11DIFFERENTIATION - Department of Mathematics

750 11 DIFFERENTIATION
11.5 Higher-Order Derivatives
That is, when the level of production is already 3000 VCRs, the actual cost of
producing one additional VCR is approximately $41. Next,
R(3000) 0.04(3000) 300 180
That is, the actual revenue to be realized from selling the 3001st VCR is approximately
$180. Finally,
P(3000) 0.000009(3000) 2 0.04(3000) 100
139
That is, the actual profit realized from selling the 3001st VCR is approximately $139.
2. We first solve the given demand equation for x in terms of p, obtaining
x f(p) 50p 15,000
f(p) 50
Therefore,
E(p) pf(p)
f(p)
p
50p 15,000 (50)
p
(0 p 300)
300 p
Next, we compute
E(100)
100 1
1
300 100 2
and we conclude that demand is inelastic when p 100. Also,
E(200)
200
2 1
300 200
and we see that demand is elastic when p 200.
H IGHER-ORDER D ERIVATIVES
The derivative f of a function f is also a function. As such, the differentiability
of f may be considered. Thus, the function f has a derivative f at a point
x in the domain of f if the limit of the quotient
f(x h) f(x)
h
exists as h approaches zero. In other words, it is the derivative of the first derivative.
The function f obtained in this manner is called the second derivative
of the function f, just as the derivative f of f is often called the first derivative
of f. Continuing in this fashion, we are led to considering the third, fourth,