derivative of position (or displacement) is
velocity, whose derivative is acceleration, whose
derivative is jerk, whose derivative is snap, whose
derivative is crackle, whose derivative is pop.
(linear) density is the derivative of (linear) mass
the word "marginal" in economics always means "derivative
of"
any "rate of change" is a derivative of some underlying
quantity
Related Rates; when in doubt, use the method:
clearly define all variables, with words and also (if possible) a
picture; notes:
there must be enough variables to do the next two steps
variables are things that vary -- no need to have a
variable for a quantity which does not change during the problem
(e.g., in "a 2m tall man walks..." you don't need a
variable for the man's height)
but don't put numbers on the picture (or into any
equations until the very end of you work) for quantities which
have a particular, stated value only at one instant (e.g., in
"how long is the man shadow when he is 3m from the wall"
you should not put that 3 in your diagram nor into your
calculus/algebra until the very end)
state the given information in terms of your variables
pose the problem's question in terms of your variables
find a "relationship equation" (or several) between the variables
in the previous two steps; these usually come from some standard
mathematics or other science, such as:
the Pythagorean Theorem
basic trigonometry (SOHCAHTOA)
the Law of Cosines or Law of Sines
forumlæ for the area or volume of some standard geometric
object (a triangle, circle, rectangle, sphere, pyramid,
rectangular parallelepiped)
some basic physics such Newton's Second Law or Law of Universal
Gravitation, or the relationship between position, velocity,
acceleration, jerk, snap, crackle, and pop.
use the above three steps to solve the problem, usually by
differentiating the relationship equation (use the Chain Rule
a lot!) and plugging in the known information to solve for the
unknown
restate the solution in the same terms with which the problem was
posed.
You do not have to follow those steps in exactly that order or
self-consciously announcing the steps as you do them. You do
have to have essentially all of the above content, and to explain what
the content is as you provide it. In particular, student most
frequently loose points for failing to define their variables or to
have a complete and coherenet explanatory narrative of their work.
linear approximations: approximate the value of a function at some
x by using the equation of the tangent line
y=f(a) + f′(a)(x-a) based at some nearby point
(a,f(a)).
max/min in the abstract:
definition of local (or relative) and global (or
absolute) maximum, minimum, and extremum
-- note the endpoint of an interval of definition of a function can
never be a local extremum. [Note the plural of the word extremum is
extrema).]
the Extreme Value Theorem: A continuous function defined on a
closed interval has a global maximum and global minimum in that
interval.
Fermat's Theorem: If a function f(x) has a local
extremum at some point c where f′(c)
exists, then necessarily f′(c)=0.
a critical number for a function f(x) is a value
c in the domain of f for which either the derivative
f′(c) does not exist or, if it exists, it equals
0.
the Closed Interval Method to find the extrema of a function
on an interval [a,b]:
find the critical numbers of f(x) inside the interval
(a,b) by computing the derivative
f′(x) and seeing where it fails to exist or
exists but equals 0 (Always check for both kinds of C#!)
for x in the interval
compute the values of f at each such critical number
and at the endpoints of the interval
pick off the largest and smallest values just computed: these are
the global extrema of f on the given interval
a pair of "existence theorems":
Rolle's Theorem: If f(x) is a function which
is continuous on some interval [a,b],
is differentiable on (a,b), and
satisfies f(a)=f(b),
then there exists some c in (a,b) such that
f′(c)=0.
the Mean Value Theorem [MVT]: If f(x) is a function
which
is continuous on some interval [a,b], and
is differentiable on (a,b),
then there exists some c in (a,b) such that
f′(c)=(f(b)-f(a))/(b-a).
When you use either of these theorems, you must check (and explain
why you believe) all of their hypotheses.
what first and second derivatives tell about the shape of the graph of a
function:
the Increasing/Decreasing Test:
If f′(x)>0 on some interval, then
f(x) is increasing on that interval.
If f′(x)<0 on some interval, then f(x)
is decreasing on that interval.
the First Derivative Test:
If f′(x) changes from positive to negative
at some value x=c, then f has a local maximum at
c.
If f′(x) changes from negative to positive
at some value x=c, then f has a local minimum at
c.
If f′(x) does not change sign at x=c,
then c is neither a local maximum nor local maximum for
f.
If, on some interval, the graph of f(x) lies above all of its
tangents, then we say that f(x) is concave upwards on
that interval; if below, we call it concave downwards.
the Concavity Test:
If f′′(x)>0 on some interval, then
f(x) is concave upwards on that interval.
If f′′(x)<0 on some interval, then
f(x) is concave downwards on that interval.
If the concavity of a function changes at a point, we call that point
an inflection point for the function.
the Second Derivative Test: If x=c is a critical number
for a function f(x) and f′′(x) exists
and is continuous near c, then:
if f′′(c)>0, then f has a local
minimum at c;
if f′′(c)<0, then f has a local
maximum at c;
if f′′(c)=0, then c might be a
local extremum for f, but it also might not.
limits to infinity:
we say the limit as x goes to ∞ of a function
f(x) is the number L if we can make the values of
f(x) as close to L as we like, by making x
sufficiently large (similarly if we are taking the limit to -∞,
except we make x sufficiently large and negative)
a theorem about powers:
if r>0 is a rational number, then the limit as x
goes to ∞ of 1/xr is 0;
if r>0 is a rational number such that xr
is defined for all x, then the limit as x
goes to -∞ of 1/xr is 0;
if r>0 is a rational number, then the limit as x
goes to ∞ of xr is ∞.
we generally compute limits to ±∞ by working with a
fraction, dividing the top and bottom by the appropriate power of
x (often the highest power, counting powers appropriately if
they are inside roots), using the Limit Laws and the above theorem
about powers until we get something with only legal arithmetic in
it
arithmetic which we can do with infinity:
(±c) ∞ = ± ∞, where c is
a finite, positive constant
∞ + ∞ = ∞
∞ . ∞ = ∞
arithmetic which we should absolutely avoid with infinity:
∞ - ∞
∞/∞
0 . ∞
graphing functions:
a strategy (or checklist) for graphing a function f(x)
from the book:
Domain
Intercepts (x and y)
Symmetry/periodicity
Asymptotes (horizontal, vertical, and slant)
slant asymptotes occur only in rational functions where the
numerator has degree exactly one more than the denominator;
in this case, the SA can be found by long division of
polynomials
Intervals of Increase and Decrease
Local Extrema
Concavity and Points of Inflection
Sketch the curve
Note that it is permissible in our class to estimate all of the
above features based on the graph from your calculator, but, unless
specifically told not to, you should always also use calculus
techniques to find exact values.
pitfalls with calculators: mostly, it amounts to finding the right
window, so it helps to use calculus to find local extrema using
calculus, for example, and then to choose a window that contains them
optimization problems
a method can be very similar to the above method for Related Rates
problems
clearly define all variables -- here, the criteria usually is to
have enough variables to describe completely the problem scenario
and how it can vary
find any given information in terms of your variables; also, find
an admissible interval for the quantities you can vary, if there
is one
state the goal, to maximize or minimize some objective or
target function
find as many relationship equations between your variables as you
can: often, there will be one or more equations which yield a
constraint equation for your input (independent) variables
solve: often, this involves using the constraint equation(s) to
eliminate all but one independent variable in your target
function, so that you can then either use
the Closed Interval Method (if there was an interval) or
techniques from the graphing section, above, to find a global
extremum even when there is no closed interval
state the answer in the terms in which the problem was posed.