# Problem 1

Use implicit differentiation to find an equation of the tangent line to the graph of the given equation at the given point.

*xy*^{2 }= 3*x*+*y*at point (2*,*2).*y*^{1/2}*x*^{3/2 }+*xy*^{1/3 }= 12 at point (2*,*8).- Show for a) that you get the same tangent if you differentiate with respect to
*y*instead of*x*. In this case you’ll get a slope d*y/*d*x*and you’ll need to use an appropriate line equation.

# Problem 2

- a) A balloon is filled at a rate of 0
*.*001*π*m^{3 }per second. At what rate is the radius of the balloon increasing when the radius is 20cm? Be aware of units! - b) An airplane flying horizontally at a height of 8000m with a speed of 500m
*/*s passes directly above an observer on the ground. What is the rate of increase of distance to the observer 1minute later?

# Problem 3

- Show that

darccos(*x*) 1

= −√

d*x *1 − *x*^{2}

(The function *y *= arccos(*x*) is the (locally) inverse function of *x *= *cos*(*y*).)

Find all critical points (points where *f*^{0}(*x*) = 0) for the following functions, and characterize whether they correspond to a local minimum, a local maximum, or neither.

*f*(*x*) = 2*x*^{3 }− 6*x*+ 9- b)
*g*(*x*) = 2*x*^{3 }+ 6*x*+ 9

b) *h*(*t*) = sin(*ωt*) with constant *ω *6= 0

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