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Unformatted text preview: M119 Notes, Lecture 11 2.4 Introduction to the second derivative The second derivative is a little more difficult than the derivative because it has 2 roles: The second derivative of fa) [concavity of 1(a)]
The derivative of f” (x) [rate of change of f’(x)] Thus for ﬁx) we have: 4M Ex. This will give us the followin_ table: ﬁx) is the height=y
f(x) is the slope of the tangent line=m
h}? ’(x) is the concavity (no standard letter) Means f 3:
Is above the xaxis
is below the xaxis
Has an xinterce t o . sitive 7'? n ative "0 o  a) «a «o  g “0'0“0'
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. . . . wk 9%
decreasm neyatlve 9 Is decreasm  0,25
9 has a critical o . int
concave u  increasin m
concave down decreasin ls concave down
neither a (has possible inﬂection
concavi chan . in . : oint
'F‘t’v) I s “‘1 “N"
If x) is decreasing, then (“no is unduemin
If :90 is increasmg,l:hen x “on 1:. 995mm. qael fee. is ”‘9‘“ U?
x is no ative. t en .
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(3 5. Y s i w. mewv _..7 Q‘ Posiiive. a? x We“ oma\ “gob M im‘miuﬂ —""7
M‘s—W‘— ‘7 Z é‘nxéq Q00 Wst‘ﬂﬂ n7 Q‘p{a$ Estimating the second derivative from a table (5 pt method)
To estimate the second derivative at a point, ﬁnd the derivative at the points to
either side, and then ﬁnd the average rate of change (slope) between those derivatives. If f(x) is given by the following table, ﬁnd f " (1 0) . 2.5 Marginal Cost and Revenue
As we stated before, the m in marginal should remind us of slope, and since gives us slope.
MR = R'(9‘);MC = 0(9) Local Linear Approximation
From last lecture, we learned that we can approximate:
f (x + Ax) e f (x) + f'(x)Ax . If we do this same formula with the cost function, we would have:
CG? + A91M" C(9)+MC(¢I) “‘39 Ex. At a production level of 2000 units, the total cost is $50000 and the marginal cost is $15/unit. Estimate the (total) costs in producing the following number of items:
A 2001 items 50909 "5‘ '~' 39.0% B) 1999 items Foam 4, 5‘1“” o—l items = “.99: c (.1400 +i)~¢caaeoh+c'c1mc>>
C (MD) ammo) a: 50w”* “'3’ Maximizing Profit L‘>'Tuw,, cow (bi. out: we F3059“?
If we look at the graph of the proﬁt function, we see that at the top of the graph, the slope is 1T(q) = 0 4: R'(q) —C'(q) = o c; MR = MC Ex. Cost and revenue functions are given below. Approximate what quantity maximizes proﬁt. Approximate the proﬁt at this point. 4000 lB000 Ex (10) The following table gives the cost and revenue, in dollars, for different production levels, q.
A) At approximately what production level 15 proﬁt maximized? Ll B) What is the price charged per unit for this product? I 00 OMKS Ci \€L\ 3 MIAIL. ?fﬁé~‘5 “Ml3W d‘.
C) What is the ﬁxed costs of production? Amp} W't Y re 'e <— “S P“; ”mm— 2500
minimum 1900 Ex. Same question, just a VERY different table The following table gives the cost and revenue, in dollars, for different production
levels, g. D) At approximately what production level is proﬁt maximized? B) What is the price charged per unit for this product? F) What is the ﬁxed costs of production? p1: Pq $700 Alla” ﬂ?) (1036' of:
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 Fall '11
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