Calc 2.6.notebook
November 07, 2011
2.6 Related Rates
Steps for Related Rates Problems
1. Draw and label a picture.
Identify all given quantities.
2. Identify what you are being asked to find.
(Usually a rate, expressed as a derivative)
3. Write an equation that relates the variables.
Remember
Position = ft
Velocity = ft/sec
Acceleration = ft/sec2
4. Differentiate implicitly with respect to "t".
5. Evaluate using the numerical information given.
Height in/sec
Area in2/sec
Volume in3/sec
What is being related in the following examples?
dy/dt
dh/dt
dr/dt
dV/dt
dA/dt
dD/dt
Nov 5­7:26 AM
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Calc 2.6.notebook
November 07, 2011
2.6 Related Rates
1. Suppose x and y are both differentiable functions
of t and are related by the equation y=x2+3.
Find dy/dt when x=1, given that dx/dt=2
Nov 4­4:56 PM
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Calc 2.6.notebook
November 07, 2011
2. Suppose x and y are both differentiable functions
of t and are related by the equation x2+xy=24 .
Find dy/dt when x=2, given that dx/dt=8
Nov 4­4:59 PM
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Calc 2.6.notebook
November 07, 2011
POINT MOVING ALONG A GRAPH
3. A point is moving along the graph of the given
function y=sin x such that dx/dt =2 cm per second.
Find dy/dt for the given x value.
Nov 5­7:43 AM
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Calc 2.6.notebook
November 07, 2011
CIRCLE AREA Area Formula?
4. A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The
radius, r, of the outer ripple is increasing at a
constant rate of 2in/sec. When the radius is
6 in. long, at what rate is the total area A of
the disturbed water changing?
.
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Calc 2.6.notebook
November 07, 2011
VERBAL STATEMENTS
distance between the origin
and a moving point
D=√x2+y2 Area of a circle
A=πr2
4 3
V= πr
3
Volume of a sphere
Volume of a cube, when side =x
Surface Area of a cube with
side x
Velocity of a car after traveling
for 1 hour is 50 miles per hour.
Volume of a cone
V=x3
1 2
h
V= πr
3
S=6x2
x= distance traveled
d (x)= dx = 50 when t=1
dt
dt
Water is being pumped into
a swimming pool at a rate of
10 cubic meters per hour
V= volume of water in pool
d (V)= dV
= 10 m3/hr
A gear is revolving at a rate
of 25 revolutions per minute
(1 revolution = 2π rad)
θ= angle of revolution
d
(θ)= d θ = 25(2π) rad/min
dt
dt
dt
dt
Nov 5­7:50 AM
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Calc 2.6.notebook
November 07, 2011
POINT MOVING ALONG A GRAPH
5. Find the rate of change of the distance
between the origin and a moving point on
the graph of y=x2-3 if dx/dt = 4 cm/sec.
D = √x2+y2
­3
Nov 5­12:08 PM
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Calc 2.6.notebook
November 07, 2011
Volume of a Sphere
6. A spherical balloon is inflated with gas at the rate
of 400 cm3/min. How fast is the radius of the
balloon increasing at the instant the radius is 30 cm.
r
Nov 5­12:12 PM
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Calc 2.6.notebook
November 07, 2011
Volume of a Sphere
7. The radius of a sphere is increasing at a rate of
2in/min. Find the rate of change of the volume
when r=6in.
r
Nov 5­12:15 PM
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