You will have a test on the summer assignment material on the second day of school. (no calculator or reference material allowed)
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1 Summer Assignment AP CALCULUS AB June 017 Dear Student: Your AP calculus exam will be on Wednesday May 15, 018. That may seem like a long time, but it leaves us with less than 150 days of class next year. With half days, snow days, exams, assemblies, school activities, etc, we can expect to have only days of class before the exam. This summer assignment is designed to provide a review of algebraic and trigonometric concepts. By covering this material before school begins in the fall, we will be able to begin differential calculus on the third day of school. I included the reference page-minimum that you must know in order to succeed in Calculus AB. You will not be allowed to use this reference page on any tests or quizzes. You will have a test on the summer assignment material on the second day of school. (no calculator or reference material allowed) The summer work is separated into 3 parts. You may work with another student, or students. The important thing is that YOU understand the work. You may find the websites below helpful to review Algebra and Precalculus concepts. prop.htm Due Date for Assignments is August 3 017, 9am. Assignments not turned in by these dates will be marked late. 1) You may drop your assignments off in my mailbox at the high school. I believe the high school is open from 8 am to 3 pm Monday-Thursday. ) You may mail your assignment to the school at: North Brunswick Twp. H.S. Route 130 and Raider Road North Brunswick, NJ 0890 Attn: Ms. Goncharova (AP Calculus AB summer assignment) 3) You may scan your assignment and to me at agoncharova@nbtschools.org ( pdf files only) 4) Finally, you may fax your assignment with a cover page Attn: Ms. Goncharova (AP Calc AB summer assignment).
2 me if you have any questions at I will not be available from 7/1-8/3 Assignments Unless noted, each problem should be done without a calculator. All solution processes and answers must be clearly and neatly stated. Answers without work or explanations will not be given credit. Your work must be done in the packet. Part I: Equations and Functions due 8/3 Part II: Trigonometric Functions due 8/3 Part III: Potpourri due 8/3 Part IV: Concepts Worksheet complete and bring to our first class in September Part V: Sample Exams with the answers: Practice on your own to study for the summer assignment test on the second day of school. ****Remember the goal is for YOU to understand the material thoroughly so that we can focus on the calculus material immediately when the school year begins. Failure to turn in a section will result in you having to drop the course. Additionally, unsatisfactory performance on this assignment and/or the summer assignment test will result in you having to drop the course. If you are not willing to work on your own, to hand in material on time or you can not demonstrate an understanding of the material on this assignment, you will not succeed in this course.
3 Summer Assessment Due 8/3 11 pts Last Name First Part I: Equations and Functions 1) a. Determine all points of intersection 3pts each x 1x 5 and y= y= b. Solve: 1 1 x 5 0 x ) Find an equation of a line, in y-intercept form, that satisfies the given conditions: pts each passes through the point (, -4); perpendicular to the line 5x y 3 3) Find an equation of a line, in point-slope form, that satisfies the given conditions: pts each passes through (, -4); parallel to the line 5x y 3 If a and h are real numbers, find and simplify ( ) ( ) ( ), ( ), f a h f a f a h f a h pts each f(a)= f(a+h)= a) f x x x ( ) 3 f ( ah) f( a) = h
4 f(a)= f(a+h)= b) f( x) 1 x f ( ah) f( a) = h 4 4) Let f ( x) 3 x and gx ( ) x 5. Find the sum, product, and quotient of f and g. Specify the domain of each result. (Do NOT simplify the sum to a common denominator.) 3pts 5) Let f ( x) sin x and gx ( ) x x +1. Find pts each a. f gxand the domain of f g. b. g f xand the domain of g f. 4x 3 6) Find the domain of f( x) x 4 pts 7) Solve the following equations for y ' a. x xy' y3 y y' 0 pts each b. x 5 y ' y y'
5 pts 8) Determine the inverse relation of f( x) x, x 0. Is 1 f a function? Explain. 9) Find the inverse of 4x f( x). pts x 1 10) Sketch each function: State the domain and the range 3pts each x a) y e 3 b) y ln( x 3) c) y x13 D: D: D: R: R: R: d) y 3 x e) y 9 x D: R: D: R:
6 Part II: Trigonometry 1pt each 1) Find the exact value: (Show the triangle inside the unit circle you used to compute the values) a. sin b. tan c. cos 3 4 d. sec e. 3 csc f. cot 6 g. sin 1 3 h. cos 1 1 i. tan 3 ) Find all exact solutions to the equation on [0, π). 3pts each a.. sin tsin t 0 b. cos x3sinx3
7 3) Graph each of the following functions. State the domain and range. pts each a) y sin x b) y cos x c) y tan x D: D: D: R: R: R: 4) Complete the following formulas: (I expect you to know these without reference) 1pt each Double Angle Formulas Power Reducing Formulas a. sin( ) sin u b. cos( ) u c.. u d.. cos u 5) If 4 tan x find 5 sec x from the firs quadrant.(no calculator) 1 pt. 6) Find the missing side of each triangle with sides a and b and hypotenuse c. (Include a drawing of each triangle on a separate piece of paper and label each side.) 1pt each Side a Side b Hypotenuse c a 3 X b 49 x 7 c 37 x 37 d 9x 1 1 e 4 4x
8 Part III: Potpourri of Mathematics Name: Notes: A function, f ( x ), has a limit if the value of f ( x) is the same as you approach x from both sides. Definition: lim f ( x ) L if f (x) equals L as x approaches a from the positive and xa negative directions. 3 x 1 Example: Find lim, x 1. x1 x 1 Method 1: Table form: x must be SMALL x f(x) error We look at the table as x approaches one from the negative and positive direction. The values of f(x) approach 3. Therefore the limit is 3. or you can use Method Graphic form: Using your graphing calculator, graph the function. Zoom in around x = 1, and you can see, using the trace function, the function approaches 3. (You should get a graph that resembles a parabola) Use the notes, to answer the following questions. Create a table with values on both sides of the value x approaches. Also, draw a graph to confirm your limit for each question. 1) Find the limit of the following functions: pts each x a. lim x x x b. sin x lim x0 x x c. lim x x 4 d. 1 lim x3 x 3 e. lim x 0 x 3 3 x f. lim x 1 x
9 ) Find the zeroes of the function (algebraically): ( ) x x f x x e 6xe pts 3)Evaluate or simplify: (no calculator-use properties of exponents an logarithms) 1pt each a. ln Ce at 1 b. log4 16 3log5 6 c. 5 4)Write as one term: 1pt each a) xln x( x1) ln( x1) b) eee 4 x c) ln xln( x1) 5)Solve leaving answers in terms or e or ln. 3pts each a) 5 x e b) 3 x x 3
10 6) An open box is to be made form a rectangular piece of cardboard 10 inches by 0 inches by cutting out identical squares of area x from each corner and turning up the sides. 3pts a. Express the volume, V, of the box as function of x. b. Use your graphing calculator to find the value of x that gives the maximum volume. Include a graph with your answers. c. What is the maximum volume of the box? 7) A field is to be marked off in the shape of a rectangle of area 11 square feet. pts a. Write an expression for the perimeter, P, as a function of the length x. b. Use your graphing calculator to approximate the dimensions for which the perimeter is a minimum. Draw the graph with your answers. 8) Expand and simplify. pts each 3 n a) b) n0 4 n1 1 n End of the Assignment to be Handed In
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