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Honors Calculus Summer Assignment 2016 The following packet contains topics and definitions that you will be required to know in order to succeed in Honors Calculus this year. You are advised to be familiar with each of the concepts and to complete the included problems by September 1, 2016. All of these topics were discussed in either Algebra II or Precalculus and will be used frequently throughout the year. All problems are expected to be completed. Honors Calculus Summer Assignment Please look over and complete problems on the following review topics. These problems will help you prepare for your Honors Calculus course next year. This assignment will be counted towards your grade in Honors Calculus. Please give it your serious attention. Section 1: Equation of a line. A) Point-Slope formula:_______________________ B) Slope-intercept form:______________________ C) Standard form:______________________ Complete the following and write your answer in slope-intercept form. 1) Write the equation of the line through the points (-1,5) and (2,7). 2) Write the equation of the line through the points (-5,6) and (3,6). 3) Write the equation of the line through the points (-4,1) and (-4,16). Complete the following and write your answer in slope-intercept form. 4) 5) 6) 7) 8) Write the equation of the line parallel to y = -3x + 1 through the point (2,6). Write the equation of the line perpendicular to y = 5x – 3 through the point (-1,2). Write the equation of the line perpendicular to 3x – 2y = 8 through the point (0,4). Write the equation of the line perpendicular to y = 4 through the point (2,-5). Write the equation of the line perpendicular to x = -10 through the point (6,7). Section 2 – Simplifying Expressions Eliminate compound fractions or combine terms to one single term for the following. 1) 𝑥−3 + √𝑥−8 √𝑥 − 8 1 1 − 𝑥 + 2 2 2) 𝑥 Section 3 – Trigonometry Fill in the table. Answers should be exact. No decimals. Degree 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 Radians SINE COS TAN CSC SEC COT Section 4: Trigonometric Functions and their graphs. 1) Graph one period of the six trigonometric functions and identify the following for each. a. Domain b. Range c. Period 2) Graph the six inverse trig functions and identify the following for each. a. Domian b. Range Section 5: Trig identities 1) The following identities will be used quite often this year. Please commit them to memory. a. Pythagorean identities i. sin2x + cos2x = 1 ii. 1 + tan2x = sec2x iii. 1 + cot2x = csc2x b. Reciprocal identities i. sin x = 1/csc x ii. cos x = 1/sec x iii. tan x = 1/cot x iv. csc x = 1/sin x v. sec x = 1/cos x vi. cot x = 1/tan x c. Quotient identities i. tan x = sin x / cos x ii. cot x = cos x / sin x d. Double Angle Formulas i. sin 2x = 2 sin x cos x ii. cos 2x = cos2x – sin2x = 2cos2x – 1 = 1 – 2sin2x e. Power Reducing Formulas 1−cos 2𝑥 i. 𝑠𝑖𝑛2 𝑥 = 2 ii. 𝑐𝑜𝑠 2 𝑥 = 1+cos 2𝑥 2 Section 6 – Inverse Functions Find the inverse function of the following algebraically and graph both the original function and the inverse. 1) y = 3x-5 2) 𝑦 = √𝑥 − 3 3) 𝑦 = 2𝑥−1 𝑥+3 Section 7 – Properties of Exponents and Logarithms Properties of Exponents: 1. Whole number exponents: 2. Zero exponents: 𝑥 𝑛 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ … ∙ 𝑥 (n factors of x) 𝑥 0 = 1, 𝑥 ≠ 0 3. Negative Exponents: 1 𝑥 −𝑛 = 𝑥 𝑛 𝑛 √𝑥 = 𝑎 → 𝑥 = 𝑎𝑛 4. Radicals (principal nth root): 5. Rational exponents: 𝑥 1⁄ 𝑛 6. Rational exponents: 𝑥 𝑚⁄ 𝑛 𝑛 = √𝑥 𝑛 = √𝑥 𝑚 Operations with Exponents: 1. Multiplying like bases: 𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑚+𝑛 2. Dividing like bases: 𝑥𝑚 𝑥𝑛 3. Removing parentheses: (𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛 Simplify the following. 1) 3𝑎−2 𝑏5 (5𝑎6 𝑏−3 𝑐 −4 ) 2) 5𝑥 −2 𝑦 5 𝑧 3𝑥𝑦𝑧 −7 3 3𝑥𝑦 −1 𝑧 2 ) 3 4 −3 4𝑥 𝑦 𝑧 3) ( 4) 93⁄2 2 3 5) (3) − 123⁄2 = 𝑥 𝑚−𝑛 𝑥 𝑛 𝑥𝑛 (𝑦) = 𝑦𝑛 (𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚 Properties of Logarithms 1. 2. 3. 4. 𝑥𝑦 log𝑒 𝑥 = ln 𝑥 log 𝑎 𝑥 𝑛 = n log 𝑎 𝑥 log 𝑎 (𝑥𝑦) = log 𝑎 𝑥 + log 𝑎 𝑦 𝑥 log 𝑎 (𝑦) = log 𝑎 𝑥 − log 𝑎 𝑦 1) ln 𝑧 2 3 2) ln ℎ √ℎ − 1 𝑥 2 −1 3) ln ( 𝑥3 4 ) 4 4) ln 𝑒 Write the following as the logarithm of a single quantity. 5) 3 ln 𝑥 + 2 ln 𝑦 − 6 ln 𝑧 6) 1 5 [5 ln(𝑥 + 3) + ln 𝑥 − ln(𝑥 2 + 4)] 7)2 ln 5 − 1 2 ln(𝑦 − 2)