AEM 303 Applied Mathematics for Economics and Social Sciences Past Paper PDF

Applied Mathematics for AEM 303 -Oh Economics and Social Sciences Department of Agricultural W h y & Farm Management a y - H Economics R. A. Sanusi and T. O. Oyekale...

Applied Mathematics for AEM 303 -Oh Economics and Social Sciences Department of Agricultural W h y & Farm Management a y - H Economics R. A. Sanusi and T. O. Oyekale 1  CLASSWORK i. Linear algebra however, can be applied only to systems of linear equati True or False …… - O h hy ii. Many exponential and power functions are readily convertible to li y -W functions and can be handled by linear algebra. True or False …… iii. Rows is the numbers H a in a vertical line of a matrix. True or False … iv. Columns is the numbers in a horizontal line of a matrix. True or False … v. Square matrix is a matrix having the number of rows equals to number of columns. True or False …… O h vi. Column vector is a matrix composed of a single row. True or False …… - vii. The dimensions of a column vector isygiven h a y - W H as r x 1. True or False …… viii. Row vector is a matrix having a single column. True or False …… ix. The dimensions of a row vector is given as 1 x c. True or False …… x. When two matrices (A and B) are conformable for multiplication, product (AB i.e. matrix C) is said to be undefined. True or False …… - O h xi. Null matrix is a matrix composed of all 0s and can be of any dimension. h y W True or False …… a y - H xii. Coefficient matrix (A): the matrix whose elements are the variables in system of (linear) equations. True or False …… xiii. Solution vector (X): the matrix whose elements are the coefficients in system of (linear) equations. True or False …… xiv. Determinant: is a single number or scalar and is found only for sq matrices. True or False … - O h h y xv. A vanishing determinant is one expressed as ࡭ ≠ 0. True or False … y - W Ha xvi. A non-singular matrix is one whose determinant is expressed as ࡭ = 0 True or False … xvii. If ρ(A) < n, A is singular and there is linear dependence. True or False … xviii. Minor matrix is a determinant with a prescribed sign. True or False … O h Triangular matrix is a matrix with zero elements everywhere above or be - xix. hy the principal diagonal. True or False … a y -W xx. H Adjoint matrix is the transpose of a minor matrix. True or False … xxi. The determinant of a matrix equals the determinant of its transpose: A Aᇱ. True or False … xxii. If all the elements of any row or column are zero, the determinant is one. True or False … - O h y xxiii. If two rows or columns are linearly dependent, the determinant is zero. W h True or False … y - Ha xxiv. Symmetric matrix is a matrix which when transposed equals the original matrix. True or False … xxv. In matrix algebra, an identity matrix is similar to number 1 in ordinary algebra. True or False …  Applied Mathematics for Economics and Social Sciences AEM 303 Department of Agricultural Economics & Farm Management R. A. Sanusi and T. O. Oyekale O h h y- 1 a y -W H REVIEW OF MATHEMATICAL TERMS AND CONCEPTS Exponents Polynomials Equations and Functions Graphs O h h y- 2 a y -W H  If n is a positive integer, xn implies that x is multiplied by itself n times. x = base n = exponent Conventionally, an exponent of 1 is not expressed: x1 = x e.g. 81 = 8. By definition, any non-zero number or variable raised to the zero power is equal to 1: ▪ x0 = 1 ▪ 30 = 1 ▪ 00 is undefined. O h h y- 3 a y -W H Exponents: O h hy- a y -W H  O h hy- a y-W H  O h hy- 6 a y-W H Polynomials 3 Given 5x : x = variable (because it can assume any number of different values) 5 = coefficient of x Monomials = 5x3 Polynomials = addition or subtraction of monomials. Term = each of the monomials comprising a polynomial. are called Like Terms = terms having the same variables and exponents O h h y- 7 a y -W H  Rules of adding, subtracting, multiplying and dividing polynomials: Like terms can be added or subtracted by adding their coefficients, unlike terms cannot be so added or subtracted:- 4x5 + 9x5 = 13x5 12xy + 3xy = 9xy (7x3 + 5x2 – 8x) + (11x3 – 9x2 – 2x) = 18x3 – 4x2 – 10x (24x – 17y) + (6x + 5z) = 30x – 17y + 5z O h h y- 8 a y -W H  O h hy- 9 a y-W H Equations and Functions Equation: a mathematical statement setting two algebraic expressions equal to each other. Function (f ): is a rule which assigns to each value of a variable (x) - the argument of the function - one and only one value [f(x)] i.e. value of the function at x. Domain of a function: the set of all possible values of x. Range: the set of all possible values for f(x). Common functions in economics:- O h h y- 10 a y -W H  O h hy- 11 a y-W H GRAPHS O h hy- 12 a y-W H  Determine the market price and quantity, if the supply and demand of carrot is given by the following equations. Qs = -5 + 3P and Qd = 10 – 2P (NB: price in ₦‘000 and quantity in tons) Solution:- ▪ Market price occurs at equilibrium i.e. Qs = Qd ▪ Solving for P:- ▪ ⇒ -5 + 3P = 10 – 2P O h h y- 13 a y -W H ▪ ⇒ 5P = 15 ▪ ⇒P=3 ▪ Substituting P 3 in either of the equations, ▪ Qs = -5 + 3P = -5 + 3(3) = 4 ▪ Qd = 10 – 2P = 10 – 2(3) = 4 ▪ Market price = ₦3,000 and Market quantity = 4 tons O h h y- 14 a y -W H  Assume a simple two-sector economy where Y = C + I, C = C0 + bY and I = I0. Assume further that C0 = 85, b = 0.9 and I0 = 55; determine the equilibrium level of income. Solution:- ▪ Y = C0 + bY + I0 = 85 + 0.9Y + 55 ▪ Y – 0.9Y = 140 ▪ 0.1Y = 140 ▪ Y = 1400 O h h y- 15 a y -W H  Applied Mathematics for AEM 303 -Oh Economics and Social Sciences Department of Agricultural W h y & Farm Management a y - H Economics R. A. Sanusi and T. O. Oyekale 1 Application of Mathematical Methods in Economics
Linear Equations - O h hy a y-W H
Non-linear Equations
Mathematics of Finance  In algebra, letters are used to represent numbers. - O h In pure mathematics, the most common letters used are x and y. y -W h H a y It is helpful in applications to choose letters that are more meaningful. Thus, letters such as Q (for quantity) and I (for investment) might be used  In this wise, economic problems can be stated mathematica and solved mathematically. - O h hy a y -W Some economic expressions include that of demand/supp H production and financial equations; examples of these are: Q = ƒ(P); Q = f (K, L); TR = PQ and 1 = TR − TC. Linear Equations   The expression P
 can be used to determine how mo
 in a savings account grows over a period of time. - O h hy y -W where P is the original sum invested (principal), r the interest r H a and n the number of years (money is saved). To evaluate, the letters need replacement by actual numbers but a there is need to understand the various conventions associated w this kind of expressions. An instance is the expression of demand equation; ordinarily, quantity dema is defined as a function of price. - O h So mathematically, this definition can be given as Qd = ƒ(P). h y a y - W H However, conventionally in economics, price is always placed on the x-axis plotting the demand curve. Thus, the inverse function is used in presenting the demand function.  Assuming a demand function is given as Q = -½P + 25; conventionally it is - O h ½P = 25 – Q y -W h ⇒P = 50 – 2Q H a y Following are examples to further illustrate the use of mathematic methods in economics:  Since the beginning of 2020, the price of whole-wheat bread at Modu Mall has been rising at a constant rate of ₦2/month. By November 1, the price had reached ₦150 per loaf. What is the price of the bread as a function of time and what was the price at the beginning of the year? - O h hy y -W Solution a  Hthat have elapsed since January 1 x = number of months  y = price of a loaf of bread (₦).  Since y changes at a constant rate with respect to x, the function relating y to x must be linear and its graph is a straight line. - O h Because the price y increases by 2 each time x increase by 1, the slope o y  -W h the line must be 2.  H a y Then, we have to write the equation of the line with slope 2 and passes through the point (1, 150).  By the formula:-  y – y0 m=x–x 0 ⇒y – y0 = m(x – x0) - O h y ⇒y − 150 = 2(x − 1) h a y -W Hyear, x = 0; thus:- ⇒y = 2x + 148 At the beginning of the y = 2(0) + 148 = 148. Therefore, the price of bread at the beginning of the year was ₦148/loaf.  Another issue in mathematical expression for economic phenomenon inequality. h In pure mathematics, a number line can be used to decide whether or not o Thus, it can be said that a number a is y - O number is greater or less than another number. right of b on the line; written as-aW h Ha y greater than a number b if a lies to > b. Likewise, a is less than b if a lies to the left of b, written as a < b. Economic problems can be written as inequality and then resolved.  For example: A cashew processing firm’s Human Resources department has a budge O h ₦25,000 to spend on training and laptops of new employees. Training cou h y - cost ₦700 and new laptops are ₦1,200. y - W HLalaptops? (a) what is the inequality expression for E and L if the department train employees and buys (b) if 12 employees attend courses, how many laptops could be bought?  Solution  (a) The cost of training E employees is 700E and the cost of buying L laptop 1200L; the total amount spent must not exceed ₦25,000:- - O h ⇒700E + 1200L ≤ 25000 hy a y -W  (b) Substituting E = 12 into the inequality gives:- H 8400 + 1200L ≤ 25000; ⇒1200L ≤ 16600 5 ⇒L ≤ 136 ∴ a maximum of 13 laptops could be bought. Non-linear Equations In reality most economic variables do not have linear relationship(s) but non-linear - O h Thus, economic phenomena cannot be described with straight line curves but y -W h curves that are curvilinear in nature. H a y The simplest non-linear function is known as quadratic function and takes the f ƒ(x) = ax2 + bx + c In fact, even if an economic function (e.g. demand function) is linear, functions de from it (such as total revenue and profit) turn out to be quadratic. 1  In determining the value of x in a quadratic equation (model), three approaches usually employed. h These are factorization, method of completing the square and the quadratic formu - O hy a y -W The quadratic formula: H      x= An illustrative application in economics is as below: 1  A retailer sells (poultry) eggs for ₦24. If a customer orders more than 100 eggs, retailer is prepared to reduce the unit price by 4kobo on eggs bought above 100 pi but up to a maximum of 300 pieces in a single order. (a) How much does it cost to buy 130 pieces? - O h y (b) If the cost is ₦5,324; how many pieces were ordered? - W h a y  Solution H 30 eggs is 24 − (0.04 × 30) = ₦22.80 (a) The first 100 eggs cost ₦24 each, so total cost for these is 100 × 24 = ₦2,400 The unit cost of the remaining ⇒the total cost of the extra 30 eggs is 30 × 22.80 = ₦684 The cost of the complete order is ₦(2,400 + 684) = ₦3,084 1 (b) The total cost of the eggs bought in excess of 100 is ₦(5,324 − 2,400) = ₦2,924. If x denotes the number of eggs above 100 pieces, then the unit price of each is - O h y 24 − 0.04x - W h ⇒ 24x − 0.04x = 2924 a H y ⇒total cost is (24 − 0.04x)x = 24x − 0.04x2; 2 ⇒ 0.04x2 − 24x + 2924 = 0. This quadratic equation can then be solved using the quadratic formula: 1    ±   .   ±   . ⇒x = = . . O h ± ±
. -
.
 y ⇒x = = h . . y - W Ha this gives two solutions, x = 170 and x = 430 the maximum permissible order is 300 pieces; therefore x = 170 this implies that the total order is 100 + 170 = 270 eggs 1  Considering the classical (Cobb–Douglas) production function of th form Q = AKαLβ (for some positive constants, A, a and b). - O h y Such functions are homogeneous of degree α+β β because -W h ƒ(K, L) = AKαL β and K and L are increased by a factor λ then: λK, λL) = ƒ(λ A(λ a λK)α(λ y H = Aλ K λL)β α α λβLβ = AλαλβK α Lβ = λαλβAK α Lβ λK, λL) = λα+ββ(AK αLβ) = λα+ββf (K, L) ⇒ƒ(λ 1  Consequently, Cobb–Douglas production functions exhibit:-  decreasing returns to scale, if α+β β 1  constant returns to scale, if α+β β=1 increasing returns to scale, ifW y - Ha  α+β In general, a function of the form Q = ƒ(K, L) is said to be homogeneous  λK, λL) = λnƒ(K, L) for some number, n ƒ(λ 2  This means that when both variables K and L are multiplied by λ, all of th λs can be pulled out as a common factor i.e. λn.  The power, n, is called the degree of homogeneity. - O h h y W Thus if:- n < 1, the function is said to -   H a y display decreasing returns to scale  n = 1, the function is said to display constant returns to scale  n > 1, the function is said to display increasing returns to scale. 2  However, not all production functions are of this type. It is not even necessary for a production function to be homogeneous. - O h hy y -W For illustration, if Q = 100K¼L½; what is the degree of homogeneity of t a H function? Solution: λK)¼(λ Q = 100(λ λ L) ½ 2  λ¼K¼λ½L½ = 100λ λ¼λ½)(100K¼L½) = (λ - O h hy -W = λ¾(100K¼L½) H a y Thus, the output gets scaled by λ¾, which is smaller than l since the pow ¾, is less than 1. Hence, this production function can be described as that which exhib decreasing returns to scale. 2 Mathematics of Finance A wide variety of mathematical methods are used in social science courses a studies. - O h hy -W These methods are applied in economics, business and management studies y a address issues such as discounting and investment appraisal. H These methods include percentage, series and sequence. An illustration is given below: 2  (a) The value of a plough depreciates by 25% in a year. What will a ploug currently priced at ₦43,000 be worth in a year’s time? - O h hy -W (b) After a 15% reduction in sale, an egg is ₦39.95. What was the price before the sa y H a began? (c) The value of a farm’s deep well appreciates by 10% in a year; what will its va be, in a year’s time, if the well is currently valued at ₦190,205? 2  Solution  (a) Since the plough depreciates, current value will be less than 100% of the origina - O h hy -W  y thus, the scale factor is 1 – = 0.75 a 100 H× 0.75 = ₦32,250 (forwards in time, so multiply) so the new price is 43000  (b) Since there is reduction in egg price, the current price will be less than 100% the original; 2 
 thus, the scale factor is 1 – = 0.85 100 so the original price was 39.95 ÷ 0.85 = ₦47 (backwards in time, so divide) - O h hy a y -W  (c) Since the well appreciates, current value will be more than 100% of the original; thus, the scale factor isH
 1+ = 1.10 100 so the new price is 190205 × 1.10 = ₦209,225.50 (forwards in time, so multiply) 2  A firm determines that the cost of tanning leather for manufacturin men’s belts is ₦2/unit plus ₦300 per day in fixed costs. Th company sells the tanned belt’s leather for ₦3 each. What is th h break-even point? y - O -W h Solution H a y  The break-even point occurs where revenue and cost are equal. 2  By letting x = the number of belts manufactured in a day: - O h the revenue function is R(x) = 3x, and hy a y -W H the cost function is C(x) = 2x +300 at break-even, C(x) = R(x) 2  ⇒3x = 2x + 300 - O h hy -W ⇒x = 300 H a y  So, the firm must sell 300 belts/day to break even and more than 30 belts/day to make a profit. 3  Applied Mathematics for AEM 303 -Oh Economics and Social Sciences Department of Agricultural W h y & Farm Management a y - H Economics R. A. Sanusi and T. O. Oyekale 1  THE DERIVATIVE AND THE RULES OF DIFFERENTIATION
Limits - O h hy
Continuity a y -W H
Derivative
Rules of Differentiation
Marginal Concepts  Limit L is the one and only one finite real number to which the funct values ƒ(x) of a function ƒ draw closer for all values of x as x draws close from both sides, but does not equal a. L is defined as the limit of ƒ(x) as x approaches a. The figure below illustr h y - O -W h H a y  From the graph, as the value of x approaches 3 from either side value of ƒ(x) approaches 2. - O h Since the limit of a function as x approaches a number depe y -W h only on the values of x close to that number, the limit exists. H a y This means that the limit of ƒ(x) as x approaches 3 is 2 written:-  lim  = 2  →  lim  = L → If lim  and lim  exist, the rules of limits are:- h → → y - O -W h lim =K (K = constant) y i. a → ii. lim  = an → H (n = positive intege iii. lim  = K lim  (K = constant) → → iv. lim   ±   = lim  ± lim  → → → h lim  .   = lim . lim  O v. - → → → h y y - W Ha vi. lim   ÷   = lim  ÷ lim  lim  ≠ → → → → vii. lim  = lim  (n>0) → →    = 9 Rule 1 → - O h   = 62 = 36 Rule 2 y h → a y -W → →  H   =   = 23 33 = 54 Rules 2 &   !) =   !   = 34 ! 3 32) = 22 Rule 4 →  → →    ! " # )] =   ! ".   ! ) → → → = 4 ! 8. 4  – 5) = -12 Rul - O h hy y - W   ' 3 2 '  Ha  ' →  34  " ')  = = = = 2.8 Rul → (   (  ( *) →    #  =    #  → → = 322 – 523 = 23 = 8 Rul  Find the limits of the following rational functions:  '/ ,-.  '/ /'/ ) i. ,-.  = →/  = 2 = = 0  →/  (  ,-.  (  " h →/ 7 (  y - O -W h    ,-. ,-.1 ,-.2 y ii. (x ≠ 0) = = ∞ or = -∞ ∞; the limit does not ex a →)  →) ) →) ') iii. ,-. →∞ ∞  & ,-.  →'∞  H= ∞ = 0; the limit exist in both cases.  / /  '/ '  '∞ ∞ = ')  iv. ,-. = ,-. * = * = →∞∞  '* →∞∞ '  '∞ ')  Continuity Continuous function: a function which has no breaks in its cur O h A function ƒ is continuous at x = a if: = -a, ƒ(x) is defined, i.e. exists, at x y - W h H a y i. ii.  f(x) exists, and → iii.  f(x) = ƒ(a). → 1   '/  '/ * ƒ(x) = = =  '  (/ ' / (/ * * h ƒ(7) = = ; ƒ(x) is defined, i.e. exists, at x = 7 O = y- /(/ * - W h Ha y  '/ * * *   =  = →/  '   →/  ( / /(/ * ⇒  f(x) exists, and →/ Since  f(x) = ƒ(7), ƒ(x) is continuous. →/ 1 Derivatives Differentiability and Continuity: a function is differentiable at a point if derivative exists (may be taken) at that point. - O h To be differentiable at a point, a function must: y - W h Ha y i. be continuous at that point, and ii. have a unique tangent at that point. It should be noted that continuity alone, however, is not a sufficient condi for differentiability. 1  ∆3 Recall that the slope (or gradient) of a line is defined as S =. ∆ This is the slope at an interval, which usually applies to a straight line. - O h hy y -W However, for a curve; the slope (gradient) is taken at a point. H a To obtain the slope at a point, the slope of the tangent at that point is t and it becomes a point slope (usually referred to as derivative). If y = ƒ(x), the derivatives can usually be written as: 1 i. f‘′′(x) ii. y′′ - O h dy hy iii. dx a y-W H df iv. dx d v. dx [ƒƒ(x)] vi. ƒ(x)] Dx[ƒ 1  ∆y In the limit, as ∆x tends to zero, the slope of the chord (of a straight line), , is e ∆x to that of the tangent. - O h y ∆y h This limit is written as: ,-. -W ∆→) ∆x H a y Thus, it can be deduced that the formal definition for derivative is: dy ∆y = ,-. dx ∆→) ∆x 1  Given a function y = ƒ(x), the derivative of the function ƒ at x is defined a f(x + ∆x) − f(x) ƒ’(x) = ,-. ∆ if the limit exists; ∆→) ) - Find the slope of the function ƒ(xy O h h Illustration:- y - W Ha 2 = 2x using the argument of derivative. i. Employ the function ƒ(x) = 2x2 and substitute in the arguments:-   2(x + ∆x) − 2x ⇒ƒ’(x) = ,-. ∆ ∆→) 1 ii. Simplify the result:-   2x + 2x(∆x) + (∆x) − 2x  ⇒ ƒ’(x) = ,-. ∆ ∆→) - O h hy - W iii. Divide through by x:- !y Ha ⇒ƒ’(x) = ,-. ∆ ∆→) iv. Take the limit of the simplified expression:- ⇒ƒ’(x) = 4x + 2(0) = 4x 1 
Rules of Differentiation Differentiation: the process of finding the derivative of a function. Instead of using the limit approach, a number of rules can be applie O h If ƒ(x) = k, where k is a constant:- ƒ’(-x) = 0 y Constant Function Rule: h y - W ƒ(x) = 8, ƒ’(x) = 0 Ha Linear Function Rule: If ƒ(x) = mx + b:- ƒ’(x) = m ƒ(x) = 3x + 2:- ƒ’(x) = 3 1  Power Function Rule: ƒ(x) = kxn :- ƒ’(x) = k·n·xn-1 h ƒ(x) = 4x3, ƒ’(x) = 4·3·x3-1 = 12x2 y - O -W h ƒ(x) = 8, ƒ’(x) = 8·x1-1 = 8·x0 = 8·1 = 8 a y If ƒ(x) = g(x) ± h(xH Sums and Differences: ):- (x) = g‘(x) ± h‘(x) f(x) = 12x5 – 4x4:- f‘(x) = 60x4 – 16x3 f(x) = 9x2 + 2x – 3:- f‘(x) = 18x + 2 1  Product Rule: If f(x) g(x) · h(x):- f‘ (x) = g(x)·h‘ (x) + h(x)·g‘ (x) - O h hy then, f(x) = 3x4(2x – 5), let g(x) = 3x4 and h(x) = 2x – 5 W and g‘ (x)-= y Ha ⇒g‘(x) = 12x3 2 ⇒f‘ (x) = 3x4(2) + (2x – 5)(2) ⇒ f‘ (x) = 6x4 + 24x4 – 60x3 = 30x4 – 60x3 2 Quotient Rule: f(x) = g(x) ÷ h(x), where g(x) and h(x) are differentiable functions and h(x) ≠ - O h hy g(x) hx·g‘(x) − g(x)·h‘ -W f(x) = h(x):- f‘ (x) = y h(x)  5 x3 H a f(x) = 4x + 3 where g(x) = 5x3 and h(x) = 4x + 3, g‘(x) = 15x2 and h‘ (x) = 4, t 2 3 ( 15x '5x   ⇒f‘(x) =  4x + 3 2  3 2 3 60x (45x '20x ⇒f‘(x) =  4x + 3 - O h hy 2 40x3(45x a y -W ⇒f‘(x) = H 4x + 3  5x2"( ⇒f‘(x) = 4x + 3  2  Generalized Power Function Rule: If f(x) = [g(x)]n, where g(x) is a differentiable function and n is any real num then:- O h (x-) = 3x ; f(x) = (x + 6) , let g(x) = x + 6, then g‘y f‘ (x) = n[g(x)]n-1·g‘(x) - W h ⇒f‘(x) = 5(x + 6) ·3xHa y 3 5 3 2 3 5-1 2 ⇒f‘(x) = 5(x3 + 6)4·3x2 ⇒f‘ = 15x2(x3 + 6)4 2  Chain Rule: If y = f(u) and u = g(x), then y = f [g(x)]; therefore, 63 63 67 = · 6 67 6 O h 63 67 - If y = (5x2 + 3)4, then let y = u4 and u = 5x2 + 3; therefore = 4u3 and 6 = hy 67 y -W Hence:- H a 63 = 4u3·10x 6 63 ⇒ = 40xu3 6 Thus, 63 = 40x(5x2 + 3)3 6 2 Marginal Concepts The concept of the margin is important because an economist is usually interested in unit change in economic variables. O h Thus, the common marginal variables of interest in economics - hy are: a y-W H  marginal product,  marginal cost,  marginal revenue,  marginal propensity to consume/save. 2  Given the demand function P = 30 – 2Q, what is the price; to marginal and average revenue when 5 units of Q is produced? TR = PQ = (30 – 2Q)Q = 30Q – 2Q2 h y - O ⇒P = 30 – 2(5) = 30 – 10 -=W h If Q = 5: y H a 20 TR = 30(5) – 2(5)2 = 150 – 50 = 100 689 MR = = 30 – 4Q 6: 2  ⇒MR = 30 – 4(5) = 30 – 20 = 10 89 ;: AR = = =P : : - O h hy -W ) ': : a y AR = = 30 – 2Q = 30 – 10 = 20 OR 30(5) – 2(5) H150 – 50 : 2 AR = = = *)) =20 OR    *)) AR = = )  2  Find the MC, TC and estimate the effect on TC given a 3 increase in Q if the current output of a firm is 15 units and firm’s average cost function is given as: O h * - AC = 2Q + 6 + hy Q a y-W H Solution TC AC = Q * ⇒TC = (AC)Q = (2Q + 6 + )Q Q 2  ⇒TC = 2Q2 + 6Q + 13 Differentiating gives:- - O h hy dTC) d -W MC = dQ = 4Q + 6 When Q = 15:- H a y MC = 4(15) + 6 = 66 TC = 2(15)2 + 6(15) + 13 = 2(225) + 90 + 13 = 553 2  Change in TC is ∆TC ≡ d(TC) and ∆Q ≡ dQ ∴d(TC) = dQ(MC) ≡ ∆(TC) = MC × ∆Q - O h So, if Q increases by 3 units then ∆Q = 3:- y -W h H a y ⇒d(TC) = 66 × (3) ⇒d(TC) = 198 ∴TC increases by 198 units approximately. ⇒ the new TC = 553 + 198 = 751 3  Applied Mathematics for AEM 303 -Oh Economics and Social Sciences Department of Agricultural W h y & Farm Management a y - H Economics R. A. Sanusi and T. O. Oyekale 1 Application of Derivative
Changing Functions - O h hy -W
Concavity and Convexity
Relative ExtremaH a y
Optimization of Functions Increasing and Decreasing Functions Since the first derivative measures the rate of change and slope of a function: - O h hy y -W i. a positive first derivative at x = a indicates the function is H a increasing at s:- ƒ ’(a)>0: increasing function at x = a ii. a negative first derivative indicates it is decreasing at s:- ƒ ’(a)0 ƒ ’(a)0 ƒ ’’(a)>0 Convex at x = a  y y - O h hy -W x x y a a H a ƒ ’(a)>0 ƒ ’(a)