Transcript 2 6 Related Rates

2-6: Related Rates
Objectives:
Assignment:
1. To implicitly differentiate
• P. 154: 1-7 odd, 11, 13,
an equation with respect
15, 16
to time
2. To solve real-life related
rate problems
• P. 154-157: 18-20, 22-25,
31-35 odd, 40, 44, 45, 46
Warm-Up
Oil is leaking from a pipeline on the surface
of a lake and forms an oil slick whose volume
increases at a constant rate of 2000 cubic
centimeters per minute. The oil slick takes
the form of a right circular cylinder with both
its radius and height changing with time.
Warm-Up
At the instant when the radius of the oil slick
is 100 centimeters and the height is 0.5
centimeter, the radius is increasing at the
rate of 2.5 centimeters per minute. At this
instant, what is the rate of change of the
height of the oil slick with respect to time, in
centimeters per minute?
Objective 1
You will be able to implicitly differentiate an
equation with respect to time
Words Into Notation
Related rate
problems
involve
derivatives
with respect
to time.
𝑑𝑣
𝑚
= 3 𝑠𝑒𝑐
2
𝑑𝑡
To become proficient at solving
these problems, you must be
able to translate a bunch of
words into sound mathematical
notation using the correct units.
The velocity of a particle is increasing at a
rate of 3 meters per second per second.
Exercise 1a
Translate each sentence into proper calculus
notation indicating the appropriate units.
1. The area of a circle is increasing at a rate of 6
square inches per minute.
2. The volume of a cone is decreasing at a rate of
2 cubic feet per second.
Exercise 1b
Translate each sentence into proper calculus
notation indicating the appropriate units.
3. The population of Denton is growing at a rate of
3 people per day.
4. The height of a tree is increasing at a rate of ½
foot per year.
Exercise 2a
Translate the following symbols into a bunch of
words, one of which is “rate.”
1.
𝑑𝑠
𝑑𝑡
= −50 meters per second
2.
𝑑𝑣
𝑑𝑡
= 11 feet per second2
3.
𝑑𝑎
𝑑𝑡
𝑚𝑖
= −2 𝑠𝑒𝑐
3
Exercise 2b
Translate the following symbols into a bunch of
words, one of which is “rate.”
4.
𝑑𝐶
𝑑𝑡
5.
𝑑𝑆
𝑑𝑡
6.
𝑑𝐴
𝑑𝑡
= 2 inches per day
=5
𝑦𝑑2
𝑚𝑖𝑛
= −4
𝑓𝑡2
𝑠𝑒𝑐
Exercise 2c
Translate the following symbols into a bunch of
words, one of which is “rate.”
7.
𝑑𝑟
𝑑𝑡
8.
𝑑𝑉
𝑑𝑡
= −10
9.
𝑑ℎ
𝑑𝑡
𝑖𝑛
= 6 𝑠𝑒𝑐
𝑐𝑚
= 40 𝑚𝑖𝑛
𝑚3
ℎ𝑟
With Respect to Time
To become proficient at solving
these problems, you must be
able to implicitly differentiate an
equation with respect to time.
Take the derivative of the area of a
circle with respect to time
Related rate
problems
involve
derivatives
𝐴 = 𝜋𝑟 2
with respect
to time.
𝑑
𝑑
𝐴 =
𝜋𝑟 2
𝑑𝑡
𝑑𝑡
Note that the area is
increasing at a constant
rate while the radius
increases at a variable rate.
𝑑𝐴
𝑑𝑟
= 2𝜋𝑟
𝑑𝑡
𝑑𝑡
𝑑𝑟
1 𝑑𝐴
=
∙
𝑑𝑡 2𝜋𝑟 𝑑𝑡
Exercise 3
Take the derivative of the volume of a cylinder with
respect to time.
The rate at which
the volume is
changing is
related to the
rates at which the
radius and height
are changing.
𝑉 = 𝜋𝑟 2 ℎ
𝑑
𝑑
Product Rule
𝑉 =
𝜋𝑟 2 ℎ
𝑑𝑡
𝑑𝑡
𝑑𝑉
𝑑𝑟
𝑑ℎ
2
= 2𝜋𝑟 ∙ ℎ + 𝜋𝑟
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑉
𝑑𝑟
𝑑ℎ
2
= 2𝜋𝑟ℎ + 𝜋𝑟
𝑑𝑡
𝑑𝑡
𝑑𝑡
Exercise 4
Take the derivative of the Pythagorean Formula
with respect to time.
𝑎2 + 𝑏 2 = 𝑐 2
𝑑 2
𝑑 2
2
𝑎 +𝑏 =
𝑐
𝑑𝑡
𝑑𝑡
𝑑𝑎
𝑑𝑏
𝑑𝑐
2𝑎
+ 2𝑏
= 2𝑐
𝑑𝑡
𝑑𝑡
𝑑𝑡
The rate at which
the hypotenuse is
changing is
related to the
rates at which the
lengths of the legs
are changing.
Time Exploration 1
In this exploration, you
will be taking the
derivative of a number of
formulas with respect to
time. First identify the
formula, and then find
the derivative of the
equation with respect to
time.
Exercise 5
Suppose 𝑥 and 𝑦 are both differentiable
functions of 𝑡 and are related by the equation
𝑦 = 𝑥 2 + 3. Find 𝑑𝑦/𝑑𝑡 when 𝑥 = 1, given
that 𝑑𝑥/𝑑𝑡 = 2 when 𝑥 = 1.
Exercise 6
2
𝐴 = 𝜋𝑟 ; The radius is increasing at a rate of
𝑑𝐴
0.5 cm per second. Find
when the radius
𝑑𝑡
is 3 cm.
Time Exploration 2
In this exploration,
substitute the given
quantities and the given
rates into the
appropriate equation,
and then solve for the
indicated rate, writing
your answer using the
correct units.
Objective 2
You will be able to solve
real-life related rate
problems
Exercise 7
Pebble is dropped into a calm pond, causing
ripples in the form of concentric circles The
radius of the outer ripple is increasing at a
constant rate of 1 foot per second. When the
radius is 4 feet, at what rate is the total area
𝐴 of the disturbed water changing?
Related Rate Algorithm
Step 1
Draw a picture of the situation
described in the problem.
Make a list of the given information
and of the required quantities.
Step 3
Step 2
Label each quantity that varies with
time using an appropriate symbol and
each quantity that does not vary with
its constant value.
Related Rate Algorithm
Relate the variables in an equation
or a formula.
Step 5
Step 4
Check to see that all necessary
values are accounted for. If not,
make an adjustment to your
strategy.
Differentiate with respect to time.
Step 6
Related Rate Algorithm
Step 7
Substitute the known quantities into
the result of Step 6 and solve for
the unknown quantity.
Check your answer for
reasonableness and make sure
you have given appropriate units.
Step 8
Exercise 8
Air is being pumped into a spherical balloon
at a rate of 4.5 cubic feet per minute. Find
the rate of change of the radius when the
radius is 2 feet.
Exercise 9
An airplane is flying on a flight path, at an
altitude of 6 miles, that will take it directly
over a radar tracking station. If the distance
from the plane to the tracking station is
decreasing at a rate of 400 miles per hour
when that distance is 10 miles, what is the
speed of the plane?
Exercise 10
A television camera at ground level is filming
the lift-off of a space shuttle that is rising
vertically according to the position equation
𝑠 = 50𝑡 2 , where 𝑠 is measured in feet and 𝑡
is measured in seconds. The camera is
2000 feet from the launch pad. Find the rate
of change in the angle of elevation of the
camera at 10 seconds after lift off.
Exercise 11
In a particular engine, a 7-inch connecting rod is
fastened to a crank of radius 3 inches. Crankshaft
rotates counterclockwise at a constant rate of 200
revolutions per minute. Find the velocity of the
𝜋
piston when 𝜃 = .
3
2-6: Related Rates
Objectives:
Assignment:
1. To implicitly differentiate
• P. 154: 1-7 odd, 11, 13,
an equation with respect
15, 16
to time
2. To solve real-life related
rate problems
• P. 154-157: 18-20, 22-25,
31-35 odd, 40, 44, 45, 46