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1 Name: Calculus AP - Summer Assignment 2017 All questions and concerns related to this assignment should be directed to Ms. Eisen on or before Wednesday, June 21, If any concerns should arise over the summer, please both the teacher and supervisor listed below: Teacher: Ms. Eisen deisen@dumontnj.org Supervisor of Mathematics & Science: Ms. Warnock dwarnock@dumontnj.org This assignment is due by Friday, September 8th. Anything handed in after that will be considered late and you will lose points. It will be graded and returned to you the following class period. This assignment will be graded and count as 1/3 of your Summer Assignment Test grade (which will also include a calculator section test grade and a noncalculator section test grade, each of which also counting as 1/3 of the Summer Assignment Test grade). You will most likely take the non-calculator section of the Summer Assignment Test on the third day of class and the calculator section on the fourth day of class. CLASS SITE: On this site, you can obtain scanned copies of all the relevant textbook pages referenced in this summer assignment by going to the AP Calculus links on the left and clicking on the AP Summer Assignment link, which will take you directly to the following page on the site:

2 Summer Assignment Checklist All of the problems you are required to complete are in this packet. However, if you would like to look up topic explanations and/or some of the solutions, go to the class site to view or download the packet(s) or scanned textbook pages from where all of the questions came. Review Topics: Algebra II and PreCalculus! 1) Algebra II: Review of Algebra Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: Pages 10-11: 2, 10, 12, 15, 85-88, 92, 94, 80, 100, 19, 21, 22, 24, 26, 27, 65, 63 (Note: problems #4a, 4b, 4e, 4f, 4g, 4h, and all of #5 in this section came from sources other than the one posted online) I think I understood the part about bringing two sharp pencils to class every day.! 2) Algebra II: Piecewise Functions Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: Page 51: 1-4! 3) PreCalculus: Trigonometric Functions Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: Page 1029: 9, 12, 13, 14, 21, 23, 25, 29 Page : 1, 2, 3, 5, 19, 20, 22, 23, 7-12, 31-34, 48, 51, Chapter 1: Limits and Their Properties (from your calculus textbook)! 4) Section 1.2: Finding Limits Graphically and Numerically Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: Pages 54-56: 1, 3, 8, 9-18, 49, 51, 52! 5) Section 1.3: Evaluating Limits Analytically Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: Pages 65-67: 4, 6, 12, 16, 20, 25, 28, 33, 36, 37, 40, 41, 44, 46, 49, 51, 60 Chapter P: Preparation for Calculus (from your calculus textbook) *You worked, and were quizzed, on the following three sections in Ms. Maslow s PreCalculus Honors class. Therefore, I will not be collecting or grading any problems from these sections. However, you are still responsible for knowing the topics they cover because these concepts will be included on the summer assignment test in September. I would recommend at least looking over the following problems to make sure that you still understand how to do them. Though numbered differently in this packet, these questions refer to the following problems in the scanned pages on the class site: 6) Section P.1: Graphs and Models Pages 8-9: 1-9, 11, 17, 18, 25-28, 63, 64, ) Section P.2: Linear Models and Rates of Change Pages 16-18: 1-4, 7, 9, 10, 15-18, 23-30, 35, 50, 54, 55, 59, 80, 95, 96 8) Section P.3: Functions and Their Graphs Pages 27-29: 1, 4, 5, 7, 11-17, 29-32, 52, 59-62, 73-76

3 Algebra II: Review of Algebra 1) Expand and simplify each of the following expressions. a) (2x 2 y)( xy 4 ) b) x(x 1)(x + 2) c) (2 + 3x) 2 d) (1+ 2x)(x 2 3x +1) 2) Use exponent rules to simplify each of the following. Completely simplified answers should not include negative exponents. (You should be able to do all of the problems without a calculator.) a) x9 (2x) 4 b) a 3 b 4 x 3 a 5 b 5 c) an a 2n+1 a n 2 d) x 1 + y 1 (x + y) 1 e) f) (x 5 y 3 z 10 ) 3 5 g) xy x 3 y h) 4 r 2n+1 4 r 1

4 3) Perform the indicated operations and simplify. a) 1 x x 3 b) u +1+ u u +1 c) 2 a 2 3 ab + 4 b 2 d) x y z e) a bc b ac f) 1+ 1 c c 1 4) Solve each of the following equations using any method. a) x 3 = 3x 2 b) 4b 3 8b 2 21b = 0 c) 3x 2 + 5x +1= 0 d) x 2 + 9x 1= 0

5 e) t 4 21t = 0 f) g +1 = 2g 7 g) 4 (1 7u) 2 3 = 0 h) 1 n n 2 9 = 2 n 3 5) Factor out the repeated term in each of the following expressions, then simplify. Simplified answers should not include negative exponents. a) x 5 2x 2 b) 4x 6 +12x 4 + x c) 3x 3 2 9x 1 2 d) (x + 2) 4 x(x + 2) 3 3(x + 2) e) (x2 x) 2 ( 6x) ( 6x 2 )(x 2 x)(2x 1) (x 2 x) 4 f) (x2 +1) 1 2 x 2 (x 2 +1) 1 2 x 2 +1

6 Algebra II: Piecewise Functions 1) The graph the following piecewise function is shown to the right. $ x 2 + 2, for x < 2 & f (x) = % 2x +1, for -2 x < 0 & ' x 2 + 2, for x 0 Based on the graph and equation above, evaluate each of the following: a) f (5) b) f ( 2) c) f ( 4) d) f (0) 2) Sketch each of the following piecewise functions. " x + 2, for x 1 a) f (x) = # $ 5, for x >1 # x, for x < 0 % b) f (x) = $ x 2, for 0 x < 2 % & 2x, for x 2 3) The graph below represents some particular piecewise function. Based solely on the graph, evaluate each of the following: a) f (0) b) f ( 3) c) f ( 1) d) f (1) e) f (2) f) f (3)

7 4) Fill in the blanks to define the piecewise function that is represented by the graph below. # x + 3, for x % f (x) = $, for % &, for PreCalculus: Trigonometric Functions 1) Express the angles in radian measure as multiples of π AND as decimals accurate to three places. a) 30 b) 210 c) 315 d) 120 2) Express the angles in degree measure. a) 5π 2 b) 7π 3 c) π 12 d) 19π 6 3) Determine all six trigonometric functions for the angle θ. a) b) c) d)

8 4) Determine the quadrant in which θ lies. a) sinθ < 0 and cosθ > 0 b) sinθ > 0 and cosθ < 0 c) cotθ < 0 and cosθ > 0 d) cscθ > 0 and tanθ < 0 5) Evaluate the trigonometric function. a) sinθ = 1 2 b) sinθ = 1 3 cscθ =? tanθ =? c) cosθ = 4 5 d) secθ = 13 5 cotθ =? cotθ =? e) cotθ = 15 8 f) tanθ = 1 2 secθ =? sinθ =?

9 6) Evaluate the sine, cosine, and tangent of each angle without using a calculator. a) 60 b) 2π 3 c) π 4 d) 5π 4 e) π 6 f) 150 g) π 2 h) π 2 7) Find two solutions of each equation. Express the results in radians ( 0 θ < 2π ). Do not use a calculator a) cosθ = 2 2 b) cosθ = 2 2 c) tanθ =1 d) cotθ = 3 8) Solve the equation for θ ( 0 θ < 2π ). a) 2sin 2 θ =1 b) tan 2 θ = 3 c) tan 2 θ tanθ = 0 d) 2cos 2 θ cosθ =1

10 Section 1.2: Finding Limits Graphically and Numerically 1) For each problem, complete the table and use the result to estimate the limit. a) lim x 2 x 2 x 2 x 2 x f (x) b) lim x 0 x x x f (x) c) lim x 0 cos x 1 x x f (x) 2) Use the graph to find the limit (if it exists). If the limit does not exist, explain why. a) lim(4 x) b) lim(x 2 + 2) x 3 x 1

11 c) lim f (x) x 2 # f (x) = $ 4 x, x 2 % 0, x = 2 d) lim f (x) x 1 " f (x) = x2 + 2, x 1 # $ 1, x =1 e) lim x 5 x 5 x 5 1 f) lim x 3 x 3 g) lim x π 2 tan x h) lim x 0 sec x i) lim x 0 cos 1 x j) lim x 1 sinπ x 3) Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. a) If f is undefined at x = c, then the limit of f (x) as x approaches c does not exist. b) If f (c) = L, then lim f (x) = L. c) If lim f (x) = L, then f (c) = L.

12 Section 1.3: Evaluating Limits Analytically 1) Find each of the following limits. a) lim x 3 x 2 b) lim x 1 (3x 3 2x 2 + 4) c) lim x 3 2x 3 x d) lim x + 4 x 4 e) lim t 4 t t 4 f) lim t 1 t t 4 2) If f (x) = 4 x 2 and g(x) = x +1, find each of the following limits. a) lim x 1 f (x) b) lim g(x) x 3 c) lim g( f (x)) x 1 3) Find the limit of each of the following trigonometric functions. a) lim x π tan x b) lim sin x x 5π 6 " c) limsec π x % $ ' x 7 # 6 & 4) Use the information given to evaluate the limits. a) lim f (x) = 2 lim g(x) = 3 i) lim[5g(x)] ii) lim iii) lim iv) lim b) lim f (x) = 27 i) lim 3 f (x) f (x) ii) lim 18 [ f (x)+ g(x)] iii) lim[ f (x)] 2 [ f (x)g(x)] iv) lim[ f (x)] 2 3 f (x) g(x)

13 5) Use the graph to determine the limits visually (if they exist). Then write a simpler function that agrees with the given function at all but one point. a) g(x) = 2x2 + x x i) lim g(x) x 0 ii) lim g(x) x 1 iii) Simpler Function: x b) f (x) = x 2 x i) lim f (x) x 1 ii) lim f (x) x 0 iii) Simpler Function: 6) Find the limit (if it exists). a) lim x 5 x 5 x 2 25 b) lim x 3 x 2 + x 6 x 2 9 2x 2 x 3 c) lim x 1 x +1 d) lim Δx 0 (x + Δx) 2 x 2 Δx