Section 2.6: Related Rates

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Section 2.6 Related Rates
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Introduction to Related Rates

• We have seen a lot of relations (whether implicit
  or explicit) that involve two variables
  (frequently x and y). It is possible these two
  variables are themselves functions of another
  variable, such as t. For instance

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Introduction to Related Rates

• Lets investigate what occurs when t changes

t x y Equation





As x and y change, their rates of change are
related to each other. But how are they related?
Notice how when t changes, both the x and y
change in relation to the value of t.
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Introduction to Related Rates

• In order to take the derivative of the relation
  using x and y , it must be done with the respect
  to t. For instance

Differentiate both sides
Chain Rule Twice
Now we know how the rate of change for x and y
are related to each other.
In our exercises, we will not need to know the
exact relations.
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Example 1

• Suppose x and y are both differentiable functions
  of t and are related by . Find when x
  10, if when x 10.

Find the derivative by differentiating both sides.
Chain Rule
Substitute the known information
Solve for the unknown
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Example 2

• Suppose x and y are both differentiable functions
  of t and are related by . Find when x
  9, if when x 9 and ygt0.

Find the derivative by differentiating both sides.
Find other important values
x
Chain Rule
Substitute the known information
Solve for the unknown
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Example 3

• A spherical balloon is being filled with a gas in
  such a way that when the radius is 2ft, the
  radius is increasing at the rate 1/6 ft/min. How
  fast is the volume ( ) changing at
  this time?

Find the derivative by differentiating both sides.
Chain Rule
Substitute the known information
ft3 per minute
Solve for the unknown
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Related Rates Guidelines

1. Draw a figure, if appropriate, and assign
   variables to the quantities that vary. (Be
   careful not to label a quantity with a number
   unless it never changes in the problem)
2. Find a formula or equation that relates the
   variables. (Eliminate unnecessary variables)
3. Differentiate the equations. (typically
   implicitly)
4. Substitute specific numerical values and solve
   algebraically for any required rate. (The only
   unknown value should be the one that needs to be
   solved for.)

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Example 1

• A person 6 ft tall is walking away from a
  streetlight 20 ft high at the rate of 7 ft/s. At
  what rate is the length of the persons shadow
  increasing?

Find the rates by differentiating both sides.
20 ft
6 ft
Chain Rule
x
y
Substitute the known information
Using similar triangles, the equation is
Solve for the unknown
ft/s
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Example 2

• A bag is tied to the top of a 5 m ladder resting
  against a vertical wall. Suppose the ladder
  begins sliding down the wall in such a way that
  the foot of the ladder is moving away from the
  wall. How fast is the bag descending at the
  instant the foot of the ladder is 4 m from the
  wall and the foot is moving away at the rate of 2
  m/s?

Find the rates by differentiating both sides.
5 m
y
Ladder
Chain Rule
Substitute the known information
x
Find other important values
Using The Pythagorean Theorem, the equation is
Solve for the unknown
m/s
x
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Example 3

• A trough 10 ft long has a cross section that is
  an isosceles triangle 3 ft deep and 8 ft across.
  If water flows in at the rate 2 ft3/min, how fast
  is the surface rising when the water is 2 ft
  deep?

Find the rates by differentiating both sides.
10 ft
8 ft
Nothing is known about b
b
3 ft
h
Using similar triangles
Chain Rule
Substitute the known information
Using the volume of a prism, the equation is
Solve for the unknown
ft/min
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Example 4

• A rocket launches with a velocity of 550 miles
  per hour. 25 miles away there is a photographer
  filming the launch. At what rate is the angle of
  elevation of the camera changing when the rocket
  achieves an altitude of 25 miles?

Find the rates by differentiating both sides.
This is x and there is no x in the derivative
x
Chain Rule
Substitute the known information
T
25 mi
Use x to find other important values
Using The Trigonometry, the equation is
Solve for the unknown
rad/h