EDEL 204 NUMERACY SKILLS FOR ENVIRONMENT AND DEVELOPMENT MODULE UNIVERSITY OF KWAZULU-NATAL PIETERMARITZBURG. January 2002

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1 EDEL 04 NUMERACY SKILLS FOR ENVIRONMENT AND DEVELOPMENT MODULE UNIVERSITY OF KWAZULU-NATAL PIETERMARITZBURG January 00 Compiled by: School of Statistics and Actuarial Science University of KwaZulu-Natal Pietermaritzburg Campus Private Bag X01 Scottsville 309, South Africa 1

2 EDEL04 - NUMERACY SKILLS FOR ENVIRONMENT AND DEVELOPMENT MATHEMATICS SECTION ALGEBRA Objectives The student should be able to do the following after studying this section. Distinguish between real, integer, rational and irrational numbers. Understand addition, subtraction, multiplication and division operations of real numbers. Understand and use various mathematical functions such as exponents, roots and squares, fractional expression, completing the squares, scientific notations, etc. 1.1 Real numbers Three important subsets of the real numbers are integer numbers, rational numbers, and irrational numbers. The most familiar subset of the real numbers is the set of counting numbers 1,, 3, 4, 5,..., also called positive integers or the natural numbers denoted by N. The negative integers 1, -, -3, -4, -5,..., together with the positive integers and the number 0 make up the set of integers I. A real number is said to be rational if it can be expressed as a quotient b a, where a is any 3 17 integer and b is any nonzero integer. Examples -,, 3, and. A set of rational 7 1 numbers is commonly, denoted by Q. Rational number can also be written in decimal form. There are two types of decimals terminating and infinite repeating decimals. Example 5 = 0.4 is a terminating decimal and 6 1 = Another form of expressing the repeating decimal is to place a bar on top of the repeating number 6 1 = =

3 Irrational numbers are real numbers, which cannot be expressed as a ratio of two integers. Examples, 5,π, etc. Thus, irrational numbers can be thought of as numbers that cannot be expressed as a ratio of two integers or as numbers whose decimal representation is not terminating and not infinite repeating. The rational and irrational numbers are mutually exclusive and together form the set of real numbers. 1. Operations with numbers Parentheses, brackets and braces are used to group numbers and to indicate the precise order in which arithmetic operations are to be performed. For instance, 5+(3*) indicates that the multiplication 3* is performed first and then added to 5 to give 11, while (5+3)* means that 5 is added to 3 before multiplying by to give 16. The parentheses indicate the order in which calculations are done. Begin with brackets, division, multiplication, addition and subtraction (commonly referred to as BODMAS). If grouping symbols are contained within one another, begin with the innermost parentheses. Example 1.1 a) 3+{6+(+[7-])} = 3+{6+(+5)} = 3+{6+7} = 3+13 = 16 b) (4 8)*5 15 = 3*5 15 (Start the operation from left to right) = Exercise 1.1 Solve the following expressions: a) 7+(6+3) b) (4+3)-5 c) (8-4)+(5-) d) 10+(7-5+4) e) 10-{9-1(3[7-]-3[5-3])+8} f) 3*18 9 g) 5*6-3* Basic laws Commutative law of addition: x + y = y +x Example: 8+7=7+8. 3

4 Commutative law of multiplication: x*y =y*x Example: 5*3=3*5. Associative law of addition: (x+y)+z = x+(y+z) Example: (7+5)+3 = 7+(5+3). Associative law of multiplication: x*(y*z) = (x*y)*z Example: 1(3*4)=(1*3)4. Distributive law of multiplication over addition: x*(y+z)=x*y+x*z Examples: i) *(8+5) = *8+*5; ii) 3x+x = (3+)x =5x; iii) + = * +3* = (+3)* 7 5 = 7 Note: There is no commutative law for subtraction or division. For instance, 8 4 = which is not equal to 4 8 = 1. Similarly, 8 (4 ) = 4 which is different from (8 4) = Signed numbers The following operations are important to note when operating the numbers. Multiplication of signs + *(+) = + + *(-) = - *(+) = - - *(- ) = + 4

5 Example 1. i) (+3) + (+5) = +(3+5) = + 8 ii) (-5) + (-7) = -(5+7) = - 1 iii) (-4) + (+9) = +(9-4) = + 5 iv) (-8) + (+ ) = -(8-) = - 6 v) (-3) (- 7) = (-3) + (+7) = + (7-3) = + 4 vi) (-)(-3) = + (*3) = + 6 Division of signs Example 1.3 ( + ) ( + ) = + ( ) ( ) = + ( ) ( + ) = - ( + ) ( ) = - ( + 5) ( 9) = - ( 9 5 ) = Exercise 1. Simplify the following expressions: a) (-) + (- 3) b) 3 (- 7) 5

6 c) 9 + (- 6) d) (-4)(4-7) e) 8 4 ( 1) f) 8( ) 1.5 The real number line Graphical presentation of real numbers is called the real number line, which helps us to visualize the number system. For example: Real numbers are ordered such that any two distinct real numbers a and b are different where one number is larger than the other. If a is greater than b we write a > b and define this by a > b if a b is positive. Likewise, a < b if b a is positive. Order in natural numbers: If a and b are natural numbers, then exactly one of the following relation holds: Either a = b or a < b or a > b. Example 1.4 i) 9 > 5 since 9-5 = 4 is positive. ii) 0 > - since 0 (-) = is positive. Transitivity: For natural numbers a, b, and c, if a < b and b < c then a < c. Addition property: If a < b, then a + c < b + c. Multiplication property: If a < b then ac < bc. 6

7 1.6 Absolute value The value of a real number independent of its sign is called the absolute value of the number. The absolute value of a real number a denoted by a is defined by Example 1.5 i) 19 = 19 since 19 > 0. ii) - = -(-) = since < 0. a = a if a 0 and a = -a if a < Sequences A sequence, denoted by {S n } is a function that associates each positive integer n to a number S n. Series: A series is an indicated sum of the terms of a sequence, say, a n = be finite or infinite. Example 1.6 i) (Finite series) n a i i= 1. It can either ii) (Infinite series) Arithmetic sequence: Is a sequence in which the difference obtained by subtracting any term from its successors is always the same. The difference is called the common difference of the sequence designated by letter d. For instance, if a 1 is the first term of a sequence we get a 1 = a 1 a = a 1 + d a 3 = a + d = a 1 + d + d = a 1 + d... a n = a 1 + (n-1)d Sum of n terms, S n = n [a1 + (n-1)d] 7

8 Geometric sequence A geometric sequence is a sequence in which the quotient obtained by dividing any term by its predecessor is always the same. The quotient is called the common ratio designated by letter r. If a 1 is the first term then a a 1 = r Thus, a = ra 1 a 3 = ra = r(ra 1 ) = r a 1... a n = a 1 r n-1 Sum of n terms of the series. Let S n = a + ar + ar ar n- + ar n-1 rs n = ar + ar + ar ar n-1 + ar n Then rs n S n = (r-1)s n = ar n a a( r n 1) S n = where r 1 r 1 We can write algebraic identity as 1 a n+1 = 1 + a + a a n + 1, a 1. 1 a Thus, can express an infinite series as 1 1 a = 1 + a + a a n +..., - 1 < a < 1 If the sequence, S n has a limit as n becomes large, we say that the series is convergent. If the limit of S n is S, we say that S is the sum (or value) of the series. If the series is not convergent we say that it is divergent and we do not assign it any sum. Example of a convergent series if -1 < a < 1 8

9 = 3 3 Otherwise, the series diverges for a - 1 or 1 a. Another example of a divergent series is Exercise 1.3 Discuss the behaviour of the following series: a) b) c) n d) n 3 n 1.8 Exponential notation Exponential expression is used for repeated multiplication of the same number. For instance, x.x.x is written as x 3 which reads x cubed or x to the third power. In general, if n is a positive integer, we have x.x.x x.x, n of them denoted by x n, and reads x to the nth power. The letter x is called the base and n the exponent of x. Example 1.7 i)... = 6 < is the base and 6 is the exponent > = 64 ii) (5x + 4)(5x + 4)(5x + 4)(5x + 4) = (5x + 4) Algebraic expressions Any grouping of numbers and symbols generated by the elementary arithmetic expressions or by taking powers or extracting roots is called an algebraic expression. Variables are symbols that represent arbitrary elements of a set. The set of all values of a variable for which the expression itself is a real number is called the domain of the expression. For instance, the 9

10 domain of x 1 is the set of all x such that x 1 normally called a set of permissible values. The following groups x 3, 6x y, (x-4y) and 7 are terms of the expression x 3 + 6x y - (x-4y)+7 Each number and symbol of a term is called a factor. For instance, 6, x, and y are factors of 6x y. If a term is expressed as a product of two or more factors, each factor is called a coefficient of the other factors. For instance, in expression x + y - 6y is the coefficient of x, 1 is the coefficient of y and 6 is the coefficient of y. Terms differing only in their coefficients are said to be similar and are combined simply by adding their coefficients. Example 1.8 i) 8x + 3x - x = (8 +3 )x = 9x ii) 5(8x + 7) 7-3(8x + 7) 7 = (5-3) (8x + 7) 7 = (8x + 7) 7 Exercises 1.4 Simplify the following expressions a) (3y 5xy 3 +7) + (x + 3xy 3-7x +) b) 3x xy + 5x xy + 9 4x + 3xy 17 xy c) (3a + ab 9) (a 5ab ) (5ab + 6) d) 3A B (B +A 1) where A = (x + y 7) and B = (x 3y) e) x (y x +[x x + 5(y-x ) + 3x] 3x) 3x f) T(3T + )-[(4 T)T (T 3)] g) x + (1- (1- (1-x))) 10

11 1.10 Multiplication and division of algebraic expressions An algebraic expression written as a product of other algebraic expressions is said to be in a factored form. To expand an expression in factored form, we use the distribution law. For example, x (3y + x 5 ) = x (3y) + x (x 5 ) = 3x y + x 7 Hence, the general rule for multiplying exponential factors having the same base and positive integer exponents is as follows: x m. x n = x m+n Example 1.9 x 4. x 1 = x 5 Another rule for positive integer exponent is (x. y) n = x n. y n. Example 1.10 i) (x) 3 = 3. x 3 = 8x 3 ii) (5xy) = 5. x y = 5x y Exercise 1.5 Factorise the following expressions: a) 3x (7x 5y + x y 3 ) b) (x y)(x 3 y 3 ) c) (x + 5x 3)(x 3 x 7) 1.11 Division of algebraic expressions We express division of two numbers x and y by either x/y or y x, commonly known as quotient of the two numbers. For example 11

12 x x 5 = x. x. x. x. x x. x = x.x.x = x 3 More generally, x m = x m-n+n. m m n+ n x x Thus, = n n x x m x Therefore, = x m-n n x m n n x.x = n x = x m-n. Exercise 1.6 Simplify: 7 ( ab) a) ( ab) 10 ( x y) b) 5 ( x y) 3 5 5a b 7a c) 0a b x y 3x y + x d) xy 4 Division process Suppose we wish to divide 4x 3 + 5x 9 by x 3. Arrange both the dividend and divisor in descending powers of the variable. Leave space between given terms for missing powers of x. Divide the first term of the dividend by the first term of the divisor. This gives the first term of the quotient. Multiply the divisor by the first term of the quotient and subtract the product from the dividend. 1

13 Divide the first term of this difference by the first term of the divisor. This gives the second term of the quotient. It is multiplied by the divisor and subtracted from the first difference. Repeat these steps until the remainder is either zero or of lower degree than the divisor. x + 3x x 3 4x + 5x 9 <Divided> 4x 3-6x 6x + 5x -9 6x 9x 14x 9 14x 1 1 <Remainder> Exercise 1.7 Divide x 3 + 3x 4 + x + x 5 by 3 x + x 1.1 Special products An algebraic expression involving the sum or difference of two terms is called a binomial. The product of two binomials can be expanded as: (ax + by)(cx + dy) = ax(cx + dy) + by(cx +dy) = acx + adxy + bcxy + bdy = acx + (ad + bc)xy + bdy The following are some of the special products in factored and expanded form: a) (x + y) = x + xy + y b) (x y) = x xy + y c) (x + y)(x y) = x y d) (x + y) 3 = x 3 + 3x y + 3xy + y 3 e) (x y) 3 = x 3 3x y + 3xy y 3 13

14 f) (x + y)(x xy + y ) = x 3 y 3 g) (x y)(x + xy + y ) = x 3 y 3 Factoring can be achieved by reversing the above products. It involves recognising common factors. Example 1.11 i) Expand (x - 7)(x + 7) Using (c), where we let y = 7, we get (x - 7)(x + 7) = x -7 = x 49. ii) Expand (3z + 5) Using (a) where we let x =3z and y = 5, we get (3z + 5) = (3z) + (3z)(5) +5 = 9z + 30z + 5 Exercise 1.8 Expand a) (x 3y) 3 b) (a + 5b)(a 5ab 5b ) c) (x + y)(5x 4y) d) (x + y)(x y)(x y) e) (x + x 3) (x + 3) 1.13 Completing the square An expression, x + 6x + 9 is said to be a perfect square since it factors into (x + 3). Similarly, x + 3 x + 9 is a perfect square since it equals (x - 3 ). Completing the square 4 method is used when an expression cannot be factored easily. The process is based on the rule (x + k) = x + (k)x + (k) 14

15 Consider a square of side x + k units as shown below: k k x x x k To complete the area of the square we need to add the area k to the area x + kx. The general approach: Look for common factors. Look for special factors. See if you can factor by grouping. Factor by completing the square. Example 1.1 Factor x + 6x + 4 by completing the square. We can make x + 6x a perfect square by adding ½ (6) = 9. Therefore, we add and subtract 9 in the given quadratic to get x + 6x + 4 = x + 6x = x + 6x = (x + 3) ( 5) = (x )(x ) Exercises 1.9 Factor the following expressions: a) 3x 3-5x + 15x b) 13y 7 7y 3 c) z 4z d) 9 x +6x +1 e) 5x 8x

16 f) x + 4x + 1 g) y 3 9y + 7y Fractional expressions If both the numerator and the denominator of a fraction are multiplied or divided by the same nonzero quantity, the value of the fraction is unchanged. Thus a b = a b = - b a An algebraic expression is in simplest form when the numerator and the denominator have no common factors. Example 1.13 Reduce x + 3x + x 1 to its simplest form. x + 3x + x 1 = ( x + )( x + 1) ( x 1)( x + 1) = ( x ( x + ) 1) (which is valid for x ± 1) The product of two fractions is the product of the two numerators divided by the product of the two denominators. Thus, a c * = b d a * c b * d ac = bd Any common factors in the numerator and denominator of the product should be cancelled. Example x y Simplify 4 5a a 16x y * 5 5 xy *. a 16

17 3x y 4 5a a 16x y * xy 45a x y * = a 80a x y 9 = 3 16a x y Exercise 1.10 Simplify a) 3( x x + x y ) * 6 x + 3x x + y b) x 3x 4 x 1 x x Integral exponents The basic rules for manipulating exponents are: a m. a n = a m+n (a m ) n = a mn (ab) n = a n b n a ( ) n n a = b n b a a m n a a m n = a m-n, if m>n 1 = n m, if m<n a (a n b m ) p = a np b mp a 0 = 1, a 0 a n-n = a n * a -n = a 0 = 1 17

18 a -n 1 = n, a 0 a ( b a ) -n = ( a b ) n Example 1.15 i) x 3 x 4 = x 3+4 = x 7 ii) (x ) 3 = x 6 iii) (3xy) = 3 x y = 9x y Exercise 1.11 Simplify r r 7 a) 11 b) xy 3 3xb c) 0 0 a + b 0 ( a + b) d) x - y x + y e) 1 ( x + y) f) (a -1 b -1 ) -1 g) (( - ) - ) -1 1 ( xy) h) 1 1 x y 18

19 1.16 Scientific notation The following is a form of expressing the real number as a product of a number between 1 and 10 and a power of 10: x = m. 10 c where m is between 1 and 10 and c is an integer. For example, represents the weight of an oxygen molecule. In scientific notation this is 5.3 x 10-3 The decimal point is moved to the right 3 places to obtain 5.3, which is between 1 and 10. A scientific calculator would express the same number as 5.3E-3. Example 1.16 i) 578 = 5.78 x 10 3 ii) = x 10 7 iii) = 1.7 x 10-4 These numbers would appear in a calculator as follows: i) 5.78E3 ii) 3.791E7 iii) 1.7E-4 Note: Most of the scientific calculators have a button that you can push to display the numbers in scientific notation. Exercise Write the following in scientific notation. a) b) c)

20 d) 46 followed by 38 zeros e) 54 followed by 13 zeros. Write the following without scientific notation. a) x 10 9 b) x 10-7 c). x Roots and squares Note If n is an even number and a > 0, then n a is the positive nth root of a. If n is an even number and a < 0, then n a is the nth root of a is not a real number. If n is an odd number and a < 0, then n a is the negative nth root of a. If n is an odd number and a > 0, then n a is the positive nth root of a. n a n b = n ab n n a = n b b a Example 1.17 i = 3 but not ± 3 5 ii. 3 = - iii x = 3 3 ( 8x )(x ) = -x 3 x 0

21 1.18 Exponential and logarithmic functions If a is a positive real number and a 1, then the function f(x) = a x is an exponential function. For example, suppose y = f(x) = x for -3 x 3 8 y y= x x The graph of y = x will never touch x- axis. The x- axis is called an asymptote for this curve. In general, y = a x (a>1) will always cut y- axis at (0, 1). The special function is y = e x, where e is a fixed irrational number (approximately ). The number e is defined by e = lim (1+ a) 1/a a 0 In words, the expression is read as limit as a approaches 0, the final value is e. For a > 0 and a 1, the logarithmic function is of the form y = log a x (pronounced log of x base a) provided x > 0. The exponential form is a y = x. In general, common logarithm is denoted by log (x) which means log 10 x; and natural logarithms is denoted ln(x) which means log e x. 1

22 Certain properties log a a =1. Example: log = y. The expression y = can only hold true when y =1. log a 1 = 0. Proof: log a 1 = y. The expression a y = 0 holds true only when y = 0. If a > 0 and a 1, then log a a x = x, for any real number x. Proof: The exponential form of y = log a a x is a y = a x, so y = x. That is log a a x = x. Example: log = 3 since 4 3 = 4 3. log If a > 0 and a 1, then a a x Proof: The logarithmic form of y = a = x, for any positive real number x. log a x is log a y = log a x, so y = x. If a > 0, and a 1, and M and N are positive real numbers, then i) log a (MN) = log a M + log a N Proof: Let u = log a M and v = log a N, then the exponential forms are a u = M and a v = N. Thus, log a (MN) = log a (a u. a v ) = log a (a u+v ) = u + v = log a M + log a N ii) log a (M/N) = log a M - log a N. If a > 0, and a 1, M is a positive real number, and N is any real number, then log a (M N ) = N log a M. Example: Simplify log 3 (9 ). log 3 (9 ) = log 3 9 = log 3 3 = * log 3 3 = **1 = 4

23 If a 1, b 1, a > 0, and b > 0, then log b x = base formula. log log a a x. This is referred to as change of b Example: Evaluate log Log 7 15 = ln15 ln 7 = = Exercise 1.13 a) Present the graphs of i. y = e x ii. y = -x iii. y = e -x iv. y = e -x v. y = 3 x-1 vi. y 1 = x vii. y = + e -x b) Use a calculator to evaluate i. b) Evaluate ii. e iii. e -3 iv. e -1.5 i. y = log 8 ii. y =log 3 9 iii. y = log 5 (1/5) iv. log 5 37 v. log vi. e ln 3 c) Graph i. y = log x ii. y = ln x 3

24 d) Simplify log 6 x 6 i. ii. log iii. 1 ln ( e ) iv. log 1 e) Use properties of logarithms to write each expression as a single logarithm. i. ln x ln y ii. log 3 (x + 1) + log 3 (x 1) iii. log (x + 1) 1/3 log(x + 1) 4

25 . MATRIX ALGEBRA Objectives The student should be able to do the following after studying this section. Understand various computational operations related to matrices. Express and solve system of linear equations..1 Definition Consider the following information presented in a table Town Jan. Feb. Mar. W X Y Z Consider only the numbers. We get the following rectangular arrangement called a matrix of dimension 4 x 3. A = In general a matrix, say A with m rows and n columns is represented by A = a a a m 1 a a... 1 a m a 1n a... a n mn 5

26 The matrix is said to be of dimension m x n, and a 11, a 1,..., a mn are called entries or elements of the matrix. We denote the number of rows by m and the number of columns by n. Two matrices are said to be equal if and only if they are of the same dimension and each entry in the first matrix is equal to the corresponding entry in the second matrix. Consider another matrix B of dimension 4 x 3. B = The sum of the two matrices denoted by C, of dimension 4 x 3 is C = = The addition of matrices is only possible if the dimensions are the same. The entries of C are direct addition of the entries of matrices A and B. For instance, c ij = a ij + b ij i=1,, 3; j = 1,, 3, 4. Consider 4 = , 31 = 8 + 3, etc. Two matrices are said to be conformable for addition if they are of the same dimension. The subtraction operation would also apply through the same line. Say, A minus B produce a matrix D of the same dimension but different entries obtained as d ij = a ij - b ij i=1,, 3; j = 1,, 3, 4. In general, the sum of matrices A and B of dimension m x n is C also of dimension m x n. Under the addition of matrices, commutative and associative laws apply. That is i) A + B = B + A ii) A + (B + C) = (A + B) + C If A is an m x n matrix, then the negative of A, written A, is the m x n matrix, each of whose entries is the negative of the corresponding entry in A. The product of a number (scalar) k and an m x n matrix A is the m x n matrix B in which each entry in A is multiplied by k. Consider A multiplied by a constant k =4, we get, 6

27 6 ka = 4A = = Multiplication of matrices Consider the following two matrices A with dimension x 3 and B with dimension 3 x. The two matrices can only be multiplied as A x B if the columns of A equals the rows of B and the product C has the dimension corresponding to the rows of A and columns of B. 15 Suppose A = B = Then A x3 * B 3x = C x That is 15(15) + 19(138) + 1(14) C = 18(15) + 3(138) + 6(14) 15(167) + 19(145) + 1(136) 18(167) + 3(145) + 6(136) 7506 = In general, the product of matrix A, of dimension m x n, and matrix B, of dimension n x p, is matrix C of dimension m x p. The entry in the i th row and j th column is the sum of the products of the corresponding entries in the i th of A and j th column of B..3 Matrix operations A matrix can be re-defined as an array of elements called scalars. A scalar will always be a number unless otherwise stated. The matrix A has elements denoted by a ij, i.e. entry of the i th row in the j th column. A = [ a ij ] m x n, where i =1,,..., m and j = 1,,..., n. Transpose The matrix A (or A T ) obtained by interchanging the rows and columns of A is called the transpose of A. Thus, 7

28 A = [a ji ] n x m. Example.1 Similarly, 15 A = , and A == i) ( AB ) = B A ii) ( ABC ) = C B A Multiplication Given the matrices A = [ a ij ] m x n and B = [ b ij ] m x n, the product is obtained by AB = C = [c ij ], where c ij = a is b sj, i= 1,,..., m. n s= 1.4 Matrix properties Scalar multiplication Suppose, A = [ a ij ], then, ka = [ka ij ], for some scalar k. Null matrix A null matrix denoted by 0, is one with all its elements zero. For instance, consider a 3x3 matrix A. Then A = [a ij ] = , i=j=1,,3 0 Identity matrix A square matrix is said to be unit or identity matrix if all its diagonal elements are unity and all off-diagonal elements are zero. Suppose D is a 3x3 diagonal matrix. Then, 8

29 D = [d ij ] = , i=j=1,, 3. 1 Rank Consider a matrix as a set of rows (or columns) vectors written in a particular order. The rank of a matrix is defined as the number of independent rows (or columns) it contains. Suppose we consider a 3x4 matrix, say A, of the following form A = The first column can be expressed as a sum of the last three columns. Thus, column one can be expressed as a linear combination of the other three. So, the rank of matrix A is 3, written as rank(a) = 3. It can also be stated as the number of independent columns. Inverse Let A be a square matrix. If there exists a matrix B such that AB = I Then matrix B is called the inverse of A and is denoted by A -1. Let A -1 = [b ij ], if A = [a ij ] then A A -1 = A -1 A = I (Identity matrix, if A -1 exists) That is [a ij ][b ij ] = I a 11 b a 1m b m1 = 1 a 1 b a m b m1 = 0 a 1 b a m b m = 1 a m1 b a mm b m1 = 0 a ii = 1, a ij = 0 i = j Singular and non-singular If a matrix A has an inverse, it is said to be non-singular, otherwise it is called singular. 9

30 Determinants The determinant of a square matrix A = [a ij ] of order m is a real valued function of the elements a ij defined by A = ± a 1. a 1... a mp, (sometimes it is denoted by det(a)) where the summation is taken over all permutations (i, j,..., p) of (1,,...,m) with + or - sign if (i, j,..., p) is even and odd, respectively. The co-factor of the element a ij, denoted by A ij is defined to be (-1) i+j times the determinant obtained by omitting the i th row and j th column. Note: A = 0 if and only if rank(a) m A = ari Ari for any row i = ari Ari for any column r A If A 0, A -1 ij = ( ) A Matrix Z = [A ij ] is called adjoint matrix of A. For a x matrix, say A = a a 11 a1 a 1 the A = (a 11 * a - a 1 * a 1 ) For a 3 x 3 matrix, say A = a a a a a a 1 3 a a a a Det (A) = a 11 a3 a a a 1 a a 1 31 a a 3 33 a1 + a 13 a31 a a 3 Trace The trace of an n x n matrix A, which we write as tr(a) is defined to be the sum of the diagonal elements of A. That is 30

31 n a ii i= 1 tr (A) = Consider the above 3 x 3 matrix. The tr(a) is a 11 + a + a 33. Let A and B each be n x n matrices, the tr (AB) = tr(ba).5 System of linear equations Let v = (b 1, b,..., b n ) be a given vector, then v is the subspace spanned by {u 1, u,..., u m } if and only if, there exist real numbers, x 1, x,..., x m such that x 1 (a 11, a 1,..., a n1 ) + x (a 1, a,..., a n ) x m (a 1m, a m,..., a nm ) = (b 1, b,..., b n ) where, u 1 = (b 1, b,..., b n ), u = (a 1, a,..., a n ),..., u m = (a 1m, a m,..., a nm ) Hence, there exist a solution of the system of linear equations a 11.x 1 + a 1.x a 1m.x m = b 1 a 1.x 1 + a.x a m.x m = b.. (1). a n1.x 1 + a n.x a nm.x m = b n Equation (1) can be written in matrix form as follows: a a a n 1 a a... 1 a n... a... a a 1m m nm x x... 1 x m = b1 b... b n The solution of () is given by A nxm X mx1 = B nx1 () 31

32 X = A -1 * B if A -1 exists. A is called matrix of coefficients. The problem of solving such systems of n linear equations in m unknowns is widespread in many branches and applications of mathematics. Example. Consider the following system of equations i) -x 1 + x = 4 ii) 4x 1-3x = -7 Thus, A = a a 11 a1 a 1 1 = 4 3 B = 4, and A -1 = Therefore, X = A -1 * B = = 5 1 Hence, x 1 = 5, x = -1. 3

33 Exercises.1 1. Consider the following matrices in answering parts (a) to () A = , B = a) A + B b) A B c) A 4B d) 5(A +B) e) A*B f) B*A g) (A*B) T h) Det(A) i) Det(B*A) j) Rank(A*B) k) Tr(A). Consider the following matrices which are identical except for the first row, and for that row the relationship c ij = a ij + b ij ; j=1,, 3; holds. A = , B = , C = a) Compute i. A + (B + C) ii. B (A + C) iii. A*B*C iv. B*A*C v. A -1 vi. (A T ) -1 vii. (A -1 ) T b) Show that det(a) + det(b) = det(c) 33

34 3. DERIVATES Objectives The student should be able to do the following after studying this section. Provide various definitions related to derivatives, differentiation and continuity of functions. Understand how to apply derivative rules such as product, quotient, and chain rules. Handle derivatives related to natural logarithm, higher order and polynomial functions. 3.1 Derivative of a function Consider the following situations as a motivation: a) If a firm receives R in revenue during a 30 day month, its average revenue per day is divide by 30 which is equal to R This does not necessarily mean that the actual revenue was R1 000 on any one day, just that the average is R1 000 per day. b) Similarly, if a person drives a car 10 Km in an hour s time, the car s average velocity is 10 Km per hour, but the driver could have gotten a speeding ticket for travelling 140 Km per hour on the highway. When we say a car is moving at a velocity of 10 Km per hour, we are talking about the velocity of the car at an instant in time (the instantaneous velocity). If a car travels in a straight line from position y 1 at time x 1 and arrives at position y at time x, then it has traveled the distance y y 1 in the elapsed time x x 1. We define y y 1 as y and x x 1 as x. The smaller the time interval, the nearer the average velocity will approximate the instantaneous velocity. Let P(x 1, y 1 ) be a fixed point on the curve. If Q(x 1 + x, y 1 + y ) is another point on the curve, then for some function y = f(x) y 1 + y = f(x 1 + x ) y = f(x 1 + x ) - f(x) Thus, y x = f ( x x f x 1 + ) ( 1) x Where x denotes small change in the direction of x axis, and y denotes the corresponding change in y- axis. 34

35 If y approaches a constant value as the point Q moves along the curve towards P, i.e. as x x becomes small, then we call this constant value the limit (sometimes called rate of change of dependent variable with respect to the independent variable) and define it as f (x) = lim x 0 f( x1 + x) f( x1) x The limits are common in areas where interest is in determining: The marginal profit given profit; marginal cost given total cost; marginal revenue given total revenue; Or rates of change, such as rates of populations and velocity. Note: The limit may or may not exist for some values of x. At each point, x 1 where it exists the function f is said to have a derivative. f(x 1 ) is said to be the derivative of f at x 1. The process of finding the derivative of a function is the fundamental operation of differential calculus. Definition Differential calculus can be defined as that branch of mathematics, which deals with: i) Given a function f, determine those values of x at which the function possesses a derivative. i) Given a function f and an x at which it exists, find f(x). Since the derivative exists at all but a few values of x, that x, can be any of the nonexceptional values of x. We henceforth write x instead of x 1. This implies f (x) = lim x 0 f ( x + x) f ( x) x If the limit exists, then it provides a rule for associating a number f (x) with the number x. The set of all pairs of numbers (x, f (x)) that can be formed by this process is called the derivative function. 35

36 Let f(x) be defined on an open interval containing c, except perhaps at c. The statement lim f(x) = L x c is read the limit of f(x) as x approaches c equals L and means that for all x sufficiently close to c, but not equal to c, the value of f(x) can be made as close as possible to the single number L. When the values of f(x) do not approach a single finite value as x approaches c, we say the limit does not exist. The limit is said to exist only if the following conditions are satisfied: i. The limit L is a real number. ii. The limit as x approaches c from the left equals the limit as x approaches c from right. Example 3.1 Evaluate i) lim (x 3 x) x 1 Let f(x) = x 3 x Thus, lim f(x) = (-1) 3 (-1) =1 x 1 ii) lim x 4x x 4 x Let f(x) = x 4x, and g(x) = x. g(4) 0. Then lim f ( x) x 4 g( x) = 4 4 4(4) = 0 =0. Example 3. Find f (x) for 36

37 i) f(x) = mx + b f (x) = m, which is the slope of the tangent to the graph of f(x) = mx + b. ii) f(x) = x + 1 x (where x 0). f (x) = x - 1 x Notation: The derivative of y = f(x) can be denoted by f (x), or y or dy dx or D xy If, we replace x by a single letter h, then the derivative of f(x) at any value x denoted f (x), is f (x) = lim h 0 f ( x + h) f ( x), h if this limit exists. If f (x 0 ) exists, we say f is differentiable at x 0. Example 3.3 Find the derivative of i) f(x) = 4x, using the definition. Solution: The change of x from x to x + h is h. f(x+h) f(x) = 4(x+h) 4x = 4(x +xh+h ) - 4x = 8xh + 4h f ( x + h) f ( x) h = 8 xh + 4h h = 8x + 4h Thus, f (x) = lim h 0 f ( x + h) f ( x) h = lim (8x + 4h) = 8x h 0 37

38 ii) y if y = x and x>0 Solution: y = 1 x Let F(x, x ) = f ( x + x ) f ( x ) x Further, let the limit of F(x, x ) as x approaches zero be L(x). Then given say positive number ε, for say number x, there exists a positive numberδ such that whenever, 0 < x <δ F(x+ x ) - L(x) <ε This implies that L(x) = lim F(x+ x ). x 0 Differentiation: The process of finding the derivative of a function is called differentiation. Continuity: A function f is said to be continuous at the point x = c if the following conditions are satisfied: 1. f(c) exists.. 3. lim f(x) exists. x c lim f(x) = f(c). x c If one or more of these conditions are not satisfied, we say the function is discontinuous at x = c. Example 3.4 Consider the function, f(x) = 3x + 5. The function is continuous at x = because 1. f() = 11,. lim f(x) = 11, and x 38

39 3. lim x f(x) = 11 = f() The following information is useful in discussing continuity of functions: A polynomial function is continuous everywhere. f ( x) A rational function is a function of the form, where f(x) and g(x) are g( x) polynomials. If g(x) 0 at any value of x, the function is continuous everywhere. If g(c) = 0, the function is discontinuous at x = c. If g(c) = 0 and f(c) 0, then there is a vertical asymptote at x =c. lim f ( x) If g(c ) = 0 and = L, then the graph has an open circle at x = c. x c g( x) 3. Constant, power and multiple rule Suppose y = cf(x) where c is a constant. Then dy dx = cf (x) Example 3.5 i) y = x 4 dy dx = d x 4 ( ) dx = (4x 3 ) = 8x 3 ii) y = - x y = 1 x dy dx = 1 d( x) 1 = dx Exercise Evaluate the following limits 39

40 a) b) lim x 4x x x lim x 3x + x 1 x 1. Evaluate the limits if they exists i. ii. iii. iv. v. lim x + 3x + x 1 x 1 lim (4x + ) x 1 lim (x 3 1x +5x +3) x 1 lim 3 x 3x + 1 x 1 x 1 lim x 9 x 3 x 3 vi. lim x 3 x x x 18x vii. lim x 7x + 10 x 5 x 10x + 5 viii. ix. lim x + lim x x 1, Where denotes infinity x + x x 3. Is the function i. f(x) = ii. f(x) = x 1 x + 1 continuous at (i) x =1 and (ii) x =-1? 1 x 1 continuous? 40

41 4. Test the function f(x) for continuity if f(x )= x + 1 ( x 1) x if if if x < 0 0 x < x 5. The total revenue for a product is given by R(x) = 1600x x, where x is the number of units sold. Find lim R(x) x If the profit function for a product is given by P(x) = 9x x 1760, find lim P(x) x Suppose that the daily sales S (in rands) t days after the end of an advertising campaign is S = S(t) = Find 400 t + 1 a) S(0) b) lim S(t) x 7 c) lim S(t) x If the revenue for a product is R(x) = 100x 0.1x, and the average revenue per unit is Find R( x) R( x) =, x > 0. x a) lim x 100 R ( x) x 41

42 b) lim + x 0 R ( x) x 9. Find the derivative of the following a) y = -6x b) y = 3 x Suppose that the average number of minutes M that it takes a new employee to assemble one unit of a product is given by M = t, t 0, where t is the number of days on the job. t + 1 Is this function continuous at (i) t = 14? (ii) for all t 0? 11. Suppose that the cost C of removing p percent of the impurities from the waste water in a manufacturing process is given by C (p) = 9800 p 101 p Is this function continuous for all those p- values for which the problem makes senses? 1. Suppose that the cost C of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by C (p) = 8100 p 100 p Describe any discontinuities for C(p). 13. The monthly charge for water in a small town is given by f(x )= x +16 if if 0 x 0 x > 0 4

43 a) Find lim x 0 f(x ), if it exists. b) Is f(x) continuous? The sum or difference rule If y = g(x) ± h(x) or f(x) = g(x) ± h(x) then dy dx = g (x) ± h (x) or f (x) = g (x) ± h (x) Example 3.6 y = 4x 3 + x Let g(x) = 4x 3 and h(x) = x 1 1 Then g (x) = 1x and h (x) = x Thus, dy dx = g (x) + h (x) = 1 1 1x + x 3.3 Product, quotient and chain rule If y = g(x).h(x) or f(x) = g(x).h(x) Then, dy dx = g(x).h (x) + h(x).g (x) or f (x) = g(x).h (x) + h(x).g (x) Example 3.7 a) y = 3x (x + 3) Let g(x) = 3x and h(x) = (x + 3) then f (x) = 3x () + (x +3)(6x) = 6x + 1x + 18x = 18x + 18x 43

44 Quotient rule If y = g ( x ) h( x) g( x) or f(x) = h( x), then dy dx = h ( x ) g '( x ) g ( x ) h '( x ) ( h( x)) or f (x) = h ( x ) g '( x ) g ( x ) h '( x ), where h(x) 0 ( h( x)) Example 3.8 y = x + 1 x 1 Let g(x) = x + 1, and h(x) = x - 1 g (x) = 1, and h (x) = Thus f (x) = h ( x ) g '( x ) g ( x ) h '( x ) ( h( x)) ( x 1)( 1) ( x+ 1)( ) = ( x 1) 3 = ( x 1) Exercise Find the derivatives of the following equations: a) y =(x+3)(x x) b) y = (3x 1)(x 3 + 1) c) p = (3q 1)(q 3 +) d) y = (x 1 +3x 4 +4)(4x 3 + 1) e) y = (x + 1) x 44

45 3 f) y = (4 x+ )( x- x x -5 ) g) y = ( 3 4 x 5-5 x 4-1)( 3 x - 1 ) x. What is the slope of the tangent to a) y = (x + 1)(x 3 4x) at (1, -6) b) y = (x 3 3)(x 4x + 1) at (, -15) 3. Find the derivatives of the following equations: a) y = x 4 x+ 3 3x b) y = c) y = d) y = 3 x + 3x + 1 x + 4 x 4x x x 3x + x 4 1 e) y = 3 x 4 x 4x + 3 f) y = 3x g) y = x x 3 4 3x x 1 h) y = 4 x i) y = ( x + 1)( x ) x

46 3. The reaction R to an injection of a drug is related to the dosage x according to R(x) = x (500-3 x ), where 1000 mg is the maximum dosage. If the rate of reaction with respect to the dosage defines sensitivity to the drug, find the sensitivity. 4. The average number of children s toys assembled by a worker in t hours of an 8-hour shift is given by N(t) = 64 18( t + 3), 0 t 8. t + 6t + 18 Find the rate of change in the number of toys assembled when t = Suppose that the proportion P of voters who recognise a candidate s name t months after the start of the campaign is given by P(t) = 13t t a) Find the rate of change of P when t = 6. b) Find the rate of change of P when t = 1. c) One month prior to the election, is it better for P (t) to be positive or negative? Chain rule The chain rule is concerned with finding the derivative of a function that is the composite of two functions. If y = f(u) is a differentiable function of u, and u =g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x. If y = [u(x)] n Then dy du = n[u(x)]n-1 If y = [u(x)] n, then dy dx = n[u(x)]n-1. du dx Thus, dy dx = dy du. du dx 46

47 Example 3.9 y = (3x - x ) 3 Let u = 3x - x Thus, y = u 3 f (x) = 3u (3-4x) = 3(3x - x ) (3-4x) Polynomial functions A single term of the form cx n where c is a constant and n 0 is called a monomial in n. A function that is the sum of a finite number of monomial terms is called a polynomial in x. Let y = f(x) define a function. If the limit dy dx = lim y exists and is finite, we call this, the derivative of y with respect x 0 x to x and say that f is differentiable at x. Derivative of a constant is zero Let y = f(x) = c, where c is a constant. y = f(x+ x ) - f(x) = c - c = 0. This implies that y x = 0. dy dx = lim x 0 y x = 0 Derivative of x n is n x n-1. Proof: Let y = f(x) = x n 47

48 y + x = (x + x ) n = x n + nx n-1 x Exercise 3.3 Find the derivatives of the following functions: i) y = 1x - 4x + 8 ii) y = x 4 + 3x3 - x iii) y = 1x + 6 x - iv) y = 9(x + 5x) 4 3 v) y = ( + ) x vi) y = x + 4 x ( x + ) vii) y = ( 3 x 1 x 3 + ) viii) y = x 1 1 ix) y = ( x + 3x + 1) x) y = (x 4x) 6 4 x xi) y = xii) y = ( x (x ) x + 5) 3 4 xiii) y = x 1 x 48

49 3.4 Derivatives of logarithmic functions If y = ln(x) and x > 0, then dy dx = 1 x If y = ln(u) where u = g(x) and g(x) > 0, then dy dx = 1 u du dx Exercise 3.4 Find the derivatives of the following functions: i) y = ln (7x) ii) y = ln(3 -x) iii) y = x 3 + 3ln x iv) y = 3 1 ln(x 6 3x + ) v) y = ln(x + 1) x + 1 vi) y = ln [x(x 5 ) 10 ] vii) y = ln 8x + viii) y = log 4 (x 3 + 1) ix) y = ln x + ln(x 4 x + 1) x) y = ln ( 3 x + 1 ) The derivative of e If y = e x, then dy dx = ex 49

50 If y = e u, where u = g(x), then dy dx = eu du dx Example 3.10 y = e 4x, Let u = 4x, du dx = 4. Thus, dy dx = eu. du dx = 4e 4x Exercise Find the derivatives of the following functions: i) y = 10 e -3x ii) y = x e x iii) y = e 4x 3 ln x e iv) y = v) y = e -x vi) y = e 3 + e ln x vii) y = e e x x e + e x x. Suppose that the revenue in rands from the sale of x units of a product is given by x 50 R(x) = 100x e Find the marginal revenue function, that is, R (x). 3. The number of people N(t) in a community who are reached by a particular rumour at time t (in days) is given by N(t) = t e 0. 7 Find the rate of change of N(t). 4. Suppose that the spread of a disease through the student body at an isolated college campus can be modeled by 50

51 y = t e 0. 99, where y is the total number affected at time t (in days). Find the rate of change of y. 5. Suppose that world population can be considered as growing according to the equation N = N 0 (1+r) t, where N 0 and r are constants. Find the rate of change of N with respect to t. 3.5 The higher derivative The first derivative of a function provides an indication as to whether it is increasing, decreasing or stationary. The function needs to be differentiable on that point under consideration. Suppose, y = f(x). If dy dx If dy dx If dy dx > 0, at a point x= c then the function is increasing. < 0, at a point x= c then the function is decreasing. = 0, at a point x= c then the function is constant. A function f has a relative maximum at x = c if there is an interval around c on which f(c ) f(x) for all x in the interval. The first derivative f (x) = 0 or is undefined at x = c in this case. A function f has a relative minimum at x = c if there is an interval around c on which f(c ) f(x) for all x in the interval. The first derivative f (x) = 0 or is undefined at x = c in this case. The second derivative of a function provides an indication as to whether the function has a minimum, maximum or no turning point. Suppose we denote the second derivative of a function y = f(x), by d y or f (x). dx If d y < 0, then we have a maximum point. Thus, the graph is concave down on the dx interval, under consideration. 51

52 If d y > 0, then we have a minimum point. Thus, the graph is concave up on the interval, dx under consideration. If d y dx = 0, then we have a saddle or turning point (point of inflection). Exercise 3.6 a) Suppose a company has determined that its total revenue, R, for a given product is given by R(x) = - x x x R(x) is measured in rand and x is the number of units produced. i) What production level will yield a maximum revenue? ii) What will this maximum revenue be? f) A canning factory, canning different kinds of fruit, has determined that its profit, P (R1000), from the production of x (100000) cans of fruit is given by P(x) = x 3-1x + 10x + 60 The manager of the factory wants to know what production level will yield a maximum profit and also what production level will yield a minimum profit. He also wants to know what the amount of this maximum and minimum profits will be. i) Obtain the first derivative. i) Determine the x value that gives the maximum profit. ii) Which x value that gives a minimum profit? iv) Determine the second derivative. 5

53 4. INTEGRATION Objectives The student should be able to do the following after studying this section. Provide various definitions related to integration. Understand how to apply integration rules such as product, quotient, and chain rules. Handle integration related to natural logarithm, higher order and polynomial functions. 4.1 Definition of integration To integrate is to indicate the whole of, to give the sum of or total of. However, the mathematical meaning of the word will be illustrated in finding areas bounded by curves, volumes of various solids, lengths of curves, centres of gravity and other applications. The second mathematical meaning is to find a function whose derivative is given. We wish to calculate the area of a region bounded between the lines x = a and x = b, and the graph of a nonnegative continuous function say f. The solution is to use integration to find this area. We denote the area by b b Area = f( x) dx a The symbol f( x) dx is called the definite integral from a to b, where a and b are fixed a numbers. The lower limit of the integration is denoted by a and the upper limit by b. The symbol... dx, is the inverse of symbol d... The function f(x) is called the integrand, and dx the dx indicates that the integral is to be taken with respect to x. The definition of the definite integral when the function is continuous on the interval a< x < b is b f( x) dx = [F(x)] b a = F[b] - F[a], a where F[x] is the integral of the function f(x 0 ). The following figure illustrate the area of the integration: 53

54 y F(b) F(a) a b x Figure 1: Area under the curve within the interval a and b. Suppose we are given dy dx dy dx as a function say, = f(x) (1) where a< x <b, and asked to find: y = F(x) For example, find y as a function of x if dy dx = x. From experience we know one answer is y = x But y = x + 1; y = x + ; y = x + 4 are also possible answers. Indeed, y = x + c is an answer if c is a constant. An equation such as (1), which specifies the derivative as a function of x, is called a differential equation. 54

55 Example 4.1 i) dy dx = xy ii) d y dx + 6xy dy dx + x y = 0 A function y = F(x) is called a solution of the differential equation (1) if, over the domain a< x <b, F(x) is differentiable and df( x ) = f(x) dx F(x) is said to be an integral of f(x) with respect to x. If F(x) is an integral of f(x) with respect to x, then F(x) + c is also such an integral. When c is any constant then d[ F( x) + c] dx = df ( x ) dc + dx dx = f(x) + 0 = f(x). If both F 1 (x) and F (x) are integrals of f(x), then df ( 1 x ) = df ( x ) = f(x) dx dx Thus, d[ F1( x) F( x)] dx = 0. Hence, F 1 (x) - F (x) = c (constant) Example 4. Solve the differential: i) dy dx = 3x, y = x 3 + c. Solution: dy = 3x dx 55

56 But d(x 3 ) = 3x dx (from experience). Hence, y = 3xdx 3 = d( x ) = x 3 + c iii) dy dx = x y Solution: dy = x y dx y 1 dy = x dx But, 1 d( y ) = y 1 dy Similarly, d( x 3 3 ) = x dx This implies that, x c 1 = y + c 1 y = x c 1 - c = x c. 1 Integration as given above, requires the ability to guess the answer, but the following formula help reduce the amount of guesswork in many cases. du = u + c c is a constant adu = a du a is a constant udv = - vdu ( du ± dv) = du ± dv n udu= u n n c ( n -1) < Antiderivative of un > 56

57 A function F is called an antiderivative of a function f if, for every x in the domain of f, F (x) = f(x). The process of finding an antiderivative is called integration. The function that results when integration takes place is called an indefinite integral or more simply, an integral. Example 4.3 i) ( 5 x x + ) dx = 5xdx - xdx+ dx = 5 x + c 1 - x c + x + c 3 = 5 x x3 + x + c. Exercise Evaluate the following integrals. 9 a) 4x 1 dx 7 b) ( x + 3x) dx 0 1 c) ( x ) dx 1 d) x( x + 1) dx e) 3 1 x dx 4. Indefinite integrals Fundamental theorem of calculus: Let f be a continuous function on the closed interval [a, b]. Then the definite integral of f exists on this interval, and 57

58 b f( x) dx = F(b) - F(a), a where F is any function such that df ( x ) = f(x) for all x in [a, b]. dx Some properties of definite integrals b [ f ( x) ± g( x)] dx = f( x) dx a b a a a b b a b ± a g ( x) dx kf ( x) dx = k f( x) dx, where k is a constant. f ( x) dx = 0 a a If f is integrable on [a. b], then f ( x) dx = - f( x) dx b a If f is continuous on the same interval containing a, b, and c where c need not to be between a and b, then b b f( x) dx = a c a b f ( x) dx + c f ( x) dx Example i) ( x + 4 ) dx = x x + c 4 = (4) + c () + c 4 = 68 < Note that the c s subtract out> 3 ii) ( 3x + 6x) dx = x 3 + 3x = ( (3 )) ( (1 ) = 50 58

59 Exercise Evaluate 5 a) ( x 9 + ) xdx 3 b) ( 4x 6x 5x) dx 0 4 c) xdx d) ( x + 1) xdx 0 3 e) ( x x ) 4 ( 1 x) dx f) 0 0 x dx 3x Compare 3 3. Show that x and 3x dx 3 dx x + 4x dx 4 dx = 4x dx 4.3 Finding the area between two curves If f and g are continuous functions on [a, b] and if f(x) g(x) on [a, b], then the area of the region bounded by y = f(x), y = g(x), x = a, and x = b is b A = [ f( x) g( x)] dx The average value of a continuous function y = f(x) over the interval [a, b] is a Average value = 1 b a b a f ( x) dx 59

60 Example 4.5 i) Find the area of the region bounded by y = x + 4, y = x, x = 0, and x = y=x y=x Since y = x + 4 lies above y = x in the interval x = 0 to x = 3, the area is 3 A = [( x + 4 ) x] dx 0 = x x - x 3 0 = ( ) - ( ) = 16.5 square units. Example 4.6 Suppose that the cost function for a product is C(x) = x + 0.3x. We wish to find a) The average value of C(x) for x = 10 to x = 0 units. b) The average cost per unit if 40 units are produced. Solution a) The average value of C(x) is 60

61 0 1 1 x 0 10 ( x + 0.3x ) dx = (400x x 3 0 ) 10 = 10 1 [( ) ( )] = 485 rands. b) The average cost function is C ( x) C (x) = x = x x The average cost per unit if 40 units are produced is 400 C (40) = (40) = 3 rands. 40 Exercise Find the area between y = - x and the x-axis from x = -1 to x =1.. Find the area enclosed by y = x and y = x Find the area of the region enclosed by the graphs of f (x) = x 3 x and g(x) = x. 4. Given f (x) = x + 3x 9 and g(x) = 4 1 x a) Find the points of intersection of f(x) and g(x). b) Determine which function is greater than the other between the points found in part (a). c) Set up the integral used to find the area between the curves in the interval between the points found in part (a). d) Find the area. 5. Find the average value of each of the function over the given interval a) f(x) = x 4 over [-1, 3] b) f(x) = x x over [0, ] c) f(x) = x 3 x over [-1, 1] d) f(x) = 1 x over [-, 0] 61

62 e) f(x) = x - over [1, 4] f) f(x) = 3 x over [-8, -1] 6. Suppose that the income from a slot machine in a casino flows continuously at a rate f(t) = 100e 0.1t where t is the time in hours since the casino opened. Then the total income during the first 10 hours is given by e Find the average income over the first 10 hours. 7. A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by 0.1 t dt 18 S = 30x x x 7 Where x is in hours and 0 x 4. Find the average number of milligrams of the drug in the bloodstream for the first 4 hours after a capsule is taken. 4.4 Power and multiple rule The power rule for integration is n [ u( x)] u'( x) dx = n [ u( x)] n C, if n -1 Example 4.7 Evaluate 5 ( x 4) xdx + Let u = x + 4, n = 5 du = u = x which implies du = u dx = xdx dx 6