Calculus Drafting Module 1 PDF

INTERVENTION FOR CALCULUS DRAFTING  Algebra Trigonometry Plane Geometry Analytic Geometry and Solid Geometry ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials Topic Outcomes ▪ Identify the different subsets of...

INTERVENTION FOR CALCULUS DRAFTING  Algebra Trigonometry Plane Geometry Analytic Geometry and Solid Geometry ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials Topic Outcomes ▪ Identify the different subsets of real numbers; ▪ Plot numbers on the real number line and their notations; ▪ Define the absolute value and perform operations involving them; ▪ Use the laws of exponents to simplify exponential expressions; ▪ Perform algebraic operations on polynomial expressions; ▪ Factor polynomial expressions using different method; ▪ Perform algebraic operations on rational expressions. ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Real numbers include all the numbers that can be found on the number line. ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Imaginary Numbers Numbers that involve the square root of a negative number. The basic imaginary unit is “i" 𝑖0 = 1 𝑖4 = 1 𝑖8 = 1 𝑖 12 = 1 𝑖 16 = 1 𝑖1 = 𝑖 𝑖5 = 𝑖 𝑖9 = 𝑖 𝑖 13 = 𝑖 𝑖 17 = 𝑖 𝑖 2 = −1 𝑖 6 = −1 𝑖 10 = −1 𝑖 14 = −1 𝑖 18 = −1 𝑖 3 = −𝑖 𝑖 7 = −𝑖 𝑖 11 = −𝑖 𝑖 15 = −𝑖 𝑖 19 = −𝑖 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Complex Numbers Numbers that have both a real part and an imaginary part (𝑎 + 𝑏𝑖) Addition/Subtraction Multiplication Division (𝑎 ± 𝑏𝑖) (𝑎 ± 𝑏𝑖) ± (𝑐 ± 𝑑𝑖) (𝑎 ± 𝑏𝑖)(𝑐 ± 𝑑𝑖) (𝑐 ± 𝑑𝑖) 𝑎 ± 𝑏𝑖 (𝑎 ± 𝑐) ± 𝑏 ± 𝑑 𝑖 𝑐(𝑎 ± 𝑏𝑖) ± 𝑑𝑖(𝑎 ± 𝑏𝑖) 𝑥 (𝑐 ± 𝑑𝑖)𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒 𝑐 ± 𝑑𝑖 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Non-Integers Rational Negative Numbers Numbers Natural Real Integers Numbers Numbers Whole Numbers Irrational Zero Numbers ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Rational Numbers – can be expressed in fraction, either terminating Non-Integers or repeating decimals Rational 1 Negative 3 −3 1 Numbers = 0.50 = 0.75 Numbers − 3 = 0.3333 = 2 4 1 3Positive Real Integers Numbers Numbers Irrational Numbers – Whole cannot be expressedNatural in fraction, and have not Numbers decimals Numbers terminating and non-repeating Irrational 𝜋 = 3.14159 2 = 1.41421 𝑒 = 2.71828 Zero Numbers ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Non-Integers Rational Negative Integers – include all whole numbers both positive and Numbers Numbers negative Positive Real Integers... , −4, −3, −2, −1, 0, 1, 2, 3, 4,Numbers... Numbers Whole Natural Numbers – have fractional Non-Integers Numbersor decimal part Irrational Zero 1 3 1 Numbers , , , 0.25, −2.5, −0.1 2 4 8 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Non-Integers Rational Negative Negative Numbers – numbers less Numbers Numbers than zero Positive Real Integers... , −4, −3, −2,Numbers −1 Numbers Whole Natural Whole Numbers – non-negative Numbers Numbers Irrational numbers starting zero Numbers 0, 1, 2, 3, 4,..Zero. ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Non-Integers Rational Negative Natural Numbers – Numbers Numbers Natural set of positive Real Integers Numbers integers Numbers Whole 1, 2, 3, 4,... Numbers Irrational Zero Numbers Zero – represents absence of quantity 0 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Real Numbers Set Symbol Examples 1 1 Rational Numbers Q , 0.75, −3, 2 3 Integers Z −4, −3, −2, −1, 0, 1, 2, 3, 4,.. Whole Numbers W 0, 1, 2, 3, 4,... Natural Numbers N 1, 2, 3, 4,... ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System Examples Numbers Set Numbers Set 10 Q, Z, W, N 13 Irrational Number -6 Q, Z − 4 Q, Z 0 Q, Z, W 0.01 Q 3.14 Q 0.2525 Q 1 0.1666 Q Q 4 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Properties of Real Numbers Property Addition Subtraction Multiplication Division Closure 𝑎+𝑏∈𝑅 𝑎−𝑏∈𝑅 𝑎𝑥𝑏∈𝑅 𝑎/𝑏 ∈ 𝑅 except 0 Commutative 𝑎+𝑏 =𝑏+𝑎 𝑎−𝑏 ≠𝑏−𝑎 𝑎𝑥𝑏=𝑏𝑥𝑎 𝑎/𝑏 ≠ 𝑏/𝑎 𝑎 𝑏 Associative 𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐) 𝑎 − 𝑏 − 𝑐 ≠ 𝑎 − (𝑏 − 𝑐) 𝑎 𝑥 𝑏 𝑥 𝑐 = 𝑎 𝑥 (𝑏 𝑥 𝑐) /𝑐 ≠ 𝑎/ 𝑏 𝑐 Distributive N/A N/A 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + (𝑎𝑐) N/A Identity 𝑎+0=𝑎 N/A 𝑎𝑥1=𝑎 𝑎/1= 𝑎 1 Inverse 𝑎 + (−𝑎) = 0 N/A 𝑎 =1 N/A 𝑎 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Properties of Negative Real Numbers Property Example Less than Zero −3 < 0 Greater than Negative Infinity −5 > −∝ Sum of Two Negatives −2 + −3 = −5 Sum of Negative and Positive −2 + 3 = 1 Difference of Two Negatives −2 − −3 = 1 Difference of Negative and Positive −2 − 3 = −5 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Properties of Negative Real Numbers Property Example Product of Two Negatives −2 𝑥 −3 = 6 Product of Negative and Positive −2 𝑥 3 = −6 Quotient of Two Negatives −6 / −3 = 2 Quotient of Negative and Positive −6/3 = −2 Absolute Value −3 + −1 = 4 Negative of Negative −(−3) = 3 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Properties of Equality Real Numbers Property Example Reflexive 𝑎=𝑎 Symmetric 𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎 Transitive 𝑖𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐 Substitution 𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑑 𝑏𝑦 𝑏 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Properties of Inequality Real Numbers Property Example Trichotomy 𝑎𝑚𝑜𝑛𝑔 𝑎 < 𝑏, 𝑏 < 𝑎, 𝑎𝑛𝑑 𝑎 = 𝑏, 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑖𝑠 𝑡𝑟𝑢𝑒 Symmetric 𝑖𝑓 𝑎 > 𝑏, 𝑡ℎ𝑒𝑛 𝑏 < 𝑎 Transitive 𝑖𝑓 𝑎 ≤ 𝑏 𝑎𝑛𝑑 𝑏 ≤ 𝑐, 𝑡ℎ𝑒𝑛 𝑎 ≤ 𝑐 Compound 𝑖𝑓 𝑎 < 𝑥 < 𝑏, 𝑡ℎ𝑒𝑛 𝑥 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑏 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Perform the indicated operations on real numbers. a. −3 + (−5)2 b. −3 − 8 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Perform the indicated operations on real numbers. c. 3 (−2)3 d. −2 3 4 /2 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Perform the indicated operations on real numbers. e. 3 (−2 + 5)2 +(6 − 3)/(8 − 2) f. 12(−4) + 5(2) ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Find the perimeter and area of the plane figure. a. 𝑎 = 62 + 10 𝑎 𝑏 = 16 + 20 𝑏 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Find the perimeter and area of the plane figure. b. 𝑎 = 72 − 9 𝑎 𝑏 = −3 + 25 𝑐 = 4 + 35 𝑐 𝑏 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING  Algebra: Real Numbers, Exponents and Polynomials The Real Number System ▪ Sample Problems Find the perimeter and area of the plane figure. 3 c. 𝑎 = 216 𝑎 ENITV12D INTERVENTION FOR CALCULUS-DRAFTING