Math 1210-001 Lab 5
Spring 2016; due February 25 at the beginning of lab
Name _____________________________________________UID______________________________
1) (20 points) Implicit differentiation (see WW #7). The graph of the equation
x2 C xy C y2 = 9
is a slanted ellipse illustrated in the figure below.
1a) Find y# for points x, y on the ellipse, using implicit differentiation.
1b) Find the equation for the tangent line to the ellipse, at the point 3, 0 on the ellipse.
1c) Find the points on the ellipse where the tangent lines are horizontal.
1d) Find the points on the ellipse where the tangent lines are vertical. (Hint: the points with horizontal
tangent line satisfy y#= 0. One way to find the vertical tangent lines is to think of the original equation as
implicitly defining x as a function of y, solve for x#, and then set x#= 0.
3
y
2
1
K3 K2
0
K1
K2
K3
1
2
x
3
4
2) (20 points) Implicit differentiation drill. (See WW #2,5,6) Find y# in terms of x and y, for the
following implicitly defined functions:
2a) K6 x2 K 2 xy C 3 y3 =K171
2b) 7 cos x K y = 3 y sin x
2c) x C y = 10
2d) In equation 2c it is possible to explicitly solve the implicit equation x C y = 10, in order to find
an explicit formula for y = f x . Do this, then find f# x and verify that your explicit derivative
computation agrees with your implicit derivative computation in 2c.
3) (15 points) A metal rod has the shape of a right circular cylinder, and as the rod is heated and cooled its
height always remains 20 times its radius. As the rod is being heated its radius is increasing at a rate of
cm
0.001
. At what rate is the volume of the rod changing, when the rod has a radius of 1 cm ?
min
4) (15 points) A 20-foot long ladder was leaning against a building, but begins to slide. If the bottom of
the ladder is sliding along the level pavement away from the building at 1 ft per second, how fast is the top
of the ladder moving down the wall when the foot of the ladder is 5 feet away from the wall?
5) (15 points) A water tank has the shape of an upside-down right-circular cone, of altitude 12 feet, base
radius of 4 feet, with vertex at the bottom of the tank, and base at the top. If water is being drained from
the tank at a rate of 10 cubic feet per minute, how fast is the water level falling when the depth is 5 feet?
1
Hints: Use similar triangles. Also, recall that the volume of a cone is V = A h where A is the area of the
3
base, and h is the height of the cone.
6) (15 points) (Related rates from data; similar to text example, page 140, and omitted example from
Wednesday class notes) The town of Cougarville monitors the height of water in its (vertical) cylindrical
water tank, with an automatic recording device. Water is constantly pumped into the tank at a rate of 3000
ft3
. During a certain 12-hour period (beginning at midnight), the water level rose and fell according to
hour
the graph in the figure below. If the radius of the tank is 20 feet, at what rate was water being used at 7:00
a.m., when a lot of people must've been getting up for their work day? Hint: The rate of change of water
volume, V# t , is the difference between how fast water is flowing into the tank, and how fast it's flowing
out.