital’s Rule
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, then lim x a f(x) = f’(a) g(x) g’(a)
Examples
Other indeterminant forms are
Examples
7.6 Rates at which functions grow
f grows faster than g as x approaches infinity if
f and g grow at the same rate as x approaches infinity if
example
Show y=e^x grows faster than y= x^2 as x approaches infinity.
example
Show y= ln x grows more slowly than y=x as x approaches infinity.
example
Compare the growth of y=2x and y=x as x approaches infinity.
7.7 trig review
This is a picnic !!!!!
7.8 derivatives of inverse trig functions
7.8 integrals of inverse trig functions
7.9 Hyperbolic Functions
Def of hyperbolic functions
cosh x =
Def of hyperbolic functions
cosh x = sinh x =
Def of hyperbolic functions
cosh x = sinh x = tanh x =
Def of hyperbolic functions
cosh x = sinh x = tanh x = sech x =
Def of hyperbolic functions
cosh x = sinh x = tanh x = sech x = csch x =
Def of hyperbolic functions
cosh x = sinh x = tanh x = sech x = csch x = coth x =
Identities
cosh^2 – sinh^2 = 1
Identities
cosh^2 x– sinh^2 x= 1 cosh 2x = cosh^2 x + sinh^2 x
Identities
cosh^2 x – sinh^2 x = 1 cosh 2x = cosh^2 x + sinh^2 x sinh 2x = 2 sinh x cosh x
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
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
These are cool
cosh 4x + sinh 4x =
clearly
cosh 4x – sinh 4x =
therefore
sinh e^(nx) + cosh e^(nx) = e^(nx)
(sinh x + cosh x ) = e^x
(sinh x + cosh x ) = e^x
So ( sinh x + cosh x )^4 = (e^x)^4
(sinh x + cosh x ) = e^x
So ( sinh x + cosh x )^4 = (e^x)^4 = e^(4x)
MORE
sinh (-x) = - sinh x
MORE
sinh (-x) = - sinh x
cosh (-x) = cosh x
Derivatives of hyperbolic functions
Integrals of hyperbolic functions
Can you guess what’s next?
Of course!
Inverse hyperbolic functions
Inverse hyperbolic functions
Derivatives
Inverse hyperbolic functions
Integrals
7.5 – 7.9 Test