Intervention for Calculus - Drafting PDF

INTERVENTION FOR CALCULUS - DRAFTING ENITV12D  TABLE OF CONTENTS SECTION 1 : ALGEBRA SECTION 2 : TRIGONOMETRY SECTION 3 : PLANE GEOMETRY SECTION 4 : SOLID GEOMETRY SECTION 5 : ANALYTIC GEOMETRY SECTION 6 : INTRODUCTION TO LIMITS SECTION 7 : INTRODUCTION TO DERIVATIVES  TABLE OF CONTENT...

INTERVENTION FOR CALCULUS - DRAFTING ENITV12D  TABLE OF CONTENTS SECTION 1 : ALGEBRA SECTION 2 : TRIGONOMETRY SECTION 3 : PLANE GEOMETRY SECTION 4 : SOLID GEOMETRY SECTION 5 : ANALYTIC GEOMETRY SECTION 6 : INTRODUCTION TO LIMITS SECTION 7 : INTRODUCTION TO DERIVATIVES  TABLE OF CONTENTS SECTION 1 : ALGEBRA TOPIC 1: Algebra Part 1 TOPIC 2. Algebra Part 2 ▪ Real Number System ▪ Rational Expressions (simplifying rational expressions, rational ▪ Laws of Exponents exponents, and radicals) ▪ Algebraic Expressions (Algebraic ▪ Linear Equations Operations and Factoring) ▪ Quadratic Equations ▪ Word Problems involving Linear and Quadratic Equations  QUADRATIC EQUATIONS AND APPLICATIONS Factoring A quadratic equation in 𝑥 is an equation that can be written in the general form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 Where 𝑎, 𝑏 and 𝑐 are real numbers with 𝑎≠0. A quadratic equation in is also called a second-degree polynomial equation in 𝑥. Solving a Quadratic Equation by Factoring 1. ) 2𝑥 2 + 9𝑥 + 7 = 3 𝑆𝑒𝑡 1𝑠𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 𝑆𝑒𝑡 2𝑛𝑑 𝑓𝑎𝑐𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 2𝑥 2 + 9𝑥 + 4 = 0 2𝑥 + 1 = 0 𝑥+4=0 2𝑥 + 1 𝑥 + 4 = 0 1 𝑥 = −4 𝑥=− 2  QUADRATIC EQUATIONS AND APPLICATIONS Solving a Quadratic Equation by Factoring 2. ) 6𝑥 2 − 3𝑥 = 0 3𝑥 2𝑥 − 1 = 0 3𝑥 = 0 𝑥=0 𝑆𝑒𝑡 1𝑠𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 1 2𝑥 − 1 = 0 𝑥= 𝑆𝑒𝑡 2𝑛𝑑 𝑓𝑎𝑐𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 2  QUADRATIC EQUATIONS AND APPLICATIONS Extracting Square Roots  QUADRATIC EQUATIONS AND APPLICATIONS Extracting Square Roots 1. ) 4𝑥 2 = 12 𝑥2 = 3 𝑥=± 3 2. ) 𝑥 − 3 2 = 7 𝑥−3=± 7 𝑥 =3± 7  QUADRATIC EQUATIONS AND APPLICATIONS Completing the Square 1. ) 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 2 + 2𝑥 − 6 = 0 𝑏𝑦 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑥 2 + 2𝑥 − 6 = 0 𝑥+1 2 =7 𝑡𝑜 𝐶𝐻𝐸𝐶𝐾, 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑥 2 + 2𝑥 = 6 𝑥+1=± 7 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑜𝑛 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑥 2 + 2𝑥 + 12 = 6 + 12 𝑥 = −1 ± 7  QUADRATIC EQUATIONS AND APPLICATIONS Completing the Square 2. ) 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 2𝑥 2 + 8𝑥 + 3 = 0 𝑏𝑦 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 2𝑥 2 + 8𝑥 + 3 = 0 5 2𝑥 2 + 8𝑥 = −3 𝑥+2=± 2 2 3 𝑥 + 4𝑥 = − 2 5 3 8 𝑥 = −2 ± 𝑥 + 4𝑥 + 2 = − + 22 2 2 22 =4= 2 2 2 𝑡𝑜 𝐶𝐻𝐸𝐶𝐾, 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 2 5 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑜𝑛 𝑡ℎ𝑒 𝑥+2 = 2 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛  QUADRATIC EQUATIONS AND APPLICATIONS The Quadratic Formula Often in mathematics you are taught the long way of solving a problem first. Then, the longer method is used to develop shorter techniques. The long way stresses understanding and the short way stresses efficiency. For instance, you can think of completing the square as a “long way” of solving a quadratic equation. When you use completing the square to solve quadratic equations, you must complete the square for each equation separately. In the following derivation, you complete the square once in a general setting to obtain the Quadratic Formula—a shortcut for solving quadratic equations. QUADRATIC EQUATIONS AND APPLICATIONS  QUADRATIC EQUATIONS AND APPLICATIONS Quadratic Formula 3. ) 𝑈𝑠𝑒 𝑡ℎ𝑒 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝐹𝑜𝑟𝑚𝑢𝑙𝑎 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒 𝑥 2 + 3𝑥 − 9 = 0 𝑥 2 + 3𝑥 − 9 = 0 −3 ± 32 − 4(1)(−9) 𝑥= 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 2(1) −3 + 3 5 −3 − 3 5 𝑥= 𝑥= 2 2  QUADRATIC EQUATIONS AND APPLICATIONS Sample Problem : 1. ) A bedroom is 3𝑓𝑡 longer than it is wide and has an area of 154 𝑓𝑡 2. Find the dimensions of the room. 𝐴=𝑙𝑥𝑤 154 = 𝑤(𝑤 + 3) 154 = 𝑤 2 + 3𝑤 𝑤 = 11 𝑤 2 + 3𝑤 − 154 = 0 𝑤 = −14 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0  QUADRATIC EQUATIONS AND APPLICATIONS Sample Problem : 2. ) An L−shaped sidewalk from the athletic center to the library on a college campus as shown below. The sidewalk was constructed so that the length of one sidewalk forming the L was twice as long as the other. The length of the diagonal sidewalk that cuts across the grounds between the two buildings is 32𝑓𝑡. How many feet does a person save by walking on the diagonal sidewalk? 2𝑥 + 𝑥 − 32 = 3𝑥 − 32 = ? 2 2𝑥 + 𝑥 2 = 322 5𝑥 2 = 1024 𝑥 2 = 204.8 𝑥 = ± 204.8 𝑥 = 204.8 3𝑥 − 32 = 10.932 𝑓𝑡