Unit 3 – Applications of Differentiation Lesson 1: Related Rates

Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
Extending Implicit Differentiation: The Element of Time
Assume that both x and y below are differentiable functions with respect to t.
Given x 2 y  9 .
Find
dy
dx
when x  3 and
 0.5
dt
dt
The Infamous Taxicab Problem
Two taxicabs begin at the same time at the intersection of Park Ave. and 59th St. One taxicab heads
uptown (north) on Park Ave at 40 mph. The other heads west on 59th St. at 30 mph. Both have the
unbelievable good fortune not to be stopped by traffic lights or slowed by traffic. When the taxicab on Park
Ave. has gone half a mile, how fast is the distance between them increasing?
1) The problem asked ___________, so we’re looking for ___________.
2) The problem talks about the _____________ the cars have gone, so we need to assign variables, x and y
to this vector. What should we call the variable that deals with the __________ between the taxicabs?
How about ______.
3) The cabs are traveling at _______ _________ to each other. Therefore, the ____________
__________ applies. Also, we can use a proportional relationship to compare the __________ of the
taxicabs.
4) Because we need to find a rate, we will need to use ____________ ___________.
5) The problem is asking for the ________ at which the distance, ____, between the two taxicabs is
increasing. In calculus notation, it is asking for ______.
6) Now all we have left to do is solve the problem by ___________ in some of the given information.
Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
Example 2
A snowball melts at the rate of 10 ft3/min. At what rate is the radius changing when the snowball is 2 feet in
radius?
Example 3
Water flows into a cylinder with radius 4 at 8 ft3/min. How fast is the water level rising?
Example 4
A cone-shaped swimming pool (hey, it could happen) with height 20 and radius 5 is being filled by Mr.
Battaglia with a hose which pumps in water at 3 m3/min. When the water level is 2 meters, how fast is the
level rising?
Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
Do You Smell That?
Smells like some AP Free Response Questions!!!
1982 AB 4
A ladder 15 feet long is leaning against a building so that end X is on level ground and end Y is on the wall
as shown in the figure. X is moved away from the building at the constant rate of ½ foot per second.
(a)
Find the rate in feet per second at which the length OY is changing when X is 9 feet from
the building.
(b)
Find the rate of change in square feet per second of the area of triangle XOY when X is 9
feet from the building.
Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
1984 AB 5
1
3
The volume V of a cone (V  r 2 h) is increasing at the rate of 28 cubic units per second. At the
instant when the radius r of the cone is 3 units, its volume is 12 cubic units and the radius is increasing at
½ units per second.
(a)
At the instant when the radius of the cone is 3 units, what is the rate of change of the area
of its base?
(b)
At the instant when the radius of the cone is 3 units, what is the rate of change of its height,
h?
(c)
At the instant when the radius of the cone is 3 units, what is the instantaneous rate of
change of the area of its base with respect to its height, h?
Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
1988 BC 3
The figure above represents an observer at point A watching balloon B as it rises from point C.
The balloon is rising at a constant rate of 3 meters per second and the observer is 100 meters from
point C.
(a)
Find the rate of change in x at the instant when y = 50.
(b)
Find the rate of change in the area of right triangle BCA at the instant when y = 50.
(c)
Find the rate of change in  at the instant when y = 50.
Unit 3 – Applications of Differentiation
Lesson 1: Related Rates
2002 Form B AB6
NO CALCULATOR