2.6 Related Rates Geometric Model dx/dt? Solve on Board.
14
2.6 Related Rates Today wew illstretch even farther!
Transcript of 2.6 Related Rates Geometric Model dx/dt? Solve on Board.
2.6 Related Rates
Today we will stretch even farther!
2.6 Related Rates
2
Suppose related
by the equation
and are differentiable fu
3
Find when 1, given 2 wh
nct
en
ions of
1.
y
dy
x
dxx
x
xddt
y
t
t
2 2 0
2 1 2
3
4
dy dxx
dt dtd
x
t
y
y
d
2.6 Related Rates
is decreasing at a rate of 40
A plane is flying over a radar
0 mph 10
tracking station. If
when , mi
what is the s
of the p p need la e?
s s
2 2 36
2 2 0
2 10 400
2
s x
ds dxs xdt dt
dx
dt x
2 2 2
?
10 6
8
x
x
x
500 mphdx
dt
400ds
dt
10s
?dx
dt
Geometric Model
dx/dt?
Solve onBoard
500 mphspeed
2=50
rate of change in the of el
Given , ( in feet and in seconds),
find the of the
camera at
evation
10 seco after n lds ift-off.
h th t t
h
2=50h t t
?d
dt
10 =5000h
Variable
Constant
Variable
You cannotput in values for variablesuntil you havedifferentiated!
Geometric Model
tan2000
h
2
2
2
1sec
20001 1
cos 2000
2
c
0
o
0 0
s
d dh
dt dtd dh
dt dt
d
d
d
dtt
h
x
2 22000 5000x
2 2
10
2000cos
2000 5000
100 10 1000t
dh
dt
2 radians per second
29Implicit
Differentiation
In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when
3.
Crankshaft = 3”Connecting Rod = 7”
Piston = x”
Given Rate: 200 Revolutions per minute
200(2 ) 400d
dt
V
C
a
on
ri
stan
able
ts 3,
,
7
s x
400d
dt
2 2 2
2 2 2
Equation: Find an equation that relates and x.
Law of Cosines: 2 cos
7 3 (2)(3) cos
b a c ac
x x
Find: when =3
dx
dt
In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when .
3
Crankshaft = 3”Connecting Rod = 7”
Piston = x”
400d
dt
Do not substitute
before you
differentiate!!
Law of Cosines
Implicit Differentiation:
249 9 6 cosx x
In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when .
3
Crankshaft = 3”Connecting Rod = 7”
Piston = x”
Solution
249 9 6 cosx x
0 0 2 ( ) 6( cos ( sin ) )dx dx d
x xdt dt dt
0 2 ( ) 6cos ( ) 6 sin ( )dx dx d
x xdt dt dt
6cos ( ) 2 ( ) 6 sin ( )dx dx d
x xdt dt dt
(6cos 2 ) 6 sin ( )dx d
x xdt dt
6 sin ( )
6cos 2
dx
d td x
x dt
You could divide
every term by 2.
3 sin ( )
3cos
dxx dt
d x
d
t
Before we substitute,
we need to find a value for x.
In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when .
3
Crankshaft = 3”
Connecting Rod = 7”
Piston = x”
249 9 6 cos3
x x
2 140 6 ( )
2x x
2 240 3 0 3 40
0 ( 8)( 5)
x x x x
x x
8x
3 (
3cos
)sind
xx
ddtx
d
t
(3)(8)sin (400 )3
3(cos ) (8)3
dx
dt
3(24)( )(400 )
23/ 2 8
Length .
4018 is decreasm .
inin
gin
Diagram
2.6 Related Rates
HW 2.6/1-9odd,31-34,41-44,52
Related Rates
1. Diagram
2. Geometric Model
3. Differentiate
?4. Solve for
5. Substitute
6. Answer Question
d
dt
9.4 Law of CosinesObjective
Cosines .To use the Law of to find unknown parts of a
2 2 2
The Law of Cosines
In , 2 co
s
& SAS SS
ABC c a b a
S
b C
2 2 2
1 2 1 22 cosOPP ADJ ADJ ADJ ADJ
Problem
