Summary Of Derivatives And Other Stuff
For f, f ' and f ", places where they are zero or fail to exist are all important.
f
Set f(x) = 0. For any number, a, such that f(a) = 0, then (a, 0) is an x-intercept for f.
Consider when f(x) does not exist. These are places where you MIGHT have a vertical asymptote. For any number, a, such that f(a) does not exist, look at the limit of f as x goes to a. If that limit is either positive or negative infinity on either side then x = a is a vertical asymptote for f.
f tells us points on the graph. Given an x, (x, f(x)) is a point on the graph of f.
f '
Set f '(x) = 0. For any number, a, such that f '(a) = 0, then a is a critical number for f. Likewise, for any number, a, such that f '(a) does not exist BUT f(a) DOES exist, then a is a critical number for f. Numbers, a, such that f '(a) and f(a) do not exist are NOT critical numbers but will still be important.
f' tells us the slope (derivative) of the curve at each point. The critical numbers and other points where f' does not exist are places where the slope can change from positive to negative or vice versa.
The critical numbers are locations of POSSIBLE maxima or minima. The first or second derivative tests are the ways we will determine whether or not such points are extrema.
f "
Set f "(x) = 0. For any number, a, such that f "(a) = 0, then a is the x-coordinate for a possible inflection point for f. Likewise, for any number, a, such that f "(a) does not exist BUT f(a) DOES exist, then a is the x-coordinate for a possible inflection point for f. Numbers, a, such that f "(a) and f(a) do not exist are NOT x-coordinates for possible inflection points but will still be important.
f" tells us whether the graph of f is concave up or concave down.
The places where f "(x) = 0 or does not exist are places where the concavity of the graph of f can change.
y-intercept
(0, f(0)), provided f(0) exists,
is the y-intercept of f.
First Derivative Test
Let a be a critical number for f. Pick a number, b, to the left of a (but to the right of the next closest place where f '(x) = 0 or does not exist). Pick a number, c, to the right of a (but to the left of the next closest place where f '(x) = 0 or does not exist). If f '(b) < 0 and f '(c) > 0 then f (a) is a local minimum for f. If f '(b) > 0 and f '(c) < 0 then f (a) is a local maximum for f. If f '(b) < 0 and f '(c) < 0, or f '(b) > 0 and f '(c) > 0, then f(a) is neither a local minimum nor a local maximum for f.
NOTE: The extremum is the value of f, not f '. Also, the extremum is f(x) (that is, the y-value), not the x-value.
Second Derivative Test
Let a be a critical number for f. Evaluate f "(a). If f "(a) < 0, then f (a) is a local maximum for f. If f "(a) > 0, then f(a) is a local minimum for f. If f "(a) = 0 or does not exist, then the second derivative test fails. f(a) may be a local maximum, a local minimum or neither.
NOTE:
The extremum is the value of f, not f '.
Also, the extremum is f(x) (that is, the y-value), not the x-value.
Inflection Points
Let a be the x-coordinate of a possible inflection point for f. Pick a number, b, to the left of a (but to the right of the next closest place where f "(x) = 0 or does not exist). Pick a number, c, to the right of a (but to the left of the next closest place where f "(x) = 0 or does not exist). If f "(b) < 0 and f "(c) > 0 then (a, f (a)) is an inflection point for f. If f "(b) > 0 and f "(c) < 0 then (a, f (a)) is an inflection point for f. If f "(b) and f "(c) are the same sign, the (a, f (a)) is not an inflection point for f.
NOTE: The y-coordinate of the inflection point is the value of f, not f ".
Asymptotes
y = k is a horizontal asymptote for f if the limit of f as x goes to infinity or negative infinity is k. All you need to do here is just take the two limits.
x = k is a vertical asymptote for f if the limit of f as x goes to k is either positive or negative infinity on either side then x = k is a vertical asymptote for f. As mentioned above, look at places where f(x) does not exist. These are possible locations for vertical asymptotes.
y = mx + b is a slant asymptote for f if, as x gets infinitely (or negatively infinitely) large, f(x) approaches y = mx + b. This is most generally going to happen with rational functions (functions that are a polynomial over a polynomial) though they can happen in other cases such as
f(x) = x + (sin x)/x.
Absolute Extrema
f(a) is an absolute maximum of f if f(a)
> f(x) for all x.
f(a) is an absolute minimum of f if f(a) < f(x) for all x.
There is no
simple method that always works for finding absolute extrema. Local extrema are possible absolute
extrema. Analyzing what happens to the
slope away from local extrema can tell whether or not an absolute extremum
exists. For example, if f(a) is a local maximum of f, f '(x) <
0 for all x < a, and f '(x) > 0
for all x > a, then f(a) would be an absolute minimum. This is because the function would always be
decreasing until x = a and then
increasing from then on.
crit # x-int y-int loc min
loc max abs min abs max int conc up
int conc down int incr int decr poss infl
infl vert asy horiz asy slant asy