Numbers Content Points. Reference sheet (1 pt. each) 1-7 Linear Equations (1 pt. each) / Factoring (2 pt. each) /28

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1 Summer Packet 2015 Your summer packet will be a major test grade for the first nine weeks. It is due the first day of school. You must show all necessary solutions. You will be tested on ALL material; it is in your best interest to do your own work. Numbers Content Points Reference sheet (1 pt. each) / Linear Equations (1 pt. each) / Factoring (2 pt. each) / Polynomial Functions (2 pt. each) / Functions (1 pt. each) / Inverses (2 pt. each) / Function Operations (5 pt. each) / Radical/Exponential Functions (2 pt. each) / Logarithmic Functions (2 pt. each) /16 35 Unit Circle (complete circle) /10 Trigonometric Identities (1 pt. each) / Evaluate Trigonometric Expressions (2 pt. each) /36 38 Trigonometric Expressions (2 pt. each) /20 39 Trigonometric Equations (2 pt. each) /12 40 Introduction to Limits (1 pt. each) /6 Total /300 Comments: 1

2 Summer Packet 2015 Greetings Future AP Calculus Scholar, I am excited about the work that we will do together during the school year. I do not yet know what your math capability is, but I can assure you that your mathematical knowledge base will be expanded as you do the reviews during the summer and as we work through calculus together during the school year. Calculus is a college-level course and will take a lot of time next year, especially if you have not adequately mastered key topics in Algebra I, Algebra II, Geometry, and Precalculus. I have included many of the major before-calculus topics in this Summer Review Series. It is further recommended that you solve the problems WITHOUT the use of a calculator. However, you may verify your work (after solving a problem) by using a calculator to confirm your solution to the problem. All problems will need to be solved completely and correctly not just attempted and not just answered. It may be necessary in several problem sets to use additional paper for the problem solution. Do not crowd your solution into a space on the page that is too small. Please note also that you are being asked to provide SOLUTIONS to some problems and ANSWERS to other problems. An ANSWER can be a number, a variable, an expression. A SOLUTION, on the other hand, includes the details and process that you use to arrive at your ANSWER. Example: If you are asked to solve the equation 2x + 1 = 7 The ANSWER is x = 3 A SOLUTION is 2x + 1 = 7 2x = 6 x = 3 Each problem should be done neatly with much thought given to the concept emphasized in the problem. You are encouraged to communicate with other students, parents, or even teachers, to get a complete understanding of the concepts. Please do not rush through this assignment. Do not copy answers or solutions from another person. Each topic covered in this series will be revisited many, many times during the school year. Feel free to me if you have any questions. bcummings@greenville.k12.sc.us Ms. Cummings 2

3 Reference Sheet Fill in the following formulas or definitions. This will become a helpful reference sheet for you throughout the year. Slope Formula Slope-intercept form Point- slope form General equation of a vertical line General equation of a horizontal line Absolute value of a number Quadratic Formula Y-axis symmetry Origin symmetry Distance Formula Circle- Area and Circumference Triangle- Area (any size) Triangle- Area (equilateral) Pythagorean Theorem 3

4 Rectangle- Area and Perimeter Trapezoid- Area Rectangular Box- Volume, Surface Area Sphere- Volume, Surface Area Right circular cone- volume Right circular cylinder- volume Properties of natural log (definition) Properties of natural log (product) Properties of natural log (quotient) Properties of natural log (exponents) Properties of natural log (zero) 4

5 Linear Equations 1. Write the slope-intercept form of a line through (-3, 1) with slope Write the general form of a line that crosses the x-axis at 4 and whose slope is 3/4. 3. Write an equation of the line that passes through the two points (2,-1) and (-3, 2). 4. Write the general form of an equation of the line whose point-slope for is given by y 3 = -3(x + 2). 5. Write an equation of the line that is parallel to the line whose equation x 2y = Write an equation of the line that is normal to the line in the problem above that passes through the point (h,k). 7. Graph: 5

6 Factoring 8. Factor completely: a. 20x x x b. 5x 3 15x 2 50x c. x 2 3x 18 d. 3x 2 10x 8 e. 10x x Factor completely: a. cos 4 x sin 4 x b. x 3 + 3x 2 4x 12 c. 81b 10 2ab 6 d. mn + mp + jn + jp e. x 3 + 8y Simplify the expressions as much as possible. a. 2 sin 2 x + 3 sin x + 1 b. cos 3 x 5 cos 2 x + 4 cos x c. sec 2 x + sec x tan x d. 2 cos x sin 2 x 2 cos x sin x + sin x 1 6

7 Polynomial Functions Solve: 16. x 3 9x x 7 = x 2 35 = x 5 4x 3 + x = x 2 + 3x 1 = x 3 8 = x 2 + 4x 32 = x x = x = 6x 20. x 3 + 5x 2 + 3x + 15 = x 2 + 4x 2 = 0 7

8 Functions 21. Determine whether the graph of each function is odd, even, or neither. Give a reason. 22. Graph each parent function. y = x y = x y = ln x y = x 2 y = sin x y = 1 x y = x 3 y = cos x y = 1 x 2 y = x y = e x 8

9 Inverses 23. Find the inverses of the following functions: a. f(x) = 2x b. g(n) = n Verify these functions are inverses using composition: Function Operations 25. Let f(x) = x 2 x + 3. Find f(a), f(a + h), and f(a+h) f(a) h. 26. If f(x) = 1 f(x+h) f(x), find. x h 27. Let f(x) = 3 2x and g(x) = 3. Find the sum, product, and quotient of f and g. Specify the x 2 domain of each result. (Note: Do not simplify the sum to a common denominator). For the functions f and g in problems 4 and 5 below, find the following: a) f(g(x)), b) (g f)(x), c) f(g(0)), d) g(f(0)), e) g(g( 2)), and f) f(f(x)). Find the domain for parts a), b), and f). 28. f(x) = x + 5 and g(x) = x f(x) = x + 1 and g(x) = x 2 9

10 Radical/Exponential Functions 30. Simplify the following exponential expressions. a. x x d. 5 ( x)( x2) x 6 b. x x 3 x c. 7 x 5 4 x 3 e. ( 3 x10 4 )( x 3 7 ( x)( x 3 ) ) 31. Solve the following equations: a. 3 x = 40 c. 12 x2 = 890 b x = 81 2+x d. 8e 2x+1 =

11 Logarithmic functions 32. Expand using the properties of logarithms. a. log 5 2x 3 3 /y b. log xyz 33. Condense using the properties of logarithms. a. 2 log 4 x (3 log 4 y + log 4 z) b. 2(log 2x log y) (log 3 + 2log 5) 34. Solve the following logarithmic equations. a. log 8 (3x) = log 8 (48 x) b. log 7 (x 2 + 9x) = log 7 (4x 6) c. log 2 (2x) + log 2 (x 5) = 3log 2 (x) log 2 x d. 2log 5 x + log 5 (9) = 2 11

12 Unit Circle 35. Complete the unit circle by memory. Be sure to fill in the radians and degrees. 12

13 Trigonometric Identities Fill in the blank by completing each trig identity: The Six Trigonometric Functions (right triangle and unit circle) Reciprocal Identities sin θ = = csc θ = = sin θ = csc θ = cos θ = = sec θ = = cos θ = sec θ = tan θ = = cot θ = = tan θ = cot θ = Pythagorean Identities Quotient Identities Even-Odd Identities tan θ = cot θ = sin( θ) = csc( θ) = Sum and Difference Identities sin(α ± β) = cos( θ) = tan( θ) = sec( θ) = cot( θ) = cos(α ± β) = Cofunction Identities sin ( π θ) = csc 2 (π 2 θ) = tan(α ± β) = Double Angle Identities sin(2θ) = cos ( π θ) = sec 2 tan ( π θ) = cot 2 Half Angle Identities (π 2 (π 2 θ) = θ) = cos(2θ) = = = tan(2θ) = sin ( θ 2 ) = cos (θ 2 ) = tan ( θ ) = = = 2 Power-Reducing Formulas sin 2 θ = cos 2 θ = 13 tan 2 θ =

14 Evaluating Trigonometric Expressions 36. Evaluate the following. a. tan ( 4π 3 ) sin2 ( π 2 ) d. cot ( 2π 3 ) g. sin 2 ( 4π 5 ) + cos2 ( 4π 5 ) b. 5 sec ( 5π 6 ) e. csc 2 ( 5π 4 ) h. sin ( 11π 6 ) + 5 cos (π 4 ) c. 2 sin 2 ( 11π 6 ) + cos 2 ( 3π 4 ) f. 8 tan(π) 6 sin(4π) i. sec ( π 3 ) csc (π 6 ) j. 5tan 3 ( 3π 4 ) 4 cos(8π) Inverse Trigonometric Expressions 37. Evaluate each expression: a. arcsin ( 1 ) d. arctan( 3) g. arcsec(2) 2 b. sin (arctan ( 3 3 )) e. sin (arcsin (1 5 )) h. sin (arctan ( 3 13 )) c. cos (arccos ( 3π )) f. arccot(0) 4 14

15 Trigonometric Expressions 38. Simplifying trig expressions a. tan2 x sec x d. csc x sin x g. sin 2 x(1 + cot 2 x) b. sec x sec x sin 2 x e. cot x sec x sin x h. sin x csc x + cos x sec x c. cos 2 x sin 2 x f. tan x cot x csc x i. tan x + cos x 1+sin x j. 1 sin2 x 1+cos x Trigonometric Equations 39. Solve the following equations on the interval [0,2π]: a. 4 cos x + 3 = 2 cos x d. sin 2 x sin x 1 = 0 b. cos x sin x = sin x e. 3(1 sin x) = 2 cos 2 x c. 1 + sin 2 x = cos x f. 2 cos 2 x 5 cos x = 3 15

16 Introduction to Limits 40. Use the graph to find the following limits: 16