THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics. Mathematics 01 MTU Elements of Calculus in Economics

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1 THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics Mathematics 0 MTU 070 Elements of Calculus in Economics

2 Calculus Calculus deals with rate of change of quantity with respect to another Calculus has two branches i). Differentiation ii). Integration

3 Is the stu of a) b) c) d) e) Differential calculus Derivative (rate of change of one quantity with respect to another) Motion of objects in space e.g. Velocity and acceleration Marginal concepts in economy The growth or decay of Biological population Curve sketching and finding Maimum and minimum values of functions

4 Differentiation The derivative f of a function f is given by f () lim h 0 f h h f() Provided this limit eist. If said to be differentiable at. eists, then is The process of finding the derivative is called differentiation. a f (a)

5 Symbols for Derivatives If y f(), then the derivative of symbols used for differentiation are y, f (), y ( or f), d f(), Dy or Df() d d at Rules of Differentiation There some rules or derivative formula used for differentiation of functions

6 The derivative of Constant function If f is a function such that f() c, then d (f()) 0 d derivative of constant is zero. Eample Find the derivative of the following constant functions i) ii). f() Solution i), d ii)., f() d () f() c 0 f() c (c) 0 d d

7 The derivative of to a power n If f is a function such that f(), then f () n n but n 0 The derivative of to a power n is the eponent times to the net lower power Eample: Find the derivative of functions (i) (ii) (iii) y y y

8 Use Solution: n d f () n (i). y, d d( d ) (ii)., y d d d d( d ) 6 6 (iii)., y d d d d d

9 Derivative of sum or difference If f is a function such that f() h() g(), then f () h () g () The derivative of the sum is the sum derivative. Eample: Find the derivative of functions f() y (i). Solution f() (i)., y (ii)., (ii). 0 d d 6 6

10 Product rule If f is a function such that f() h() g(), then f () h () g() g () h() derivative of the product is the first factor times the derivative of the second plus the second factor times the derivative of the first. Eample: Differentiate the function (i) (ii) f ( ) ( )( ) y ( )( )

11 Use (i). (ii). ) )( ( f() ) )( (6 ) )( (0 d 6 d ) )( ( y h() () g g() () h () f

12 Derivative of quotient If f is a function such that, then f() h() g() f () g() h () h() g() g () The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

13 Eample: (i) (ii) Solution: (i) (ii). f() y f() ) ( ) ( ) (6 d 0 d y ) ( ) )( ( ) ( d ) ( d

14 Chain rule Chain rule is used to find the derivative of composite functions (function of function) fog() = f(g()) If f is a function such that f() = f(g()), then d Eample: Differentiate (i). y d g d g d (ii). ( ) y 7

15 Solution: Given y let ( ) g ( ) then y (g()) dg d dg g use d dg dg d d g ( )

16 Derivative of Parametric Functions If y f(t) and f(t) when t is called a parameter, such equations are parametric equations, d dt dt d Eample: Given (t ) and y t, Find Solution: dt, then d t and (t ) d dt d dt dt d d t (t ) t t

17 Derivative of Eponential function For eponential function f() e, then f () e If a, then,, y Iny Ina yina a Ina Eample: Find the derivative of y e u Solution: Let y e andu y e d e u du d d d 6 e

18 Derivative of Logarithm of function If f() In Eample: Find the derivative of i). y Solution: i). y d In(, then f () ) ii). y ii). ln( ) In( ) y ln( ) d 6

19 Higher Derivatives If f() and g(), then can be y d y differentiated with respect to to give. Eample: Find d d Solution: Given y In d d y d y d d d when g() y In d d d y d d d y d d y

20 Minimum and maimum If f is a function such that y = af(), then Maimum when Minimum when and and Eample: using d second derivative find the relative curve maimum or minimum points of the f() d d d d y d y 0 0

21 Solution When f () maimum f f f () () or () ( ) 0 relative at (-, f(-)) f( ) ( ) ( ) 8( )

22 , relative minimum f () () 0 At (, f()) f( ) ( ) 8( )

23 INTERGRATION INTERGRATION AS REVERSE PROCESS OF DIFFERENTIATION (ANTIDERIVATIVES)

24 Integration Integral is the process of summing. It involve determination of a) Areas b) Antiderivatives c) Volumes d) Length e) Moment of inertia f) Centre of gravity

25 Properties of integral Integral of zero If f is a function such that f() 0, then 0 d c The integral of zero is constant Eample: Find the integral of 0 d

26 Integral of constant If f is a function such that f() k, then Kd K c The integral of zero is X + constant Eample: i). Solution i). ii). d yd d c ii). yd y c

27 Integral of Power If f is a function such that f() n, then Increase the eponent of by one and divide by new eponent Eamples: Evaluate the following i). Solution: i). d n ii). d n n d d c ii). c d 7 7 c

28 Integral of coefficient of the function If y is a function such that y = kf(), then Kf() d K f( ) d Eample: Evaluate the following functions i) d ii) p d

29 Solution i) d d d c ii) p d p d p d 7 p 7 C

30 The integral of If f is a function such that f() = d ln c, then True only when is positive The integral of e If f is a function such that f() = e, then e d e c

31 Integral of sum and difference If f is a function such that, then The integral of the sum is the sum integral Eample: h() g() d h() d i) ii) iii) ( ) d ( ( e ) d ) d f() h() g() g() d

32 Solution i) ii) iii) d d d ) ( C d ) ( d d d e d d ) e ( C e e 6 ln d ) ( d d d ) ( C 6 6 d ) (

33 Definite integrals If the function f() is differentiable and continuous at the interval = a and = b, then

34 eamples Use the above properties to evaluate the following int egrals. d. d. 7 d. ( 6 ) d

35 Application of Calculus In business mathematics we eperience changes of variable with respects to others. a) b) c) Sales revenue changes with volumes of units sold Cost change with volume of unit produced or sold Sales demand for a product varies with its sale price

36 Marginal Functions Total cost ( C ) Total cost represents the cost of producing a specific quantity (X) of commodity The total cost consist of two types of costs. i) Fied cost these is the cost independed of quantity produced e.g. salaries, rent etc. ii) Variable cost is the cost depend on the quantity produced

37 C = Fied cost +Variable cost Eample: Give a total cost production function c() i). fied cost ii). variable cost Solution: 0,000. i). Fied cost = 0,000 ii). Variable cost = Identify the

38 Marginal cost(mc) If c() is the total cost of producing units, ' the Marginal Cost = C MC d(c) d Average cost (AC) Average cost function (AC) given the cost per unity is defined as AC Totalcost Quantity C X

39 Marginal average Cost (MAC) The marginal average cost (MAC) is defined as MAC d(ac) d Eample: Give a total cost production function 0,000. Identify the a. fied cost b. variable cost c. marginal Cost d. average cost e. marginal average cost c()

40 Solution: c() 0,000. a. fied cost 0,000. b. variable cost c. marginal Cost, MC MC 0 d(c) d 6

41 d. Average Cost AC Totalcost Quantity C X AC AC e. Marginal Average Cost MAC d(ac) d d(ac) d

42 Total Revenue functions (R). Total revenue function is the product of price per item by quantity. Total revenue R Price (P) quantity (X) Price as a function of quantity epresses the link between the price and the quantity demanded and referred to demand function. Marginal Revenue Function (MR) The Marginal Revenue (MR) Marginal revenue (MR) = R ' MR d(r) d

43 Average Revenue Function (AR) Average revenue function (AR) gives the revenue per unit. It is defined as AR R X But R PX where P is the price per units, thus P R X Then the average revenue AR is equal to the price P

44 Marginal Average Revenue Function (MAR) The marginal average Revenue (MAR) is defined as Eample: Given the total revenue function Function MAR d(ar) d R(),000. Find the (i). Marginal Revenue (ii). Average Revenue Function (iii). Price when MR = 0

45 Solution i) Marginal Revenue Function (MR) MR d(r) d 6 ii) Average Revenue Function(AR) AR R X AR

46 iii) Price when MR = 0 MR (0.-) =0 = 0 or = Since AR = P when =, P P 0. 0.( ) 6 Hence the price when MR = 0 is,6 ( )

47 Total Profit Profit (π) is the difference between the total revenue and total cost. That is π = R C The aim of any firm is to maimize profit. This involves finding value of which makes dπ 0 d Differentiating π = R C dπ d d(r) d d(c) d

48 At maimum profit There fore MR MC dπ d 0 0 hence MR MC Eample: The total cost and revenue function for a certain commodity are R andc (a) Derive the total profit function (b) Find the profit break even point (c) calculate the level and amount of demand that maimize profit

49 Solution a) Derive the total profit function b) π R π The break - even points are levels of demand which makes C π 0 = 6 or = The.0 break even 0 points 00is when 0 the demand is or 6 products

50 (c). Profit is maimized when dπ d π 0 and 0 d d Since π Then dπ d d d π. 0 0 gives maimum profit when then π.0( ) 0 ( ) 00

51 Hence the maimum profit of. is raised when the demand is

52 Thanks for listening