Calculus 7.5-7.9

7.5 Indeterminant Forms

L’Hopital’s Rule If f(a)=g(a)=0,

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

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

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.

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.

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.

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

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

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