Problem 1

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

1. xy² = 3x + y at point (2,2).
2. y^1/2x^3/2 + xy^1/3 = 12 at point (2,8).
3. Show for a) that you get the same tangent if you differentiate with respect to y instead of x. In this case youll get a slope dy/dx and youll need to use an appropriate line equation.

Problem 2

1. a) A balloon is filled at a rate of 0.001 m³ per second. At what rate is the radius of the balloon increasing when the radius is 20cm? Be aware of units!
2. 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

1. Show that

darccos(x) 1

=

dx 1 x²

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

Find all critical points (points where f⁰(x) = 0) for the following functions, and characterize whether they correspond to a local minimum, a local maximum, or neither.

1. f(x) = 2x³ 6x + 9
2. b) g(x) = 2x³ + 6x + 9

b) h(t) = sin(t) with constant 6= 0