This document discusses the concept of related rates in calculus. Related rates examine how two or more quantities that are dependent on each other change with respect to time. Examples are provided of using derivatives to determine the instantaneous rate of change of one quantity given the rate of change of a related quantity. The key aspects are that both quantities are changing, they are related to each other, and the task is to determine the relationship between their rates of change at a particular point in time.
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Calculus is the mathematics of ____________,
time
and everything changes with respect to __________.
variables
In algebra we study the relationship between ______________.
x = horizontal displacement y = vertical displacement
rate change
In calculus we study the relationship between the ________ of ______________ of variables.
Instantaneous rate of change of dy Instantaneous rate of change of
dx = vertical displacement
dt = horizontal displacement dt
with respect to with respect to
time. time.
http://mathdemos.gcsu.edu/mathdemos/relatedrates/relatedrates.html
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3. Let V = volume dV/dt = 2gal/min.
Let V = volume dV/dt = 3 cm3/s
Let h = height dh/dt = 30 yd/s
3
5. y
θ
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or
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6. h
r
dr/dt = instantaneous rate of change of radius with respect to time
dh/dt = instantaneous rate of change of height with respect to time
dV/dt = instantaneous rate of change of volume with respect to time
dV/dt = 5π/3 ≈ 5.2 in3/s
A positive value means that the volume is increasing.
dh/dt = 15/π + 1 ≈ 3.8 in3/s.
A negative value means that the volume is decreasing.
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9. Related Rates
You are given two quantities
• Both are changing
• They have something to do with each other.
You are asked to find out how their rates of change are related to each other
at some particular instant in time.
On tougher related rate problems, you will need to do some algebra
to get from what you know to what you are asked to find.
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