Basic Mathematics and Statistics (BPC2111) PDF

Basic Mathematics and Statistics (BPC2111) Basic Calculation Dr Ho Ket Li Basic Mathematics and Statistics (BPC2111) Functions Dr Ho Ket Li  Learning Outcomes By the end of this lecture you should be able t...

Basic Mathematics and Statistics (BPC2111) Basic Calculation Dr Ho Ket Li Basic Mathematics and Statistics (BPC2111) Functions Dr Ho Ket Li  Learning Outcomes By the end of this lecture you should be able to perform basic calculation involving: – Domain, co-domain and range – Quadratic and reciprocal functions – Quadratic and reciprocal equations – Graph of quadratic and reciprocal functions – Polynomial functions: division algorithm and zeroes of a polynomial function (rational zero theorem) Function Domain, Codomain and Range  Question 1 1 Let f(x) = 2 −4 a. Determine the maximal domain of f b. Determine the range of f Question 1: Answer Domain, Codomain and Range  Quadratic Equations There are three methods of solving a quadratic equation 2 + + =0 1. factorization, 2. Completing the square 3. Using the quadratic formula  Quadratic Equation: Factorisation Method 2 Solve 2 = 13 − 15 Answer 2 2 − 13 + 15 = 0 −5 2 −3 =0 3 =5 2  Quadratic Equations: Completing the Square Method 2 Solve 3 − 6 + 2 = 0 by completing the square. Answer: Quadratic Equations: Quadratic Formula Method  Quadratic Equations: Quadratic Formula Method 2 Solve 3 = 4 + 2 by using the quadratic formula Answer  Reciprocal Equation A reciprocal function of a number is equal to 1 divided by that number 1 = Reciprocal function of 0 is undefined Reciprocal function of negative or positive numbers are negative or positive respectively Reciprocal function of very big number or infinity are close to zero Graph of Reciprocal Function Graph of Quadratic Functions Graph of Quadratic Functions Graph of Quadratic Functions  Question 2 Find the minimum or the maximum value of the following function and the point where it occurs: 2 =3 −4 +1 After that, sketch the graph for this function. Question 2: Answer Question 2: Answer  Question 3 Find the minimum or the maximum value of the following function and the point where it occurs: 2 =5 −2 −3 After that, sketch the graph. Question 3: Answer Question 3: Answer Polynomial Function Polynomial Functions: Division Algorithm  Polynomial Functions: Division Algorithm 3 2 +6 −7 Use polynomial division to find the quotient and the remainder of +4 Answer Remainder Theorem  Remainder Theorem By using the remainder theorem, find the remainder when 2 3 + 2 − 10 + 2 is divided by 2x+1 Answer Polynomial Functions: Factor Theorem  Polynomial Functions: Factor Theorem 3 2 Use the factor theorem to show that 3x+4 is a factor of 3 − 11 + + 28 Answer Polynomial Functions: Rational Zero Theorem  Polynomial Functions: Rational Zero Theorem 3 2 Factorise 2 −7 + 13 − 5 Answer  Polynomial Functions: Rational Zero Theorem 4 3 2 Factorise − −7 + 5 + 10 Answer  Exercise 1 Solve the following equations: 2 a. 3 + 14 + 8 = 0 b. −3 =1 2 c. 4 + 12 = 9  Exercise 2 For each of the following functions, find the vertex by completing the square. Find the values of the intercepts. Sketch the graph. a. = 2 − 2 − 15 b. =5 −6 −2 2  Exercise 3 2 For what values of a does the equation − 4 + = 0 have real roots?  Exercise 4 a. Find the exact roots of the equation: 2 − 2 3 − 1 = 0 b. Find the stationary value of the function: = 2 2 − 16 + 22 State whether it is a maximum or a minimum and write down the coordinates of the point where it occurs  Exercise 5 Two flower beds, each of which is a square, have a combined area of 18.5 2. The sum of the perimeters of the two flower beds is 24m. Determine the length of the side of each flower bed. Exercise 6  Exercise 7 2 For the function = −2 + 8 + 11 a. Find the stationary point of f by the method of “completing the square” b. Find the x and y intercepts  Exercise 8 a. Given that the equation 2 + 8 − + 1 = 0 has a repeated root, find the possible values of the constant a. b. Use the method of completing the square to sketch the graph of the following quadratic function, showing clearly the positions of the stationary point and intercepts: = 3 + 2 − 2 2  Exercise 9 Solve the following equation, giving your answers correct to 2 decimal places: 2 2 +3 −4=0  Exercise 10 a. Given that the equation 2 − 3 + 4 + 1 = 0 has a repeated root, find the possible values of the constant b. b. Use the method of completing the square to sketch the graph of the following quadratic function, showing clearly the positions of the stationary point and intercepts: = 3 2 + 6 + 14 c. Sketch the graph  Exercise 11 Given the function = 2 2 + 3 − 8 a. Use the method of completing the square to find the stationary point of the function b. Find the values of the intercepts c. Sketch the graph of the function over the domain −3 ≤ ≤ 2 d. hence, state the range of values of y over the above domain  Exercise 12 Find the exact zeros of the function: 2 − 5 − 2  Exercise 13 2 Let =4 −8 −5 a. Find the stationary point of f by the method of completing the square. b. Find the x and y intercepts c. Sketch the graph  Exercise 14 Find the stationary point of the function = 3 2 − 6 + 5 by the method of completing the square. State whether it is a maximum or a minimum point.  Exercise 15 Given that the equation 5 + 1 2 − 8 + 3 = 0 has a repeated root, find the possible values of the constant a.  Exercise 16 Use the remainder theorem to find the remainder when 3 2 a. 3 −8 − + 19 is divided by x-2 3 2 b. 3 −2 − 12 − 4 is divided by 3x-2  Exercise 17 The expression 3 + 5 2 + − 21 gives a remainder of 55 when divided by x-4. Find the value of c.  Exercise 18 Use the factor theorem to show that a. X + 7 is a factor of = 5 4 + 35 3 − 3 2 − 17 + 28 b. 3x + 2 is a factor of =6 4 +4 3 − 18 2 + 9 + 14. In each case rewrite the polynomial p(x) in the form (ax+b)q(x)  Exercise 19 4 3 2 One of the factors of =2 + − 14 + 5 + 6 is a. X + 1 b. x+ 2 c. X + 3 d. X - 3 e. None of the above  Exercise 20 a. Find the remainder when 4 5 + 3 − 7 2 + 5 is divided by 2x + 1 b. Given that the expression 3 − 2 + + 18 is exactly divisible by 2 + − 6, find the values of a and b.  Exercise 21 The expression 2 3 − 3 2 + − 5 gives a remainder of 7 when divided by x-2. find the value of c  Exercise 22 Given that − 4 is a factor of 2 3 − 3 2 − 7 + , where b is a constant, find the remainder when the same expression is divided by 2x-1  Exercise 23 The expression 3 − 5 2 + + 9 where c is a constant, gives a remainder of 6 when divided by x-3. Find the remainder when the same expression is divided by x-4  Exercise 24 a. Use the remainder theorem to find the remainder when 5 + 3 3 + 2 + 4 2 is divided by x+2. b. If x-2 is a factor of 3 3 − 2 − 6 + 8 , find the value of a and the remaining factors of the expression  Exercise 25 The polynomial 2 3 − 3 2 + − 5 gives a remainder of 7 when divided by x-2. Find the value of a.  Exercise 26 The expression 2 3 + 3 2 + + gives a remainder of 7 when divided by x- 2 and a remainder of -3 when divided by x-1. Find the values of a and b.  Exercise 27 3 2 Factorise 6 +4 −9 −6  Exercise 28 4 3 2 Factorise +3 +5 +9 +6  Exercise 29 4 3 2 Factorise and hence solve the equation: 2 +3 − 12 − 15 + 10 = 0  Exercise 30 4 3 2 Given that x+1 is a factor of =4 + −9 + + 2. a. Find p b. Hence factorise f(x) completely.  Exercise 31 3 2 Factorise and hence solve the equation: 3 −4 −6 +8=0  Exercise 32 3 2 Solve the equation 3 +2 − 21 − 14 = 0  Exercise 33 3 2 Factorise: 2 − −2 +1  Exercise 34 3 2 Factorise 2 − −5 −2  Exercise 35 4 3 2 Factorise and hence solve the equation: +3 − 15 − 15 + 50 = 0  Exercise 36 Determine whether the following polynomial has any rational zero, justifying your answer: 4 − 2 3 − 8 2 + 12 − 4  Exercise 37 4 3 2 Factorise: +2 −7 − 8 + 12 Exercise 38  Exercise 39 3 2 Solve the equation 6 −9 − 8 + 12 = 0  Exercise 40 4 3 2 Given that x-1 is a factor of = +3 + − −2 a. Find c b. Hence factorise f(x) completely.  Exercise 41 Let = 36 − 2 A. Determine the maximal domain of f b. Determine the range of f  Exercise 42 1 Find the range and domain for = 1 2+ The End  Learning Outcomes By the end of this lecture you should be able to perform basic calculation involving: – real number system and significant figure – proportion, percent and variation – law of indices and standard form – law of logarithm – progression  Real Number System Natural number, N: { 1, 2, 3, ……… } Integers, Z: { ……, -3, -2, -1, 0, 1, 2, 3,……. } Rational number, Q: { ; , ∈ , ≠0} Irrational number, Q’: {cannot be written as a fraction, e.g. 2 }  Question 1 Determine which of the numbers below are integers, rational numbers, irrational numbers and real numbers: 13 − , 4, 13, 5 2  Question 1: Answer Integer: 4 = 2 13 Rationals: 4, − 5 Irrationals: 13, 2 13 Real numbers: 4, − 5 , 13, 2  Significant Figure Not significant 0.004004500 Significant  Significant Figure: Examples 202: 3 sig figs 1.01: 3 sig figs 1305: 4 sig figs 10001: 5 sig figs 3002: 4 sig figs 62004: 5 sig figs  Question 2 Determine the number of significant figures in the following number: 1. 328 2. 6.01 3. 0.805 4. 0.043 5. 1.510 × 10 3 6. 5000 7. 5.0 × 10 3 8. 5.000 × 10 3 9. 0.030200 10. 6.40 x 10^4  Question 2: Answer 1. 328: 3 sig figs 2. 6.01: 3 sig figs 3. 0.805: 3 sig figs 4. 0.043: 2 sig figs 5. 1.510 × 10 3: 4 sig figs 6. 5000: 1. 2, 3 or 4 sig figs 7. 5.0 × 10 3: 2 sig figs 8. 5.000 × 10 3: 4 sig figs 9. 0.030200: 5 sig figs 10. 6.40 × 10 4: 3 sig figs  Percent Percent (%) mean “in a hundred” Common fractions may be converted to percent by dividing the numerator by the denominator and multiplying by 100 e.g. 3 3 – To convert 8 to % : 8 × 100 = 37.5%  Question 3 A pharmacist had 5 grams of codeine sulfate. He used it in preparing the following: 8 capsules each containing 0.0325 gram 12 capsules each containing 0.015 gram 18 capsules each containing 0.008 gram How many grams of codeine sulfate were left after he had prepared the capsules  Question 3: Answer 5 − 0.0235 × 8 − 0.015 × 12 − 0.008 × 18 = 4.488  Proportion A proportion is the expression of the equality of two ratios. It is usually written in the following form: = a and d are known as extremes while b and c are known as meas The product of the extremes is equal to the product of the means This allows us to find the missing term of any proportion when the other three terms are known  Question 4 If 3 tablets contain 975 milligrams of aspirin, how many milligrams should be contained in 12 tablets?  Question 4: Answer 3( ) 975 ( ) 12 ( = ( ) 12 × 975 ( ) = 3( ) = 3900  Question 5 If 30 mL represent 1/6 of the volume of a prescription, how many milliliters will represent ¼ of the volume? Question 5: Answer  Variation Types of quantities – Constant – Variables Direct variation Inverse variation Joint variation Law of Indices  Question 6 6 2 × 12 3 Express 84 in the form 2 3 Question 6: Answer  Standard Form Standard form or scientific notation is useful for writing very large and very small numbers. Every number can be expressed in standard form: × 10 , ℎ 1≤ ≤ 10 ∈ Common prefixes used for small quantities – Milli (m): 10 −3 – Micro (μ): 10 −6 – Nano (n): 10 −9 – Pico (p): 10 −12  Question 7 Write in standard form: a. 1321000000 km b. 0.000000516 g Evaluate and give your answer in standard form: c. 6.4 × 10 8 − 5.2 × 10 7 d. 2.4 × 10 5 ÷ 6 × 10 8 e. 78 μg + 512 ng. Give your answer in g  Question 7: Answer a. 1.321 × 10 9 b. 5.16 × 10 − 7 c. 5.88 × 10 8 d. 4 × 10 − 4 e. 7.8512 × 10 − 5 Law of Logarithm Law of Logarithm Law of Logarithm  Question 8 Simplify: a. log 12 + log 8 – 2 log 6 b. log 100 – 2 log 0.001 64 c. Express log 36 in terms of x and y where x = log 2 and y = log 3 64 d. hence, evaluate log 36 correct to 4 d.p. given that log 2= 0.30103 and log 3 = 0.47712  Question 8: answer 8 a. log 3 b. 8 c. 4x – 2y d. 0.2499 Application of index and log: Arithmatic Progression Geometric Progression  Exercise 1 3 270 Express 45 in the form of 2 5  Exercise 2 Evaluate and express your answer in standard form: a. 4.23 × 10 17 ÷8.63 × 10 16 b. 8.5 × 10 27 ÷ 3.4 × 10 14 c. 25 nm – 89 pm. Express in m  Exercise 3 Express as a single logarithm: log 1000 + log 20 – 3 log 5 -1 Exercise 4  Exercise 5 If log 6 = p and log 3 = q, then log 5 is equal to: 1 A. 2 B. 1 – p + q C. 1 – p - q D. log 2 - 1 E. 1 + p + q Exercise 6  Exercise 7 Given that log 4 + 2 log p = 2, find the value of p without using a calculator  Exercise 8 Express as a single logarithm in its simplest form: 2 5log 2 − log − log 4  Exercise 9 Express as a single logarithm in its simplest form: 3 log 2 + 2log 18 − log 36 2 Exercise 10  Exercise 11 0.2 10 Express 3 in the form 2 5 25  Exercise 12 3 6.75 Express 90 in the form 2 5  Exercise 13 Express as a single logarithm in its simplest form 2ln − 3ln + 2ln  Exercise 14 Express as a single logarithm in its simplest form: 6ln + − 2ln −5 The End  Learning Outcomes By the end of this lecture you should be able to perform basic calculation involving: – order relation of real numbers – Linear and quadratic inequalities – Solving inequalities by factoring – Absolute values – Polynomial inequalities  Order Relation The order relation (< or >) can be defined for real numbers. It has the following properties If a < b, then a + c < b + c If c > 0 and a < b, then ac < bc If c < 0 and a < b, then ac > bc  Question 1 Solve 2 + 3 ≥ 6 Question 1: Answer 3 ≥ 2  Question 2 2 Solve < + 12  Question 2: Answer +3 −4