Calculus

1
Calculus

• Slope of a quadratic function
• Formal definition of a derivative
• Derivatives of power functions
• Derivatives of sums of power functions
• Derivatives of exponential functions and
  Logarithmic functions
• Product Rule, Quotient Rule and Chain Rule
• Applications

2
Calculus

• Mathematical way of measuring rates of change
• QbmP then m ?Q/?P
• m by how much does Q change if P changes by 1
  unit
• Q1000-10P
• For every 1 increase in P, Q decreases by 10
  units
• Q1000-P/2
• For every 1 increase in P, Q decreases by ½ a
  unit
• For every 2 increase in P, Q decreases by 1
  unit
• C1005Q
• For every 1 unit increase in Q, C increases by
  5.

3
CalculusSlope of a quadratic

• What about R1000P-10P2?
• If P increases by a small amount by how much does
  revenue change?
• Depends upon the value of P
• Linear functions e.g. QbmP, have the defining
  property that the slope doesnt change with P.

4
CalculusSlope of a quadratic

• Problem Find the slope of a nonlinear function
• R1000P-10P2, how does the slope of this function
  change with P?
• What is the slope at P30?
• When P30,R21000
• When P40, R24000
• Slope of line connecting these points is
  3000/10300

5
CalculusSlope of a quadratic

• What is the slope at P30?
• When P30,R21000
• When P35,R22750
• Slope of line connecting these points is
  1750/5350
• When P31, R21390
• Slope connecting (30,2100) and (31,21390)390

6
CalculusSlope of a quadratic

• Develop a mathematical expression which relates
  the slope between points
• (P30,R21000) and
• (P30?, R at P30? )
• to ?
• Slope ?R / ?P
• Need an expression which relates ?R / ?P to ?
• ?R to ?,
• ?P to ?

7
CalculusSlope of a quadratic

• Relating DR to d
• What is R when P30 ? ?
• R1000P-10P2
• 1000(30? )-10(30?)2
• 30000 ?1000-10(9002???30?2)
• 21000 ?400-10?2
• What is R when P30?
• R21000
• What is ?R?
• ?R 21000 400 ? -10?2 -21000
• 400? -10?2

8
CalculusSlope of a quadratic

• Relating DP to d
• DP30d-30d
• Relating the slope to d
• SlopeDR/DP
• Slope(400d-10d2)/d
• Slope400-10d

9
CalculusSlope of a quadratic Slope at
point?Slope of Tangent

• Slope of curve at P30 equals slope of tangent
• Tangent ? d?0
• Slope at P30?
• What happens to the slope as d?0?
• Slope400-10d
• As d?0, slope ?400
• If R1000P-10P2, the slope of this curve at P30
  is 400
• Interpretation?
• If price is currently 30 then for the rate of
  increase of revenue, (with respect to price) is
  400.

10
CalculusSlope of a quadratic

• What about the slope at P20?, P40? P60?
• Slope DR/DP
• DR is the change in R when P? Pd.
• When price is at P R1000P-10P2
• When price is at Pd R1000(Pd)-10(Pd)2
• 1000P1000d-10(P22Pdd2)
• 1000P1000d-10P2-20Pd-10d2
• DR1000P1000d-10P2-20Pd-10d2-1000P10P2
• 1000d- 20Pd-10d2
• DP is the change in P when P? Pd DPd

11
CalculusSlope of a quadratic

• Slope (1000d- 20Pd-10d2 )/d
• 1000-20P-10d
• Slope as d?0?
• Slope ? 1000-20P
• Slope at P30?
• 1000-20 ? 30400
• At P40?
• 1000-20 ? 40200
• At P70?
• 1000-20 ? 70-400
• At P50?
• 1000-20 ? 500

12
CalculusSlope of a quadratic

• For all functions of the form
• yax2bxc
• Slope2axb
• We call the slope of a function yf(x) the first
  derivative of f(x), and write it as
• dy/dx
• ?y/?x
• f?(x) where yf(x)

13
CalculusSlope of a quadratic Examples

• Rule Slope2axb
• y3x24x-7
• dy/dx6x4
• yx2-1
• dy/dx2x
• y5x2
• dy/dx5
• y2x2-10x1
• dy/dx4x-10
• yx2/2-x
• dy/dxx-1
• y3
• dy/dx0

14
Calculus Slope of a quadratic Examples

• Q If R400P-12P2 and the current price is 22
  what is the rate of change of revenue?
• dR/dP400-24P
• When P22 dR/dP400-24 ? 22-128
• A The rate of change of revenue is -128
• Q If profit, Pr is given by Pr-8P2250P-330,
  what will be the effect on profit of small
  increase in price , if P10
• dPr/dP250-16P
• 250-16 ? 10
• 90
• A The rate of change of Profit is 90

15
Calculus Formal Definition
16
CalculusDerivatives of power functions

• Power functions definition
• yaxm
• If yf(x)axm, then
• e.g. y2x, i.e. y is a linear function of x
• m1, a2
• dy/dx2
• y5x2, i.e. y is a quadratic function of x
• m2, a5
• dy/dx10x

17
CalculusDerivatives of power functions

• y2x3
• a2,m3
• dy/dx6x2
• y3?x
• a3, m1/2
• y1/(4x)
• a1/4, m-1

18
CalculusDerivatives of Sums of functions

• If yx2 3x then
• dy/dx2x3
• Let ux2 and v3x then
• dy/dxdu/dx dv/dx
• In general if yf(x)g(x) then
• dy/dxf ?(x)g?(x)
• In words, the derivative of a sum of functions is
  the sum of the derivatives

19
CalculusDerivatives of Sums of functions

• Example
• If y3x24x-1
• dy/dx6x4
• If y5x4-12x
• dy/dx20x3-12
• If y x2 1/x
• dy/dx2x-1/x2
• If yx3/2x2-5x1
• dy/dx3x2/22x-5

20
CalculusDerivatives of Sums of functions

• R240P-30P2, what is the rate of change of R when
  P7
• dR/dP240-60P -180 when P7
• Pr40P-P2-300, what is the rate of change of Pr,
  when P15 and when P20
• dPr/dP40-2P 10 when P15 dPr/dP0 when P20

21
CalculusDerivatives of Sums of functions

• If Sales (S) are related advertising expenditure
  (A) by the expression
• S1200500A 0.6, what is the rate of change of
  sales when A0.5?A10?
• dPr/dP500 ? 0.6A-0.4
• 396 when A0.5
• 119 when A10
• How would your answer change if
• S1200500A 0.06?
• dPr/dP500 ? 0.06A-0.94
• 576 when A0.5
• 34 when A10

22
Calculus Derivatives of Log functions

• If yf(x)log(x) then

23
Calculus Derivatives of Log functions

• yln(x2)
• g(x)x2 g?(x)2x
• dy/dx2x/x22/x
• yln(5x3-3x-1)
• g(x) 5x3-3x-1 g?(x)15x2-3
• dy/dx (15x2-3)/(5x3-3x-1)
• yln(52?x)
• g(x) 52?x g?(x)1/?x
• dy/dx(1/?x) ? 1/ (52?x)
• 1/ (5?x2x)
• yln(1x)3ln(x33x23x1)
• g(x)(1x)3 g?(x)3x26x3
• dy/dx (3x26x3)/ (1x)3
• 3(x22x1)/(1x)3
• 3(x1)2/(x1)33/(1x)

• If yf(x)lng(x) then
• Where g?(x)dg(x)/dx

24
CalculusDerivatives of Log Functions

• Example
• The supply of Qs units of a product at a price of
  P is given by
• Qs2510ln(2P1)
• What is the rate of change of supply with respect
  to price?

• dQs/dP20/(2P1)

25
CalculusDerivatives of Log functions

• Recall that rcln(1r), where rccontinuous
  interest rate and rordinary compound interest
  rate. Graph the relationship between rc and r
  and calculate the derivative of rc with respect
  to r

26
CalculusDerivatives of Exponential Functions

• If yf(x)Abx then
• In the special case where be then

• y5ex
• dy/dx5ex
• y100(1.01)x
• dy/dx100(1.01)xln(1.01)
• y2 ? 3x
• dy/dx2?3xln(3)

27
CalculusDerivatives of Exponential
FunctionsExample

• If yf(x)Aeg(x) then

• If ye2x then
• g(x)2x g?(x)2
• dy/dx2e2x
• If yex2 then
• g(x)x2 g?(x)2x dy/dx2xe(x2)
• If ye(x35x2-3) then
• g(x)x35x2-3 g?(x)3x210x
• dy/dx(3x210x)e(x35x2-3)
• If yeln(x)
• g(x)ln(x) g?(x)1/x
• dy/dx1/x?eln(x)x/x1

28
CalculusDerivatives of Exponential Functions

• Suppose you have 100 to invest. A contract
  specifies that you will receive 8 p.a., where
  interest is quoted as ordinary compound rate.
  What is the rate of increase of your investment
  after 18 months?
• SP(1r)T r0.08 P100
• dS/dT100?ln(1r)?(1r)T
• at T18/12 years dS/dT100?ln(1.08)?(1.08)1.5
• 8.64

29
CalculusDerivatives of Exponential Functions

• What would be your answer to the previous
  question if you converted r to rc and used the
  appropriate growth formula?
• rcln(1r)ln(1.08)0.077
• Appropriate growth formula
• SPercT
• dS/dTPrcercT100?0.077?e(0.077?1.5)8.64

30
CalculusDerivatives of Products of functions

• What is the derivative of yx2(2x-1)?
• y2x3-x2 ?dy/dx6x2-2x
• If y x2(2x-1) then y can be written in the form
  yf(x)g(x), where f(x)x2 and g(x)2x-1
• Product Rule
• If yf(x)g(x) then
• dy/dxf(x)g ?(x) f ?(x)g(x)
• If f(x)x2 and g(x)2x-1 and yx2(2x-1), then
• f?(x)2x
• g ?(x) 2
• using product rule
• dy/dx2x22x(2x-1) 2x24x2-2x6x2-2x

31
Calculus Product Rule dy/dxf(x)g?(x)f??(x)g(x)

• y3x ? ln(x2-1)
• f(x)3x f?(x)3 g(x)ln(x2-1) g?(x) 2x/(x2-1)
• dy/dx3x ? 2x/(x2-1)3 ? ln(x2-1)6x2/(x2-1)3 ?
  ln(x2-1)
• yx2e4x
• f(x)x2f ?(x)2x g(x)e4x g?(x) 4e4x
• dy/dxx2 ? 4e4x2x ? e4x2x ? e4x(2x1)
• y(4x-1)1.2x ?xln(x)
• f1(x)4x-1f1?(x)4 g1(x)1.2x g1 ?(x) 1.2x ?
  ln(1.2)
• f2(x) ?x f2?(x)1/(2?x) g2(x)ln(x)
  g2?(x)1/x
• dy/dx (4x-1) ? 1.2x ln(1.2) 4 ? 1.2x ?x /x
  1/(2?x)ln(x)
• dy/dx (4x-1) ? ln(1.2) ? 1.2x 4 ? 1.2x
  1/(2?x)ln(x)1/?x

32
CalculusDerivatives of Ratios of functions

• Quotient Rule
• If yf(x)/g(x) then
• dy/dxf ?(x)g(x) - f(x)g?(x)/g(x)2
• Examples
• yx2/(x1)
• f(x)x2 f?(x)2x g(x) (x1) g ?(x)1
• dy/dx 2x(x1)-x2/(x1)2(x22x)/(x1)2
• yex/(1ex)
• f(x)ex f ?(x)ex g(x)1exg ?(x)ex
• dy/dxex(1ex)-ex.ex/(1ex)2
• dy/dxex/(1ex)2

33
Example Quotient Rule

• Recall the hyperbolic function
• Q(80-2P)/(3P/81)
• What is the rate of decrease in Q when P5?
• Let f(P)80-2P f ?(P)-2
• Let g(P)3P/81 g ?(P)3/8
• dQ/dP-2(3P/81)-(80-2P)3/8/(3P/81)2
• dQ/dP-32/(3P/81)2
• P5? dQ/dP-32/(3 ? 5/81)2 -3.9
• P35 ? dQ/dP-32/(3 ? 35/81)2 -0.16

34
Calculus Chain Rule

• What is the derivative of y(1x2)2?
• y (1x2)2 12x2x4
• dy/dx4x4x34x(1x2)
• What is the derivative of y(1x2)-21/(1x2)2?
• Let z(1x2) then yz-2
• dy/dz-2z-3-2/z3
• dz/dx2x
• Chain Rule dy/dx(dy/dz)(dz/dx)
• dy/dx-2(1x2)-32x
• dy/dx-4x/(1x2)3

• In general if yf(z) and zg(x) then
• dy/dx(dy/dz)(dz/dx)
• y(x210)1/2
• Let zx210 ? yz1/2
• dy/dz1/(2?z) 1/(2?(x210))
• dz/dx2x
• dy/dx2x /(2?(x210))
• yln(x2x)
• Let zx2x ? yln(z)
• dy/dz1/z 1/(x2x)
• dz/dx2x1
• dy/dx 2x1 /(x2x)

35
Application Derivatives of Exponential Functions

• You have a 2 yr investment contract which
  specifies that interest for the first 6 months
  is 6.0 p.a, for each month after that interest
  will increase by 0.1 per month. The contract
  quotes interest at a continuously compounding
  rate.
• What is the value of a 1 investment at the end
  of the 2 year period and what is the rate of
  increase of the investment after 2 years?

36
Application Exponential Functions(cont)

• Contract (1), P1, T in months
• Se rc(T)T ef(T), where rc(T) is a function of
  T
• If Tlt 6months
• rc(T)0.06/120.005
• f(T)T ? 0.005 f?(T) 0.005
• If Tgt 6months
• rc(T)(0.005(T-6) ? 0.001) 0.001T0.005-0.0060
  .001T-0.001
• f(T)T ? (0.001T-0.001) 0.001T2-0.001T
• f?(T) 0.002T-0.001
• At T24 months,
• Se T(0.001T-0.001) dS/dT0.002T-0.001e T
  ?(.001T-0.001)
• Se 24(0.001 ? 24-0.001)dS/dT0.002 ? 24-0.001
  e 24 ?( 0.001?24-0.001)
• S1.74 dS/dT0.082

37
Application Exponential Functions and Product
Rule

• Recall the 2 yr investment contract which
  specified that interest for the first 6 months
  is 6.0 p.a, for each month after that interest
  will increase by 0.1 per month. Suppose you are
  given the choice of two contracts, the first
  contract is as previously described. The second
  contract quotes an initial interest rate of
  0.00001 p.a. (continuous) which doubles every
  quarter.
• What is the value of a 1 investment at the end
  of the 2 year period and what is the rate of
  increase of the investment after 2 years?
• Which investment contract would you prefer?

38
Application Exponential Functions and Product
Rule

• Contract (2), for each P1,
• Se rc(T)T ef(T), where rc(T) is a function of
  T
• rc(T)0.00001 ? 2T
• f(T)T ?(0.00001 ? 2T)
• f?(T) 0.00001 ? 2TT ? 0.00001 ? 2T ? ln(2)
• At T2 years (8 quarters)
• Se rc(T)T ef(T), dS/dT f?(T)ef(T),
• f(T)8 ?(0.00001 ? 28)
• f?(T) 0.00001 ? 288 ? 0.00001 ? 28 ? ln(2)
• S1.02, dS/dT4.37

39
Application Exponential Functions and Product
Rule
contract1
contract 2
contract 2
contract1
40
ApplicationFinding a maximum

• If QD(100-2P)/(P/101),
• RevenueQDP
• R P(100-2P)/(P/101)
• R(100P-2P2)/(P/101)
• f(P)100P-2P2 f?(P)100-4P
• g(P)P/101 g ?(P)1/10
• dR/dP(100-4P)(P/101) (100P-2P2)/10
  /(P/101)2
• dR/dP-2/10P2-4P100/(P/101)2
• dR/dP0 when -2/10P2-4P1000
• dR/dP0 when P14.5

41
ApplicationMarginal Rates Chain Rule

• A health agency found that the proportion of
  patients discharged at the end of t days was
  given by the function
• f(t)1-10/(10t)3
• Find the rate of change of the proportion
  discharged as a function of time

• Let yf(t)1-z3,where z10/(10t)
• dy/dz-3z2 -310/(10t)2
• dz/dt-10/(10t)2
• dy/dt-310/(10t)2?-10/(10t)2
• dy/dt3000/(10t)4

42
ApplicationMarginal Rates Chain Rule