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Name:_____________________________ Date:___________________ Calculus Pretest I. Linear Equations 1. Find the point-slope form of the equation of the line passing through the points (-1, 2) and (1, -2). II. Solving Equations, finding zeros, roots of functions, factoring. 1. Factor by grouping: 5cos2(x) – 5sin2(x) + sin(x) + cos(x) 2. Factor over the set of real numbers: x2 + 5x + 5. 3. Factor the polynomial: x 3 8 4. Solve for x . a. x 3 x 2 x 2 2 0 b. x 2 x 4 0 c. 3x 2 8 x 3 0 x4 2 x y 0 5. Solve the system for x and y: 2 x y 1 6. Solve for x : (x + 1)1/2 + (x – 1)1/2 = 2 III. Families of Functions (Domain, Range, Asymptotes, Intercepts, Max, Min, Etc) 1. Given the graph of f ( x) log 4 ( x) , determine the domain, range, asymptotes and intercepts. 2. Given the graph of g(x) = cos(2x + π /3), determine the domain, range, period, frequency, amplitude, and phase shift. IV. Rules of Logs and Exponentials 1. Solve for x . a. 3x+1 = 15 b. log(x-2) + log(2x-3) = 2log(x) c. -14 + 3ex = 11 d. 5 + 2ln(x) = 4 2. If 55 is approximately equal to 3000, then approximate the value of 510? V. Algebraic Substitution 4x 2 1. If f ( x) , find f (b 1) . x4 1 f ( x h) f ( x ) 2. Given f ( x) 2 , find the difference quotient, , and simplify completely. h x VI. Solve Inequalities/Polynomial/Absolute Value 1. |x – 2| +3 < 7 Name:_____________________________ Date:___________________ Calculus Pretest 2. x2 – 25 > 0 VII. Composition of Functions/ Inverse Functions 1. If r(x) = x2 – 4 and s(x) = x1/2, find s(r(x)) and specify its domain. 2. Given the graph of an inverse function f-1, sketch a graph of the function f. VIII. Simplifying Rational Expressions/ Rationalizing Algebraic Expressions 1. Reduce the expression to lowest terms: (x + 1)3(x – 2) + 3(x + 1)2 (x + 1)4 1/2 1/3 2. Reduce the expression to lowest terms: x – x x1/6 3. Simplify: x1/2 – 51/2 x–5 3. Simplify: x4 + x – 2 x-1 IX. Represent Geometric Parameters with Algebraic Expressions 1. The length of a rectangle is 3 meters more than twice its width. What is the width if the perimeter of the rectangle is 186 meters? 2. A square picture is mounted on a larger rectangular sheet of poster paper leaving a border around the picture of 2 inches on the bottom and 1 inch on each of the remaining sides. The sheet of poster paper has an area of 42 square inches. If x represents the length of a side of the square picture, then write the equation that can be used to determine the value of x and solve it. X. Trigonometric Functions and Solving Trigonometric Equations. 1. Prove: csc2(x) – cot2(x) = 1 2. If f(x) = cos(3x), then find f(π/6). 3. For which values of x is sec(x) not defined? 4. Simplify the following expression: sec(θ)cot(θ)sin2(θ) 5. Given that cos(θ) = ½, and θ terminates in quadrant I, evaluate all six trigonometric functions of θ. 6. Solve the following equation for x: 2sin(x) – 1 = 0.