Bonaventura Francesco Cavalieri
1598 – 1647
Bonaventura Cavalieri was an Italian mathematician who
developed a method of indivisibles which became a factor in
the development of the integral calculus.
Example
In conclusion,
Steps for Related Rates Problems:
I Introduction
-Read the problem carefully.
-Draw a picture (sketch), if possible.
-Introduce notation. Assign symbols to all quantities
that are functions of time.
II Known Derivative
Express the given rates in terms of derivatives.
III Unknown Derivative
Express the unknown rates in terms of derivatives.
IV Equation
Write an equation that relates the various quantities
of the problem. If necessary, use geometry of the
situation to eliminate one of the variables by
substitution.
Steps for Related Rates Problems:
V Differential Equation
Use the Chain Rule to differentiate both sides of the
equation with respect to t, also known as implicit
differentiation to find a differential equation.
VI Snap Shot
Substitute the given information into the resulting
equation and solve for the unknown derivative.
VII Conclusion
State your conclusion to the problem.
Example
A ladder 15ft in length leans against a vertical wall, with the bottom
of the ladder 5ft from the wall on a horizontal floor. If at that time
the bottom end of the ladder is being pulled away at the rate of
2 ft/sec, at what rate does the top of the ladder slip down the wall?
Example
Suppose a particle is moving on the curve x 2  xy  y 2  7. At
the point (1, 2) the particle is moving left at 3 feet per second.
How is the particle moving vertically at that point?
Example
Suppose a liquid is to be cleared of sediment by pouring it through
a cone-shaped filter. Assume the height of the cone is 16 inches
and the radius at the base of the cone is 4 inches.3 If the liquid is
flowing out of the cone at a constant rate of 2 in / min, how fast is
the is the depth of the liquid decreasing when the level is 8 inches
deep?
Example
A camera is mounted at a point 3000 feet from the base of a rocket
launching pad. Let us assume that the rocket rises vertically and the
camera is to take a series of photographs of the rocket. Because the
rocket will be rising, the angle of elevation of the camera will have to
vary at just the right rate to keep the rocket in sight. If the rocket
is rising vertically at 800 ft/sec when it is 4000 feet up, how fast must
the angle of elevation of the camera change at that instant to keep the
rocket in sight?
Example
Runners stand at first and second base in a baseball game. At the
moment a ball is hit, the runner at first base runs to second base
at 18 ft/s; simultaneously the runner on second runs to third base
at 20ft/s. How fast is the distance between the runners changing
1 second after the ball is hit? (Hint: The distance between consecutive
bases is 90 ft and the bases lie at the corners of a square.