Summer Packet Greetings Future AP Calculus Scholar,

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1 Summer Packet 2017 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 review 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 imperative that you have these concepts mastered prior to beginning your education in Calculus. 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. You will not be expected to complete this packet for a grade however, you will be tested on any material included in this packet. This test will be the first week you return to school in August. Feel free to me if you have any questions. bcummings@greenville.k12.sc.us Ms. Cummings 1

2 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 2

3 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) 3

4 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: 4

5 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 5

6 Polynomial Functions Solve: 11. 5x 2 35 = x 3 9x x 7 = x 2 + 3x 1 = x 5 4x 3 + x = x 2 + 4x 32 = x 3 8 = x = 6x 19. 8x x = x 2 + 4x 2 = x 3 + 5x 2 + 3x + 15 = 0 Functions 6

7 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 7

8 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 8

9 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 =

10 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 10

11 Unit Circle 35. Complete the unit circle by memory. Be sure to fill in the radians and degrees. 11

12 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 = 12 tan 2 θ =

13 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 13

14 Trigonometric Expressions 38. Simplifying trig expressions a. tan2 x sec x h. sin x csc x + cos x sec x e. cot x sec x sin x b. sec x sec x sin 2 x i. tan x + cos x 1+sin x f. tan x cot x csc x c. cos 2 x sin 2 x j. 1 sin2 x 1+cos x g. sin 2 x(1 + cot 2 x) d. csc x sin 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 14

15 Introduction to Limits 40. Use the graph to find the following limits: 15

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