CalcAB
Chp 46A
Related Rates, Algebra and Geometry
Notes
1.
Relatedrate problems involve movement.
The rate something changes or moves can be any
value (slow, fast, positive, negative, zero).
The RATE OF CHANGE of something for us means a
DERIVATIVE is used.
All derivatives in this section are done WITH RESPECT TO TIME; IMPLICIT
DIFFERENTIATION is needed in order to accommodate things changing with respect to time and
to accommodate more than one variable without substitution.
There is a difference between optimization problems and rate problems.
Optimization problems
focus on finding specific values that maximize or minimize some quantity; the derivatives are set
equal to zero.
Whereas, relatedrate problems are about determining a value of something at
any time, not just when a situation is at an extreme value (max/min); derivatives are used
because (instantaneous) rates are involved.
2.
Ex(algebra)
—
Find the rate y changes as x changes for each function.
(a)
Find dy/dt
when x = 1
and
dx/dt = 5
for
y = 2x
3
+ 3
(b)
Find dy/dt
when x = 3
and
dx/dt =
6
for
y x
2
= 81
3.
Ex(algebra, point moves along a function path)
—
A point is moving along the parabola
6y = x
2
.
When x = 6, the abscissa (
xterm
) is increasing
at a rate of 2 ft/sec; at what rate is the ordinate (
yterm
) changing at that instant?
Organize given information; translate the words into mathematical notation.
“x
term is increasing at a rate of 2 ft/sec
”
translates to
dx/dt = 2 ft/s
“
rate the yterm is changing
”
translates to
dy/dt = ?? ft/s
"when
x = 6"
is a value to use in the derivative after it has been determined.
In the problem stated, the x value is changing with respect to time (seconds); the
only way to solve how fast the yvalue is changing is to work with the derivative
implicitly, with respect to time (dt).
After the derivative has been determined
, any known values can be plugged in.
Use
x = 6 and
dx/dt = 2 in the derivative to solve for the unknown dy/dt.
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 Spring '10
 KNOPF
 Algebra, Geometry, Derivative, Rate Of Change, 3 ft, 1 inch, 2 inches, 2 ft, 1.3 ft

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