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Given

`f(x)=sin^(-1)(\sqrt(x^(3)+4))`

, find

`f^(')(x)`

and simplify: (4) A stone is dropped into a lake, creating a circular ripple that travels outward from where the stone hit the water. The area of the circle enclosed by the ripple is growing at the steady rate of

`4f(t^(2))/()`

second. a. Find the rate at which the radius of circle is changing when the radius is

`1ft`

?

`2ft`

?

`5ft`

? (5) 2 continued. A stone is dropped into a lake, creating a circular ripple that travels outward from where the stone hit the water. The area of the circle enclosed by the ripple is growing at the steady rate of

`4f(t^(2))/()`

second. b. Just as the stone is about to hit the water there is no circular ripple. What is the rate of change in the radius when the radius is zero

`ft`

? Explain the mathematical result. (3) Two cars are traveling away from the same intersection on mutually perpendicular roads. At a certain moment car A is

`5km`

away from the intersection and is moving at

`1.2k(m)/(m)in`

. At the same time, car

`B`

is

`8km`

from the intersection and is moving at

`0.9k(m)/(m)in`

. At what rate is the angle

`\theta `

(see diagram) changing at that instant? (5) For each function, find the critical numbers and classify them as local maximum or local minimum: a.

`f(x)=x^(3)-x^(2)+x`

. Also, find the inflection point. (5) b.

`g(t)=4t-tant`

. (5)