Rolesville High School AP Calculus Summer Assignment

Transcription

1 Rolesville High School AP Calculus Summer Assignment Dear Future Calculus Students, Welcome to AP Calculus. I am excited you have chosen to take on this challenge next year. Calculus is a rigorous, yet rewarding, math course, and I look forward to helping you succeed. Many Calculus students are surprised by how much prior knowledge is required to do well in this class. Quite often, it is Pre-Calculus, not Calculus, which gives students the most trouble. To help you prepare for this course and make it through the tedious algebra ahead, I believe it will be beneficial for you to create your own Pre-Calculus Reference Booklet this summer. In addition, you will also need to complete the Take-Home Test attached to this letter. Both the Pre-Calculus Reference Booklet and the Take-Home Test will be due on the first day of class: August 25, Per RHS policy, the summer assignment will count as 15% of your 1 st Quarter Grade. The Pre-Calculus Reference Booklet will be worth 50% and the Take- Home Test will be worth 50%. You are encouraged to use reference tools such as notes from previous classes, textbooks, or internet resources. Websites such as Khan Academy ( may be of great benefit to you. I will be available via to answer questions and provide suggestions. My address is: eanthony2@wcpss.net. Please allow 3-4 days for a response over the summer. Also, make sure to e- mail me using your WCPSS address; otherwise, I am not permitted to respond. Sincerely, Ms. Anthony 1

2 1. Calculus Reference Booklet (50%) Using your previous math notebooks and any necessary reference books, create an algebra/geometry/trigonometry reference booklet which will help you succeed in calculus. The topics are listed below. TOPICS Algebra 1. Writing the equation of a line 2. Simplifying algebraic fractions 3. Laws of logarithms 4. Solving exponential equations Geometry 1. Area formulas (circle, square, isosceles right triangle, equilateral triangle) 2. Volume formulas (sphere, cube, rectangular solid, cone) 3. Surface area formulas (sphere, cube, rectangular solid) 4. Solving similar triangles Trigonometry 1. Sum and difference formulas 2. Double-angle formulas 3. Basic trigonometric identities (Pythagorean and reciprocal) 4. Unit circle 5. Graphing sine, cosine and tangent Families of Functions* 1. Graph of y = ln(x) x 2. Graph of y = e 3. Graph of y = x 2 4. Graph of y = x 5. Determining asymptotes of rational functions 6. Direct variation 5. Graph of y = x 3 6. Graph of y = 3 x 2 7. Graph of y = x 3 FORMAT 1. Divide each page into two columns. 2. In the left column, state the topic; under the topic name, list any formulas, facts, definitions, procedures and/or equations which relate to that topic. 3. In the right column, write one or more examples which illustrate each formula, fact and equation from the left column. The examples should be worked out fully, with all steps shown. (See the sample below.) 4. *The Families of Functions topic should not have examples. It should, however, include a graph and a verbal description of the characteristics of the function, including domain/range, intercepts and asymptotes where appropriate. 5. Be neat and make your booklet easily readable. EXAMPLE ENTRY Slope The slope is the change in y over the change in x, which is sometimes expressed as "rise over run." Formula: y2 y1 m x2 x1 Procedure for finding slope: If you are given two points on the line, use the formula above. If you are given a graph, you may count the change in y and the change in x on the graph. If you are given an equation in slope-intercept form, the slope is the coefficient of x. If you are given an equation in standard form, the slope is the opposite of the ratio of the coefficients of x and y. Two points: (3,2) and ( 4,5); Graph: 1 m 2 m Slope-intercept form: y = 2x + 5; m = 2 Standard form:3x 4y = 10; 3 m 4 * Adapted from 2

3 Name Date Period: Calculus Reference Booklet Grading Rubric Please turn in this sheet with your project. Your grade for the reference booklet will be determined using the following rubric as a percentage of 50 possible points. Overall Neatness Algebra Geometry Trigonometry Families of Functions All work is Work is complete but is Work is completed but Work is incomplete and exceptionally neat and somewhat lacking in does not follow lacks neatness and readable and follows the either neatness or guidelines and is organization. guidelines provided. following the unreadable. All algebra topics are illustrated. Examples are appropriate and All geometry topics are illustrated. Examples are appropriate and All trigonometry topics are clearly explained and illustrated. Examples are appropriate and Each family of functions is correctly graphed and all characteristic information is guidelines. Most algebra topics are illustrated. Most examples are appropriate and Most geometry topics are clearly explained and illustrated. Most examples are appropriate and Most trigonometry topics are clearly explained and illustrated. Most examples are appropriate and Most families of functions are correctly graphed and some characteristic information is missing or in Approximately half of algebra topics are illustrated and examples lack accuracy. Approximately half of geometry topics are illustrated and examples lack accuracy. Approximately half of trigonometry topics are illustrated and examples lack accuracy. Approximately half of the families of functions are correctly graphed and characteristic information is missing or in Most algebra topics are unclear and lack explanations and examples. Most geometry topics are unclear and lack explanations and examples. Most trigonometry topics are unclear and lack explanations and examples. Most families of functions are correctly graphed and very little characteristic information is **The teacher reserves the right to add extra points for creativity and uniqueness. Final Grade /50 = Comments: 3

4 Name Date Period: 2. Take-Home Test (50%) These problems are a review that should help you be prepared for algebraic challenges ahead. Make sure to show all work and circle your answer. For any problems where work may not be necessary, you are required to write at least one complete sentence to justify your answer. 1. Find the x-intercepts, if any. a. y = 4x 5 b. y = (x + 3)(x 7) c. y = x 4 x+2 d. xy = 8 2. Describe the symmetry of each of the following ( x-axis, y-axis, origin?) Justify your answer. a. x 2 y x 2 + 8y = 0 b. y = x 6 x Sketch the graph of each equation. a. y = 1 2 ( x + 5) b. 1 4 x y = 1 c. y = 18 5x + x 2 d. y = 6 x 4. Solve each system of equations. 3x 4y = 8 a. { x + y = 2 b. { x y 3 = 0 y x 2 = 6 4

5 6. Create an equation for which the following information must be true: The graph has intercepts at x = 3 and x = 3 and is symmetric about the origin. 7. Find the slope of the line passing through the given points: a. (1,3) and (-4, 6) b. ( 2 3, 6) and (4, 5 3 ) 8. Use your knowledge of slope to determine the value of t so that the three points are collinear. ( 2,4), (0, t), (5, 7) 9. Write the equation of the line passing through the point with the given characteristic(s). a. (-1,4) with slope 2 d. (8,-3) parallel to the y-axis 3 b. (-3,4) parallel to the line 4x y = 8 e. (5,1) perpendicular to the line x + 2y = 0 c. (6,-7) passing through the origin 5

6 11. Sketch (on the same set of coordinate axes) a graph of f for c = -2, 0, and 2. a. f(x) = x 3 + c b. f(x) = (x c) 3 c. f(x) = (x 2) 3 + c d. f(x) = cx Simplify: a. 5( x h) 5x h 2 2 a a c. b 2 a a b b. (x 1)2 (3x 1) 2(x 1) (x 1) 4 6

7 13. Evaluate (if possible) the function f(x) = 2 at the specified values of the independent variable and simplify the x result. a. f(0) b. f(1) c. f(1 + b) f(1) 14. Evaluate (if possible) the function f(x) = { x2 + 3, x < 0 x 3, x 0 a. f( 10) b. f(0) c. f(1) b 15. Find the domain and range of each function: a. y = 25 x 2 c. f(x) = { x2, x < 0 5 x, x 0 b. y = 5 2x Given f(x) = 1 x 2 and g(x) = 4x 1, find the following. a. f(x) g(x) c. f(g(x)) b. f(x)g(x) d. g(f(x)) 17. Solve each equation: a. 2x 3 x 2 = 1 c. 3e 7a = 6 b. 5 = x+1 x 2 2x 3 x 3 7

8 Application Problems: 18. The purchase price of a new classroom set of calculators is $3,450. If the calculator set decreases in value by $235 each year, write a linear equation that gives the value (V) of the calculators t years after they were purchased. Then find the value of the calculators at the end of 4 years. 19. Ms. Anthony decides to quit her job at RHS and open up a sno-cone business. The machinery costs her $24,000 to purchase and requires an average of $4.50 per hour for maintenance and other operation fees. She pays her employee Gertrude $9.75 per hour. Business is booming, and Ms. Anthony rakes in approximately $45 for snocones every hour. a. Write an equation for the cost (C) of operating the business for t hours. b. Write an equation for the revenue (R) derived after t hours of selling sno-cones. c. Find the break-even point for Ms. Anthony s sno-cone business. (Hint: When does R=C?) 20. A wire 34 inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of x. a. Express the area (A) of the rectangle as a function of x. b. Determine the domain of the function and sketch a graph of the function over the domain. c. Determine the maximum area of the rectangle d. Give the dimensions of the rectangle that would provide the maximum area. 21. The rate at which water is entering a tank ( t 0) is represented by the given graph. A negative rate means that water is leaving the tank. State the interval(s) on which each of the following holds true: a. The volume of water is constant. b. The volume of water is decreasing. c. The volume of water is increasing. d. The volume of water is increasing fastest. *Problems adapted from 7 th Edition Calculus, Larson and www. 8