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Derivatives
are used to calculate the kinematics of a shuttle flight to orbit.
Since this is a science and mathematics class, we developed the derivative concept in
the context of kinematics, i.e., we saw the derivative arise as the
instantaneous growth rate of position. The derivative arises in many
contexts:
- Velocity is the instantaneous growth
rate of position as a function of time.
- Acceleration is
the instantaneous growth rate of velocity as a function of time.
- Slope is the instantaneous growth rate of f(x) as a
function of x. Thus, f ´ (x) is the slope of
the graph of y = f(x) at the point (x, f(x)).
- Density is the instantaneous
growth rate of mass (weight) as a function of volume.
- Force is the instantaneous growth rate of momentum as a
function of time.
- Marginal revenue is the instantaneous
growth rate of revenue as a function of level of production.
- Current is the instantaneous growth rate of electrical charge
as a function of time.
Do exercises
2-8,
2-9, and
2-10 to get
some experience with derivatives at play.
By now you have figured out (or have been informed) that for values
of x where the derivative of f(x) is positive, the function is
increasing, and for values of x where the derivative is negative,
the function is decreasing. So a high or low point (local maximum
or local minimum point) can occur only where
f ´ (x) = 0.
Do exercise
2-11 to look at
maxima and minima using a new Maple command.
Once mathematicians are onto a good thing, they try to get all they
can from it. Once they have computed the derivative of a function,
they are prone to ask: ``Since the derivative of a function is
another function, what happens when we take the derivative of the
derivative? Does this tell us anything about the original function?''
Of course, the answer to the latter question is ``yes'', because the
second derivative (as it is called, and denoted
f ´ ´ (x) or
) gives
information about how the instantaneous
growth rate is changing. So you can consider what it means for
f to be increasing at an increasing rate, increasing at a decreasing rate,
etc.. This all comes together graphically, as one sees that:
- where the second derivative of
f is positive,
the graph of the function is concave up (i.e., the driver of
a car travelling along the graph from left to right would have to
turn the wheel to the left)
- where the second derivative
of f is negative, the graph of the function is
concave down (i.e., that driver would be turning the
wheel to the right), and
- where the second derivative is zero, the graph has a ``flex''
(or a point of inflection).
For your working pleasure!!! A problem you are required to do ONLY by hand. Exercise
2-12 is a "Look, Ma, no
Maple" type problem.
Next: Differential Equations
Up: Describing How Things Change
Previous: Derivatives