Problem 1
Use implicit differentiation to find an equation of the tangent line to the graph of the given equation at the given point.
a) xy2 = 3x + y at point (2,2).
b) y1/2x3/2 + xy1/3 = 12 at point (2,8).
c) 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 dy/dx and you’ll need to use an appropriate line equation. (2 points)
Problem 2
a) A balloon is filled at a rate of 0.001π m3 per second. At what rate is the radius of the balloon increasing when the radius is 20cm? Be aware of units! (5 points) 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
a) 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.
b) f(x) = 2x3 − 6x + 9
b) g(x) = 2x3 + 6x + 9
b) h(t) = sin(ωt) with constant ω 6= 0
