Chapter 11 cont'

1
Chapter 11 (cont.)

• Differentiation

2
Review

• Last week, we covered
• Limits of difference quotients, interpreting as
  the slope or derivative a a function
• Rules for differentiation
• Constant function rule
• Constant factor rule
• Power function rule
• Sum or difference rule

3
Examples of Derivatives

• y f(x) 5 dy/dx 0
• Y f(x) x80 dy/dx 80x79
• Y f(x) (2/3)x4 dy/dx (8/3)x3
• Y f(x) 4x2 2x 3 dy/dx 8x 2
• Y f(x) x3/3 dy/dx 3x2/3 3x2
• Y f(x) -8x4 ln(2) dy/dx -32x3

4
Derivation of Rules

• All rules for taking derivatives can be derived
  from the limit of a difference quotient
• Example Let f(x) 4x2 2x 3.
• Find the limit as h ? 0 of f(x h) f(x)h
• So, the difference quotient is
• 4(xh)2 2(xh)3 4x2 2x 3/h
• 4x2 8xh 4h2 2x 2h 3 - 4x2 2x -3/h
• (8xh 4h2 2h)/h 8x 4h 2
• So, lim as h ? 0 8x 2
• Note that this is the answer on previous slide

5
Evaluating Derivatives

• y f(x) 2x(x2 5x 2)
• Evaluate the derivative when x 2
• Y 2x3 10x2 4x
• dy/dx 6x2 20x 4
• If x 2 dy/dx f (x) 24 - 40412
• Interpretation If x 2, the slope of f(x)
  equals 12 that is if x changes by one unit then
  y changes in the same direction by 12 units

6
Another Example

• Let y f(x) (3x2 2)/x
• Write as y 3x 2x-1
• dy/dx 3 (-1)2x-2 3 2x-2
• 3 2/x2
• Suppose that x 1 dy/dx 5
• Interpretation if x 1, the slope of the
  function is 5
• If x 2, the slope is 3.5 so from x 2, is x
  changes by one unit then y changes by 3.5 units
  in the same direction

7
Tangent Lines

• Let y f(x) (3x2 2)/x
• Find the equation if a (straight) line that is
  tangent to f(x) at the point x 1.
• The tangent line will have the same slope as f(x)
  at x 1.
• The tangent line will pass through the same
  y-point as f(x) when x 1.

8
Graphical Interpretation
y
f(x) (3x2 2)/x
Tangent line
x
x 0.82
9
Equation of Tangent Line

• We know that y f(x) (3x2 2)/x
• We know that dy/dx 5 if x 1
• We know that if x 1, f(x) 1
• So to find the tangent line, we need to find the
  equation of a line with slope of 5 that passes
  through the point y 1, x 1. Using
  point-slope formula
• (y 1) 5(x 1), so y 5x - 4

10
Homework

• P. 552 75, 79, 81, 83, 85

11
Applications

• Interpretations of a derivative
• Limit as h ? 0 of a difference quotient
• Slope of a function
• Rate of change in a function
• Total cost and marginal cost
• Total revenue and marginal revenue
• Total product and marginal product

12
Costs

• Suppose that c 0.1q2 3. Find marginal and
  average costs if q 4.
• MC dc/dq 0.2q. So, if q 4, MC 0.8
  Interpretation If q 4, total cost rises by
  0.80 if q increases by a very small amount
  (remember MC is the limit of the difference
  quotient for total cost as h ? 0)
• How much does total cost change if q increases
  from 4 to q 5 ?c 5.5 4.6 0.90 gt 0.80
• So, strictly speaking MC is not the incremental
  cost of a one unit increase in q instead it is
  the incremental cost of an arbitrarily small
  increase in q, the rate of change in c with
  respect to q.

13
Costs (cont.)

• Average costs If total costs are given by, c
  0.1q2 3, then average costs would be
• AC c/q 0.1q 3/q

c
q
14
Relationship Between AC and MC

• What is the relationship between marginal cost
  and average cost?
• One way to approach this question is to ask At
  what value of q (if any) is AC MC?
• Set 0.1q 3/q 0.2q (AC MC)
• Or, 0.1q2 3 so q 301/2 5.48
• If q 5.48, then AC 0.55 0.55 1.10

15
Relationship (Cont.)

• Find the slope of a line tangent to the AC curve
  at the point q 5.48
• AC 0.1q 3/q 0.1q 3q-1
• dAC/dq 0.1 3(-1)q-2 0.1 3q-2
• dAC/dq 0.1 3 (5.48)-2 if q 5.48
• dAC/dq 0.1 3/30 0
• What does this outcome mean?

16
Interpretation

AC 0.1 3/q
Tangent Line has m 0
1.10
MC 0.2q
q
q 5.48
17
Revenues

• Suppose that a demand function is given by p
  30 -2q
• Find total revenue, average revenue, and marginal
  revenue
• TR pq 30q -2q2
• AR TR/q 30 2q (AR D)
• MR dTR/dq 30 4q, so MR is twice as steep as
  D

18
Graph of D and MR

MR is twice as steep as D
MR 30-4q
p 30 2q
q
19
Relationship Between TR and MR

• Recall that TR 30q 2q2
• Find the value of TR when MR 0
• MR 30 4q
• MR 0 if q 30/4 7.50
• If q 7.50, then TR 225 112.5 112.5
• Find the slope of the tangent line to TR when q
  7.5. This slope would be dTR/dq MR 0 if q
  7.5
• What price should be charged if q 7.5? From
  demand curve p 30 2(7.5) 15

20
Relationship (cont.)

Tangent line With m 0
MR 30 4q
TR 30q 2q2
q
q 7.5
21
Relative Rates of Change

• In economic and business situations, we sometimes
  want to know the percentage increase in revenues
  and costs associated with making and selling a
  little more or a little less output.
• In general, this percentage change (or relative
  rate of change) can be calculated by finding
• f (x)/f(x) dy/dx/f(x)
• This is just the derivative divided by the value
  of the function

22
Percentage Change in Costs

• Suppose that a total cost function is given by C
  0.4q2 4q 5
• So, MC 0.8q 4
• The relative rate of change in costs for a small
  increase in q is MC/C. Find this value for q 2
• If q 2, MC/C 5.6/13.6 0.412, so from the
  point q 2, total costs increase by 41.2 if q
  increases by a small amount

23
Percentage Change in Revenues

• Suppose that a demand function is given by p
  80 5q
• Then TR pq 80q 5q2
• MR 80 10q
• Find the percentage increase in TR from a small
  increase in q, if q is now 1.
• MR/TR 70/75 0.93
• If q 4, MR/TR 40/240 0.167
• If q 8, MR/TR 0 Why?

24
Homework

• Pp 561-563 9,11,19,21,23,25,35,41,45

25
More Differentiation Rules

• Product rule
• Suppose that y f(x)g(x)
• If f(x) and g(x) are differentiable functions
• dy/dx f (x)g(x) g (x)f(x)
• So, dy/dx is the derivative of f times g plus the
  derivative of g times f
• Quotient rule
• Suppose that y f(x)/g(x)
• Then dy/dx g(x)f (x) f (x)g(x)/g(x)2

26
Examples

• Let y (x2 3x)(4x 5)
• Denote f(x) (x2 3x) g(x) (4x 5)
• Then, dy/dx f (x)g(x) g (x)f(x)
• So, dy/dx (2x 3)(4x 5)4(x2 3x)
• 8x2 22x 15 4x2 12x
• 12x2 34x 15
• Another method to obtain this derivative is to
  first multiply f(x)g(x) and then take the
  derivative

27
Another Example

• Suppose y (x2/3 3)(x-1/3 5x), find dy/dx
• dy/dx (2/3)x-1/3(x-1/3 5x)
  (x2/3 3)(-1/3)x-4/3 5)
• Note that another approach to computing this
  derivative would be to expand the product of two
  terms and then take the derivative

28
Yet Another Example

• Suppose y (x 2)(x 3)(x 4)
• dy/dx (x 3)(x 4) (x 2)(x4) (x
  2)(x 3)
• So, in general, we can use the product rule as
  follows
• (fg) fg gf
• (fgh) fgh gfh hfg

29
Cost Example

• Suppose that a cost function is of the form c
  (4q 1)(2q 3)
• Find marginal cost and evaluate MC when q 5
• MC dc/dq 4(2q 3) 2(4q 1)
• If q 5, MC 4(13) 2(21) 94
• So, to evaluate MC, no need to simplifyjust plug
  in the value for q

30
Quotient Rule

• Suppose that y f(x)/g(x), then dy/dx g(x)f
  (x) f (x)g(x)/g(x)2
• Suppose that y (4x23)/(2x- 1)
• This is in the form y f(x)/g(x)
• dy/dx (2x-1)8x (4x23)2/(2x-1)2
• 16x2 8x 8x2 6/(2x-1)2
• 8x2 8x 6/(2x 1)2

31
Example

• Suppose y 1/x 1/(x 1)
• 1/x(x 1) 1/(x 1)
• (x 1)/(x2 x 1) (x 1)/D
• dy/dx (x2x1)(1)-(x1)(2x1)/D2
• (x2x1-2x2-x-2x-1)/D2
• -(x2 2x)/D2

32
Examples

• Suppose that a demand function is
• p 1000/(q 5)
• Find marginal revenue
• TR pq 1000q/(q 5)
• MR (q5)(1000) 1000q(1)/(q5)2
• 1000q 5000 1000q/(q 5)2
• 5000/(q 5)2
• If q 45, MR 2 if q 65.71, MR 1
• Note that MR never 0 increases in q just make
  MR ever smaller

33
Consumption Function

• Let C 5(2I3/2 3)/(I 10), where C
  consumption and I Income
• What is the marginal propensity to consume?
• dC/dII 10)(3/2)(10)I1/2 -
  5(2I3/2 3)1/(I 10)2
• So, when I 100, MPC 0.536

34
Another Example

• A consol is perpetual annuity and its value today
  is V 1/r assuming that the annuity is 1 per
  year
• What is the rate of change in V for a small
  change in the interest rate, r?
• dV/dr r(0) 1/r2 -1/r2
• Thus, as r rises, V declines and vice-versa
• Suppose we wish to compute the percentage change
  in V for a small change in r.
• (dV/dr)/V -(1/r2)/(1/r) -1/r

35
Consol (Cont.)

• Suppose that r 0.05, V 20 and a small
  increase in r will reduce V by 20
• Suppose that r 0.10, V 10 and a small
  increase in r will reduce V by 10
• Suppose that r 0.11, V 9.09 and a small
  increase in r will reduce V by 9.09
• Suppose that r 0.12, V 8.33 and a small
  increase in r will reduce V by 8.33

36
Consol (cont.)
V
V 1/r
V declines by less and less with successive
increases in r
r
37
Another Example

• Suppose that we buy a zero coupon bond today that
  matures in one year for 1. How much is it worth
  today if the interest rate is r? V 1/(1 r)
• What is the rate of change in V for a small
  change in r?
• dV/dr (1r)01/(1 r)2 -1/(1r)2
• So, again we see that V varies inversely with r

38
Comparison to Consol

• Find the percentage change in V for a small
  change in r
• This is (dV/dr)/V -1/(1r)
• Suppose that r 0.05 V 0.95 and a small
  change in r will change V by 0.95
• Suppose that r 0.10 V 0.91 and a small
  change in r will change V by 0.91
• Suppose that r 0.20 V 0.83 and a small
  change in r will change V by 0.83

39
Homework

• Pp 573-574 1,15,19,21,27,45,49, 61,69,71 Do as
  many exercises as you can besides these.