Application of differential calculus The rate of change of the indicator. Taylor's schedule. Finding the maximum and minimum values. Partial derivatives. Lagrange operator More: https://ek.biem.sumdu.edu.ua/

1. Lecture 2-3. Differential and integral
calculus applied in economic.
Kashcha M.
1

2. Contents
• Introduction
• Differential calculus
• Application of differential calculus
• Maxima and minima
• Differentiating functions of more than one
variables
• Integral calculus or integration
2

3. Introduction
• Differential calculus and integral calculus
• Differential calculus – calculated how, or at
what rate , a given variable changing.
• Integral calculus is used to calculate the areas
and volumes of shapes that are bounded by
curves lines or curved surfaces.
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4. Differential calculus
• The rate of change of one variable Y, in
response to a change in another variable X, is
known as the first derivative of Y with respect
to X.
• Whether that rate of change is increasing or
decreasing or constant , i.e. whether Y is
accelerating, decelerating or changing at a
constant rate, is given by second derivative.
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5. The first derivative - the rate of change.
Y=3X and Y=3X+12
The rate of change of Y
resulting from a change in
X is given by the slope of
the line. The slope of each
of these lines is given by the
ratio of ΔY, the change in Y,
over ΔX, the change in X;
this ratio is ΔY/ΔX
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y = 3x
y = 3x + 12
0
3
6
9
12
15
18
21
24
27
30
0 1 2 3 4 5 6
ΔX
ΔY
ΔY

6. The average rate will
be less
representative of
the instantaneous
rate change the
larger the distance
between the two
points on the
curve.
And conversely, the
smaller the
distance between
the two points the
more accurate will
be the measure of
the rate of change
of Y.
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y = 2x2 + 4
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8
ΔY=6
Δ Y=14
Δ Y=22
Δ Y=26

7. • The limit is the proportional change in Y when the
change in X approaches zero.
• Recall and deductive y:
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8. 8

9. Some basic rules of differentiation
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10. Differentiating the product (quotient)
of two functions of X
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11. Differentiating a function of a function
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12. Differentiating an exponential
(logarithmic) function
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13. Second derivatives
• To determine whether the variable is
accelerating, decelerating or changing at a
constant rate we need the second derivative
of the function.
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14. Taylor series expansions
• Example: price
yield curve
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15. Approximations
• Y –point in f.
• Y+h, h – is taken to be small, can be negative,
is variable.
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16. How to interpret approximations?
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17. Applying Taylor series to estimate
bond price changes
• One-year;
• 0 coupon bond;
• 100% one year hence;
• If the yield-to-
maturity is 10% -
current price 90,909;
• What would be the
price of the bond if y
changed from 0,1 to
0,11?
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18. 18

19. Applying calculus to measure bond
price risk
1. The modified duration;
2. The convexity
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20. Volatility of bonds
• The yield to maturity of a bond is the
annualized internal rate of return that equates
the value of future cash flows to the current
bond price.
• The current value of coupon paying bond:
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21. Macualay’s duration
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Modified duration -
Used by indicator of the interest rate risk in bonds

22. Numerical example of modified
duration
• 2-years bond, pays 5 semi-annually, yield to maturity is
8% p.a.
Duration:
• 3,585/2=1,793;
Modified duration:
• If yield ↑ 0,5%
• ↓ 0,5*1,793=0, 896%

23. • Changes in dy/dx is
represented by the
second derivative of Y
which respect to X, and
indicated by d2y/dx2 .
• If y=x2 and dy/dx=2x:
d2y/dx2 =2.
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y = 2x2 + 4
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8
The second derivative – the rate of
change the rate of change

24. Application of the second derivative:
bond convexity
• Improve on our measure of bond price
sensitivity given by modified duration.
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25. 25

26. Numerical example of convexity
Convexity = 0,5 P’’ /P
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27. Maxima and minima
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A, B, C – stationary points, dy/dx=0.

28. Stationary points
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29. Finding minimum and maximum
values of a function
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30. Differentiation functions of more than
one variable. Partial diffentiation.
• Y=f(x,z)
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∂z

31. 31
• The second derivatives ∂2y/ ∂x2, ∂2y/ ∂w2, ∂2y/
∂z2 tell us how the marginal changes in Y
behave the x,w or z are changed but the other
two variables remain fixed.

32. Total differentiation
To indicate how Y will change in response to
simultaneous small changes in each of the
independent variables we simply add together
the products of each partial derivative multiplied
by the small change in its respective variable.
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z

33. Maxima and minima of functions of
more than one variable
1) local maxima:
2) local minima:
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34. Maximization and minimization subject to
constraints: Lagrangian multipliers
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35. 35

36. 36

37. 37

38. • https://www.coursera.org/learn/mathematics
-for-economists/lecture/77pUp/2-13-hessian-
matrix
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39. Integral calculus
• Action is reverse to differentiation – indefinite
integral.
• Definite integral – process to find area
bounded to functions.
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40. Definite integral
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