SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

Transcription

1 SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET Name: 1. This packet is to be handed in on Monday August 8, All work must be shown on separate paper attached to the packet. 3. Completion of this packet will be counted as your first quiz grade. 4. Place all FINAL answers on answer sheet. 1

2 Formula Sheet Reciprocal Identities: csc x 1 sin x sec x 1 cos x cot x 1 tan x Quotient Identities: tan x sin x cos x cot x cos x sin x Pythagorean Identities: sin x cos x 1 tan x 1 sec x 1 cot x csc x Double Angle Identities: sinx sin xcosx tanx tan x 1 tan x cosx cos x sin x 1 sin x cos x 1 Logarithms: y log a x is equivalent to x a y Product property: Quotient property: Power property: Property of equality: log b mn log b m log b n log b m n log b m log b n log b m p plog b m If log b m log b n, then m = n Change of base formula: Derivative of a Function: Slope-intercept form: y mx b Point-slope form: y y 1 m(x x 1 ) log a n log b n log b a Standard form: Ax + By + C = 0 Slope of a tangent line to a curve or the derivative: lim h f (x h) f (x) h

3 Answer Sheet Solution Interval Notation Graph < x 4 [ 1,7)

4 36. a. b. c. 37. a. b. c. 38. a. b. c. d. 39. a. b. c. d. 40. a. b. c. 41. d. e. f

5 x f(x) 65. x f(x)

6 Complex Fractions Simplify each of the following a a 5 a. 4 x 5 10 x x x 3 Functions Let f (x) x 1 and g(x) x 1. Find each. 4. f () 5. g(3) 6. f (t 1) 7. f g() 8. g f (m ) 9. f (x h) f (x) h Let f (x) sin x Find each exactly. 10. f 11. f 3 Let f (x) x, g(x) x 5, and h(x) x 1. Find each. 1. h f () 13. f g(x 1) 14. g h(x 3 ) Find f (x h) f (x) h for the given function f. 15. f (x) 9x f (x) 5 x 6

7 Intercepts and Points of Intersection Find the x and y intercepts for each. 17. y x 16 x 18. y x 3 4x Find the point(s) of intersection of the graphs for the given equations. 19. x y 8 4x y 7 0. x y 6 x y 4 1. x 4y 0x 64y x 4y 30x 64y Interval Notation. Complete the table with the appropriate notation or graph. Solution Interval Notation Graph x 4 1,7 8 Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 3. f (x) x 5 4. f (x) x 3 5. f (x) 3sin x 6. f (x) x 1 Inverses Find the inverse for each function. 7. f (x) x 1 8. f (x) x Prove f and g are inverses of each other f (x) x3 g(x) 3 x 30. f (x) 9 x, x 0 g(x) 9 x 7

8 Equation of a line 31. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of /3. 3. Find the equation of a line passing through the point (, 8) and parallel to the line y 5 6 x Find the equation of a line perpendicular to the y- axis passing through the point (4, 7). 34. Find the equation of a line passing through the points (-3, 6) and (1, ). 35. Find the equation of a line with an x-intercept (, 0) and a y-intercept (0, 3). Radian and Degree Measure 36. Convert to degrees: a. 5 6 b. 4 5 c..63 radians 37. Convert to radians: a. 45 o b. 17 o c. 37 Angles in Standard Position 38. Sketch the angle in standard position. a b. 30 o c. 5 3 d. 1.8 radians Reference Triangles 39. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. a. b. 5 3 c. 4 d. 30 8

9 Unit Circle (0,1) 40. a.) sin180 o b.) cos70 o c.) sin(90 o ) d.) sin e.) cos360 o f.) cos() (-1,0) (0,-1) (1,0) Graphing Trig Functions - fx = sinx fx = cos x y = sin x and y = cos x have a period of and an amplitude of 1. Use the parent graphs above to help you sketch a graph of the functions below. For f (x) Asin(Bx C) K, A = amplitude, B = period, C = phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift. B Graph two complete periods of the function. 41. f (x) 5sin x 4. f (x) sinx 43. f (x) cos x 4 Trigonometric Equations: 44. f (x) cos x 3 Solve each of the equations for 0 x. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 x. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at the end of the packet.) 45. sin x cos x cosx sin x sinx cos x 1 cosx cos x sin x cosx cosx 0 9

10 Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsinx sin 1 x Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos -1 x sin -1 x cos -1 x tan -1 x sin -1 x tan -1 x Example: Express the value of y in radians. y arctan 1 3 Draw a reference triangle. 3-1 This means the reference angle is 30 or 6. So, y = 6 so that it falls in the interval from y Answer: y = 6 For each of the following, express the value for y in radians. 53. y arcsin y arccos1 55. y arctan(1) For each of the following give the value without a calculator. 56. tan arccos sec 1 sin sin arctan sin 7 sin1 8 10

11 Circles and Ellipses r (x h) (y k) (x h) a (y k) b 1 Minor Axis b a CENTER (h, k) FOCUS (h - c, k) c FOCUS (h + c, k) Major Axis For a circle centered at the origin, the equation is x y r, where r is the radius of the circle. x For an ellipse centered at the origin, the equation is a y 1, where a is the distance from the center to the b ellipse along the x-axis and b is the distance from the center to the ellipse along the y-axis. If the larger number is under the y term, the ellipse is elongated along the y-axis. For our purposes in Calculus, you will not need to locate the foci. Graph the circles and ellipses: 60. x y x y 5 6. x 1 y x 16 y 4 1 Limits Finding limits numerically. Complete the table and use the result to estimate the limit. 64. lim x 4 x4 x 3x 4 x f(x) 65. lim 4 x 3 x5 x 5 x f(x) 11

12 Finding limits graphically. Find each limit graphically. Use your calculator to assist in graphing. 66. lim cos x x0 Evaluating Limits Analytically 67. lim x5 x lim f (x) x1 f (x) x 3, x 1, x 1 Solve by direct substitution whenever possible. If needed, rearrange the expression so that you can do direct substitution. x x 69. lim(4x 3) 70. lim x x1 x lim x0 x 4 7. lim x cos x x lim x1 x 1 x x 6 HINT: Factor and simplify. 74. lim x3 x lim x0 x 1 1 x HINT: Rationalize the numerator. Horizontal Asymptotes Determine all Horizontal Asymptotes. 76. f (x) x x 1 x 3 x f (x) 5x3 x 8 4x 3x f (x) 4x5 x 7 True or False? If you answer false, explain why the answer is false and correct if possible = If x > π, then sin 1x > If x is any real number then, x + 5 is also a real number. 81. If x = 3 then x + 7 = x Given that y = 1, if x > 100 then y < x = The function y = x 7 + 3x 5 x + is an odd function. 1