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2.1 Instantaneous rate of change
1
2.1 Instantaneous rate of change
Instantaneous rate of change of a function f at the number a (or just the rate of
change of a function f at the number a) is the limit of the average rate of change
of f over shorter and shorter intervals around a.
Recall that the phrase rate of change is replaced with velocity when the function is
the position s(t) of an object along a line at time t.
Recall that the phrase rate of change is replaced with marginal when the function
comes from economics.
Enough to look at intervals of length a > 0 of the form [a, a+a] and [aa, a]
and then let the length a be smaller and smaller, we write a 0.
The average rate of change over the interval [a, a + a] is of course
f (a + a) f (a)
a
Now let a 0 to obtain rate of change of f at a. 2
2.1 Derivative, visualization
The derivative of a function f at the number a, denoted f (a) is the same as the rate
of change of f at a.
The derivative f (a) of a function f at the point a is the slope of the tangent to the
graph of f at the point (a, f (a)).
The equation for the tangent to the graph at the point (a, f (a)) with slope f (a) is
then
y f (a) = f (a) · (x a)
or equivalently
y = f (a) · x + (f (a) f (a) · a)
When a function f is given by a graph one is able to give relative statements about
the derivative at a point a (f.ex. f (a) > 0 and at dierent points a and b (f.ex.
f (a) < f (b)).
When a function is given by a graph or partly by a table one is sometimes able to give
an estimate of the derivative. 3
2.2 The Derivative function
The derivative function (or just derivative of a function f is denoted f at its value
at the number x, is f (x).
If the derivative function f (x) > 0 on an interval the the original function f (x) is
increasing over that interval.
If the derivative function f (x) < 0 on an interval the the original function f (x) is
decreasing over that interval.
If the derivative function f (x) = 0 on an interval the the original function f (x) is
constant over that interval. 4
2.3 On units of derivatives and approximation.
Another notation for the derivative FUNCTION f (x) of a function y = f (x) is
Leibnizs notation
f (x) = dy
dx
One can thing about this as
Dierence in y
Dierence in x
The UNITS of the derivative of a function is the unit of the dependent variable divided
with the unit of the independent variable.
The derivative f (a) of a function f at the point a can be used to approximate values
f (a + a) of the function f when the input a + a is close to a, i.e., a has small
magnitude. The formula is
f (a + a) f (a) + f (a) · a 5
2.4 The second derivative.
The derivative function f (x) of a function f (x) (can) have a derivative function
denoted by f (x) and called the second derivative (function) of the function f (x).
The second derivative is thus the rate of change of the rate of change.
Thus if f (x) > 0 [< 0] on an interval then f (x) is increasing [decreasing] on that
interval. Thus
if f (x) > 0 [< 0] on an interval then the original function f (x) is concave up
[concave down] on that interval.
The unit of the second derivative f (x) is thus the unit of the dependent variable divided
by the square of the unit of the independent variable for the the original function f (x).
In the case of position s(t) as a function of time the (rst) derivative was called velocity
v(t) = s (t). The second derivative a(t) = s (t) = v (t) is called acceleration.
The second derivative in economics is then the marginal of the marginal · · ·. 6
2.5 Economics
Revenue function R(q) and cost functions C(q) need not be linear as in Chapter 1.
The denition of the prot function P (q) is unchanged P (q) = R(q) C(q), but
is then necessary NOT linear anymore.
If the graphs of R(q) and C(q) are drawn in the same coordinate system then we rec-
ognize the quantities q for which the prot is positive [negative, zero] as those quantities
q where the graph for the revenue function is above [below, intersect] the graph for the
cost function.
The quantity (quantities) q
max
corresponding to the MAXIMAL prot is that (those)
quantity (quantities) q where graph of the revenue function is above the graph of the
cost function and and where the vertical distance(s) between the two graphs is(are)
greatest.
The marginal Revenue function M R, the marginal Cost function M C, and the marginal
Prot function N P , is thus the derivatives M R(q) = R (q), M C(q) = C (q),
and P (q), respectively. The quantity q
max
corresponding to the MAXIMAL prot
can often be found among those quantities q that solves the equation R (q) = C (q).