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59. The cost of manufacturing x items is given by C(x) = 13:38
2. One way of numerically approximating a derivative is by
′
2
0.02x + 20x + 1800. Find the marginal cost function. Com- computing the slope of a secant line. For example, f (a) ≈
pare the marginal cost at x = 20 to the actual cost of produc- f(b) − f(a)
ing the 20th item. b − a , if b is close enough to a. In this exercise, we will
develop an analogous approximation to the second deriva-
60. For the cost function in exercise 59, find the value of x that tive. Instead of finding the secant line through two points
minimizes the average cost C(x) = C(x)∕x. on the curve, we find the parabola through three points
on the curve. The second derivative of this approximat-
ing parabola will serve as an approximation of the second
derivative of the curve. The first step is messy, so we rec-
ommend using a CAS if one is available. Find a function of
EXPLORATORY EXERCISES the form g(x) = ax + bx + c such that g(x ) = y , g(x ) = y
2
2
1
1
′′
1. Let n(t) be the number of photons in a laser field. One and g(x ) = y . Since g (x) = 2a, you actually only need to 2
3
3
′
model of the laser action is n (t) = an(t) − b[n(t)] , where a find the constant a. The so-called second difference approx-
2
′′
′′
′
and b are positive constants. If n(0) = a∕b, what is n (0)? imation to f (x) is the value of g (x) = 2a using the three
Based on this calculation, would n(t) increase, decrease or points x = x −Δx [y = f(x )], x = x [y = f(x )] and x =
1
3
1
2
2
2
1
′
neither? If n(0) > a∕b, is n (0) positive or negative? Based x +Δx [y = f(x )]. Find the second difference for f(x) =
3
3
on this calculation, would n(t) increase, decrease or nei- √ x + 4 at x = 0 with Δx = 0.5, Δx = 0.1 and Δx = 0.01.
′′
′
ther? If n(0) < a∕b, is n (0) positive or negative? Based on Compare to the exact value of the second derivative, f (0).
this calculation, would n(t) increase, decrease or neither? e − e −x
x
Putting this information together, conjecture the limit of 3. Forthehyperbolictangentfunctiontanh(x) = e + e −x ,show
x
n(t) as t → ∞. Repeat this analysis under the assumption that d tanh x > 0.Concludethattanh(x)hasaninversefunc-
that a < 0. dx
tion and find the derivative of the inverse function.

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318 | Lesson 4-9 | Rates of Change in Economics and the Sciences