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TO: Next Year’s AP Calculus BC Students FROM: T. Wernau, AP Calculus BC Teacher As you probably know, the students who take AP Calculus AB and pass the Advanced Placement Test will place out of one semester of college Calculus; those who take AP Calculus BC and pass the Advanced Placement Test will place out of two semesters of college Calculus. Because Calculus BC covers two semesters, you will be learning Calculus twice as fast as the students who take Calculus AB. In order to have enough time to learn the material of two collegelevel courses, we will start learning Calculus as soon as possible when school begins and will not be able to spend time reviewing the material you learned in Algebra I & II, Geometry, and Precalculus. Attached is a summer homework packet, which will need to be completed by first day of Calculus class in August. The material in the packet should be material you learned in Algebra II and Precalculus. On the fourth day of class, you will take a test on the material in the packet. My recommendation is that you look over the problems in the packet when you receive it but that you wait until the week before school starts to work the problems so that you will remember the material very well when school starts. All of the AP Calculus AB and BC classes at Travis will be using a graphing calculator. I feel that the TI-89 is the best calculator available. You will be expected to have a TI-83, TI-84, or a TI-89 (recommended) the first week of school. I am looking forward to seeing you in Calculus in August. Sincerely, Mr. Wernau Travis High School Math Department CALCULUS BC SUMMER NOTES Some of the material that was briefly covered in Pre-Calculus is knowledge that I need you to know to be able to complete the packet. Here are some notes/examples for you to use: The Limit: x3 1 . x 1 What do we know about the graph of this function? __There is a hole at x=1__ Right! Graph it. 1) Let’s take a look at the function f ( x) If we put the point of our pencil on f(0.2) and trace the curve towards f(1) our pencils approaches the y-value of 3. We denote this as : lim f ( x) 3 because as we approach f(1) from the left side (the reason for x 1 the raise – next to the 1) the value of the graph approaches 3. If we put the point of our pencil on f(1.2) and trace the curve towards f(1) our pencils approaches the y-value of 3. We denote this as: lim f ( x) 3 because as we approach f(1) from the right side (the reason for x 1 the raised + next to the 1) the value of the graph approaches 3. Because the limit from the left side is equal to the limit from the right side we can use the notation: lim f ( x) 3 (notice the lack of a raised + or –). If this is not the case, we say the limit x 1 does not exist. ______________________________________________________________ 2) We can also evaluate simple limits quite easily, without even looking at the graph. Ex 1: lim x2 3x 1 x 3 If we first think about the function f ( x) x 2 3x 1 we know that it’s graph does not have any special properties (holes/asymptotes/etc) substitute the x-value we are approaching with the x-values in the function. lim( x2 3x 1) 32 3(3) 1 17 x3 Therefore, lim( x2 3x 1) = 17. x3 Ex 2: lim x 2 x2 2 x 8 x2 x2 2x 8 we know x2 there exists a hole at x = 2 and that happens to be the x-value we want to know about, so our previous method won’t work. So we have to be a little more creative If we first think about the function f ( x) Think about the factors of x 2 2 x 8 . One of the factors is (x – 2). x2 2x 8 ( x 2)( x 4) lim lim lim x 4 Now we can substitute like we did on problem #1. x 2 x 2 x 2 x2 x2 lim( x 4) 6 x2 x2 2 x 8 6 x 2 x2 ______________________________________________________________________________ x 2 1 Ex 3: lim Try using the conjugates x 0 x Therefore, lim lim x 0 x9 9 x x 9 9 x 99 lim lim x 0 x 9 9 x 0 x x 9 9 x x x 9 9 lim x 0 1 x9 9 1 6 ______________________________________________________________________________ x3 2 x 2 5 Ex 4: lim If the limit is going off to positive or negative infinity, it is x 3 x 4 x 3 2 asking you to find the horizontal asymptote. So remember your methods to finding the horizontal asymptote. (Ratio of the leading coefficients) x3 2 x 2 5 x3 2 x 2 5 1 lim 3 3 x 3 x 4 x 2 x 4 x 3 x 2 4 ____________________________________________________________________________ lim Note: To complete the optimization problems, you should have learned how to use a system of equations, the substitution method, and your calculator to find the maximum or minimum value. Ex 5: Find two positive numbers whose product is 115 and whose sum is a minimum. Eqn 1: xy 115 Eqn 2: S x y , where x represents my first number, y represents my second number, and S represents my sum function. 115 Combine equations by solving for x or y in eqn 1, you get: S x x Put S into Y1 and use calc to find the minimum x-value. Then use that x-value in eqn 1 to solve for the y-value. In this case, x 10.724 y 10.724 CALCULUS BC SUMMER HOMEWORK This homework packet is due the first day of school. It will be turned in the first day of Calculus class and will count as a daily grade. You will take a test on the material in the packet on the third day of school. Work these problems on notebook paper. All work must be shown. Find the x- and y-intercepts and the domain and range, and sketch the graph. Do not use your graphing calculator on these. 1. y 2 x 2 10. y e x 2. y x 2 2 11. y ln x 1 , if x 1 12. y 3x 2, if x 1 7 2 x, if x 1 3. y x 1 4. y 9 x 2 x 2 1, if x 0 13. y 2 x 2, if x 0 5. y x 2 3 6. y x x 7. y sin x, 2 x 2 8. y cos x, 2 x 2 9. y tan x, 2 x 2 ______________________________________________________________________________ Find the asymptotes (horizontal, vertical, and slant), symmetry, and intercepts, and sketch the graph. Do not use your graphing calculator on these. 2 x2 9 1 14. y 16. y x2 4 x 1 1 x2 2 x 4 15. y 17. y 2 x 1 x 2 _____________________________________________________________________________ Solve. 3x 2 0 18. x 2 x 12 0 20. x4 19. x 2 x 1 x 5 0 2 3 21. 2 x 5 x 1 3 x 2 2 0 Find the limit. x 2 22. lim 4 x 2 5 x 3 23. lim x 0 24. lim x 4 2 25. x0 x 1 1 x x 5 lim 2 25 x 0 x 3x 5 x 2 27. lim x 4 x 2 1 2 3 x3 29. lim 2 x x 4 x x6 x 3 x3 x6 28. lim 2 x 6 x 3 x 18 x3 8 30. lim x 2 x 2 ______________________________________________________________________________ Use the figure to find the limit. y 31. lim f x 32. lim f x y 2 26. lim x 3 33. lim f x x 35. lim f x x 2 x0 34. lim f x x 36. lim f x x x 5 Graph of f ______________________________________________________________________________ Write a function for each problem, and use your graphing calculator to solve. Give decimal answers correct to three decimal places. Be sure to sketch the graph you used, and label it with your answer. 37. Find two positive numbers such that their product is 192 and their sum is a minimum. 38. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. 39. A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals, as shown. What dimensions should be used so that the enclosed area will be a maximum? 40. A rectangle is bounded by the x-axis and the semicircle y 25 x 2 as shown. What length and width should the rectangle have so that its area is a maximum? 41. A rectangular pace is to contain 24 square inches of print. The margins at the top and 1 bottom of the page are to be 1 inches, and the margins on the left and right are to 2 be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? 42. Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area? 43. The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area. 44. Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. How far from the left post should the stake be placed to use the least wire? Know your Trig. Values: here is a worksheet to help you. To use the hand trick: 1. Always use your left hand, palm facing you. 2. Your hand represents the first quadrant only in radians-must use reference angles for any angle out of the first quadrant. 1 3. Pinky finger is x-axis (0 radians) and thumb is the y-axis ( radians). 2 4. Whatever radian measure, pull that finger-that represents the comma in the ordered pair of trig. values (cos, sin). 5. All the fingers on top of the pulled down finger are the cosine, on bottom the sine. fingers 6. Formula: for sin, cos, csc, and sec = 2 fingers sin 7. Formula for Tangent = . However recall that Tangent is . You may need cos fingers to flip hand to palm facing away from you. Let’s try a few: Find all six Trig values for the following. Angle(radians) 0 6 4 3 2 Cos Sin Sec Csc tan Cot Solve. Give exact answers in radians, 0 x 2 . 45. 2cos x 3 0 46. 2 cos 2 x 3cos x 2 0 47. sin 2x 1 48. 2 cos 2 x sin x 1 0 49. sin 2x cos x 50. tan 2 x sec x 1 Know the following Trigonometric Identities: Pythagorean: sin 2 x cos 2 x 1 Division: sec x 1 cos x csc x 1 tan 2 x sec 2 x 1 sin x cot x 1 tan x tan x 1 cot 2 x csc 2 x sin x cos x cot x cos x sin x Double Angle: sin( 2 x) 2 sin x cos x Product to Sum: 1 sin x sin y [cos( x y ) cos( x y ) 2 1 cos x cos y [cos( x y ) cos( x y ) 2 1 sin x cos y [sin( x y ) sin( x y ) 2 Logarithms: Inverse Property: log b y x iff Recall: ln x log e x bx y Product Property: log b (mn) log b m log b n m Quotient Property: log b ( ) log b m log b n n n Power Property: log b (m ) n log b m Expand the following: x 4 53. ln 3x 52. log 9 (27m 2 ) 51. log 4 (3m) Solve the following log and exponential equations for x: x 54. 3 3 x4 x 1 55. 4 16 56. 3 9 58. log 4 x 3 59. log 2 8 x 60. log 3 9 x 1 61. 2 log 3 x 8 62. log 4 ( x 3) 2 x x x 1 57. 4 2