Page 286 - Calculus Workbook For Dummies

270 Part V: The Part of Tens

Extrema, Sign Changes,

and the First Derivative

When the sign of the first derivative changes from positive to negative or vice-versa,
that means that you went up then down (and thus passed over the top of a hill, a local
max), or you went down then up (and thus passed through the bottom of a valley, a
local min). In both of these cases of local extrema, the first derivative usually will equal
zero, though it may be undefined (if the local extremum is at a cusp). Also, note that if
the first derivative equals zero, you may have a horizontal inflection point rather than a
local extremum.

The Second Derivative and Concavity

A positive second derivative tells you that a function is concave up (like a spoon holding
water or like a smile). A negative second derivative means concave down (like a spoon
spilling water or like a frown).

Inflection Points and Sign Changes in

the Second Derivative

Note the very nice parallels between second derivative sign changes and first derivative
sign changes described in the section above.
When the sign of the second derivative changes from positive to negative or vice-versa,
that means that the concavity of the function changed from up to down or down to up.
In either case, you’re likely at an inflection point (though you could be at a cusp). At an
inflection point, the second derivative will usually equal zero, though it may be unde-
fined if there’s a vertical tangent at the inflection point. Also, if the second derivative
equals zero, that does not guarantee that you’re at an inflection point. The second
derivative can equal zero at a point where the function is concave up or down (like, for
4
example, at x = 0 on the curve y = x ).

The Product Rule

The derivative of a product of two functions equals the derivative of the first times the
d
second plus the first times the derivative of the second. In symbols, ^ uv = l uvl.
u v +
h
dx
The Quotient Rule

The derivative of a quotient of two functions equals the derivative of the top times the
bottom minus the top times the derivative of the bottom, all over the bottom squared.
d u u v - uvl
l
In symbols, c m = .
dx v v 2