related rates
TRANSCRIPT
Related Rates
Do Now: If air is being pumped into a balloon, what two aspect of the balloon are increasing?
Volume and radius
The _________________________________ are related to each other.
Rates of increase
Idea: Compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured
Procedure: Find an equation that relates the two quantities and use the chain rule to differentiate both sides with respect to time
Ex. It is easier to measure rate of increase of volume than rate of increase of radius.
• Steps1. Draw diagram if possible2. Identify given information,
using rates as derivatives3. Write equation relating
quantities. If necessary, use geometry to eliminate one variable by substitution
4. Use Chain Rule to differentiate both sides of the equation with respect to time.
5. Substitute given information into the resulting equation to solve for unknown rate
Example 1 If V is the volume of a cube with edges length x and the cube expands as time passes, find in terms of .
dt
dV
• Steps1. Draw diagram if possible2. Identify given information,
using rates as derivatives3. Write equation relating
quantities. If necessary, use geometry to eliminate one variable by substitution
4. Use Chain Rule to differentiate both sides of the equation with respect to time.
5. Substitute given information into the resulting equation to solve for unknown rate
xx
x
dt
dx
3xV
23xdx
dV
dt
dx
dx
dV
dt
dV
dt
dxx
dt
dV 23
s
cm
dt
dV 3
100 • Steps1. Draw diagram if possible2. Identify given information,
using rates as derivatives3. Write equation relating
quantities. If necessary, use geometry to eliminate one variable by substitution
4. Use Chain Rule to differentiate both sides of the equation with respect to time.
5. Substitute given information into the resulting equation to solve for unknown rate
?dt
drWhen d = 50 cm (r = 25 cm)
3
3
4rV
dt
drr
dt
dV 4
dt
dr
dr
dV
dt
dV
rdr
dV
24
dt
dr2)25(4100
dt
dr
2)25(4
100
dt
dr
25
1
The radius of the balloon is increasing at a rate of s
cm
25
1
Example 2: Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
Example 3: The power P (watts) of an electric circuit is related to the circuit’s resistance R (ohms) and current I (amperes) by using P = RI2 .
How is related to and ?
How is related to if P is constant?
dt
dPdt
dR
dt
dI
dt
dRdt
dI
dt
dIRI
dt
dRI
dt
dP22
dt
dI
dI
dP
dt
dR
dR
dP
dt
dP
0dt
dPdt
dIRI
dt
dRI 20 2
dt
dIRI
dt
dRI 22
dt
dI
I
R
dt
dI
I
RI
dt
dR 222
dt
dI
I
P
dt
dI
I
R
dt
dR
RIP
3
2
22
, If
Example 4: a. If A is the area of a circle with radius r and the circle expands as time passes, find in terms of .dt
dA
dt
dr
2rA
dt
dr
dr
dA
dt
dA
dt
drr
dt
dA 2
rdr
dA 2
Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30m?
2rA
rdr
dA 2
sm
dt
dr1mr 30
dt
dr
dr
dA
dt
dA
dt
drr
dt
dA 2
)1)(30(2dt
dA
sm
dt
dA 260
?dt
dA
The area of the spill is increasing at a rate of 60π m2/s
Example 5: If and y = x3 + 2x, find when x = 25dt
dxdt
dy
xxy 23
23 2 xdx
dy
dt
dx
dx
dy
dt
dy
dt
dxx
dt
dy23 2
70)5)(14( dt
dy