Description
Problem 1
(10 points)
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). (4 points)
b) y
1/2x
3/2 + xy1/3 = 12 at point (2, 8). (4 points)
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
(10 points)
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 20 cm? Be aware of units! (5 points)
b) An airplane flying horizontally at a height of 8000 m with a speed of 500 m/s passes
directly above an observer on the ground. What is the rate of increase of distance to the
observer 1 minute later? (5 points)
Problem 3
(10 points)
a) Show that
d arccos(x)
dx
= −
1
√
1 − x
2
(The function y = arccos(x) is the (locally) inverse function of x = cos(y).) (4 points)
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.
b) f(x) = 2x
3 − 6x + 9 (2 points)
b) g(x) = 2x
3 + 6x + 9 (2 points)
b) h(t) = sin(ωt) with constant ω 6= 0 (2 points)