AP Calculus BC Summer Assignment
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1 AP Calculus BC Summer Assignment Attached is an assignment for students entering AP Calculus BC in the fall. Next year we will focus more on concepts and thinking outside of the box. We will not have time to review basic algebraic concepts that you should already know. Therefore it is imperative that you have mastered the basic skills needed for success in this course. Many students struggle with the Algebra component of Calculus and neglect the concepts and theorems. It is not a good scene when a student understands the calculus but can t simplify, thus getting the problem incorrect. Thus the purpose of this worksheet is to help you be better prepared for next year. Also included are basic concepts you must know. You may want to use your notes from Pre -Calculus as well. Although the assignment says BC, the introductory concepts are the same. There will be a TEST over this material on the second day of school. First day I will review them with you in class. You should wait till mid-summer to begin this packet. This will give you a better indication of what you know and what you ve forgotten. Keep in mind that this packet is not to torture you or punish you for taking AP Calculus BC. This will better prepare you and allow us more time to review before the exam. Please solve all of them. I will not collect them. However there will be a test from this material. Enjoy your summer, AP CALCULUS BC SUMMER ASSIGNMENT Complete the following. I will NOT collect them, however we will have a test from this packet the second day of the class meeting. Test grade will be given.
2 Calculus BC Summer Assignment PROBLEM SET #1 Slopes ***Calculators Not Allowed*** Calculate the slope of the line containing the following points: 1. (2,8) and ( 4,6) 6. ( 3, 4) and (2,0) 2 2. ( 4, 7) and (3,0) 7. ( 11 14, 3 7 ) and ( 9 14, 5 7 ) 3. ( 3, 6) and ( 1, 6) 8. ( 8 9, 2 3 ) and (5 6, 2 5 ) 4. (4, 2) and (4,5) 9. ( 4, 3) and (0, 11) 5. ( 4 5, 6) and (3 5, 4) 10. ( 3 7, 3 8 ) and ( 1 6, 5 6 )
3 PROBLEM SET #2 Equations of Lines ***Calculators Not Allowed*** For each of the following questions, write the equation of the line given the specific information. 1. Passes through (2,3) and m = 2 6. Passes through (-1, 2) and the slope is undefined. 2. Passes through ( 2,4) and m = Passes through (-2, 2) and is parallel to 2y = 4x Passes through ( 4, 5) and (2,7) 8. Passes through (-3, 2) and is perpendicular to 15y = 10x Passes through (3, 5) and ( 3,5) 9. m = 3 5 and b = 0 5. Passes through (-1, 2) and m = m = 0 and b = 1 7
4 Problem Set #3 Functions & Graphing Functions ***Calculators Not Allowed*** 1. a) True/False 3x 2 + 5y = 7 2x is a function. b) Why or why not? 2. a) True/False 2x 2 + 3y 2 = 11 is a function. b) Why or why not? 3. a) True/False The following table represents a function. x y b) Why or why not? 4. a) True/False The following table represents a function. x y b) Why or why not? 5. Evaluate g(2) if g(x) = 3x 2 5x Evaluate g( 3) if g(x) = x 2 2x Evaluate f(x 2) if f(x) = 2x 2 3x + 11
5 8. Evaluate f(5 x) if f(x) = 4x 3 2 x 9. Write the new equation of the function y = x with the following transformations: reflection over x-axis, vertical compression of 1/2, right 3 units, and up 4 units 10. Write the new equation of the function y = 1 with the following transformations: x horizontal compression of 3, right 3 units, and down 2 units 11. Write the new equation of the function y = x 2 with the following transformations: reflection over x-axis, vertical stretch of 2, left 2 units, and down 3 units 12. Write the new equation of the function y = ln x with the following transformations: vertical stretch of 3, right 5 units, and up 2 units
6 Problem Set #4 Piecewise Functions ***Calculators Not Allowed*** 1. Graph the following piecewise function: x 1 5 x < 1 f(x) = { 2 1 x < 2 x < x 6 2. Graph the following piecewise function: f(x) = { 2x + 1 x < 1 x x 1
7 3. Graph the following piecewise function: f(x) = { 2x 6 x < 4 x x < 2 x 2 3 x 2 4. Graph the following piecewise function: f(x) = { x x < 1 x 2 x > 1
8 5. Given: f(x) = { x 2 5 x < x < 1 3x x 1 a) Find f( 7) b) Find f( 5) c) Find f(0) d) Find f(1) e) Find f(3) 6. Given: f(x) = { x x 3 4 x < 2 x 2 a) Find f( 5) b) Find f( 2) c) Find f(0) d) Find f(1)
9 Problem Set #5 Function Composition ***Calculators Not Allowed*** Use the following functions to answer questions f(x) = 3x g(x) = 2x 1 h(x) = x 3 k(x) = 4x 2 1. g(f(2)) = 9. f(k(3)) = 2. f g(3) = 10.h f( 7) = 3. h (f(g(0))) = 11. k (f(h( 5))) = 4. k g h( 2) = 12. h k g( 1) = 5. g k(x) = 13. k f(x) = 6. 5f(x) 3g(x) = 14. 2g(x) + 4f(x) = 7. g f k(x) = 15. g k f(x) = 8. f(x) g(x) = 16. g(x) k(x) =
10 Problem Set #6 Function Roots ***Calculators Not Allowed*** Find any real roots, if they exist, for questions y = x 2 2x 8 7. y = 2x 2 x f(x) = x 2 + 4x f(x) = x 2 + 4x r(t) = t 3 11t t 9. f(x) = 5x 2 + 5x y = 3x 2 10x y = 3x 2 8x 2 5. r(t) = t 3 5t t 11.q(z) = 5z 3 + 2z 2 7z 6. r(t) = t 2 6t y = 3x 3 + 6x 2 x
11 Problem Set #7 Domain & Range ***Calculators Not Allowed*** Domain: Range: Domain: Range: Domain: Range: 3. Domain: Range: 6. Domain: Range: Domain: Range:
12 7. y = 2x y = ln(x 3) Domain: Range: Domain: Range: 8. y = x 2 + 4x 32 Domain: Range: 9. y = 3x 2 + 6x + 5 Domain: Range: 10. y = x Domain: Range: 11. y = x Domain: Range: 12. y = 5x+2 x+3 Domain only: 14. y = 4 ln(x + 2) 1 Domain: Range: 15. y = x Domain: Range: y = x Domain: Range: 17. y = 2x x 2 +2x 8 Domain: Range: 18. y = 25 2x 2 +5x Domain only:
13 Problem Set #8 Inverses ***Calculators Not Allowed*** State whether the following functions are inverses. 1. f(x) = (x 1) 2 g(x) = 1 + x 2 3. g(x) = x 3 +5 h(x) = (x 5) g(z) = 3 z + 5 f(z) = 3 z 5 4. k(t) = 2t m(t) = t+1 2 Find the inverse of each function h(x) = 4 x k(t) = 5t m(x) = 7x g(z) = (z 3) 5 + 2
14 Problem Set #9 Trigonometry ***Calculators Not Allowed*** Evaluate each of the following. 1. csc 7π csc 15π 4 2. tan π cot 2π 4 3. sin 7π sin 4π 3 4. sec π csc 4π 3 5. cotπ 15. cos 11π 6 6. csc π cot 4π 3 7. sin π tan π 2 8. cos 5π cot 5π 4 9. csc 14π csc 3π tan 2π cos 5π 2
15 Evaluate: 21. cos 1 ( 3 2 ) cos 2 ( 3π 2 ) 22. sin 1 ( 0) 28. cot 1 ( 1) 23. csc 1 ( ) 29. sec 1 ( 2) 24.tan 1 ( 3 3 ) sin 2 ( π 2 ) 25.sin 1 (1) 31. cos 1 ( 1 2 ) 26. cos 1 (0) 32.csc 1 (1)
16 Problem Set #10 Exponents ***Calculators Not Allowed*** Simplify: 1. 15m11 k 5 10m 4 k ( 5x 3 y 6 z 4 ) e 3 f 14 42e 7 f 3 8. (4m 2 k 4 p) 2 3. (3x 2 5x + 2)(x 2 + 3x 1) 9. (2x 3 y 3 z) 2 (15x 10 y 4 z 0 ) 4. (2y 3 + 3y 2 4)(y 2 + 7y 3) 10. ( 5r 5 s 2 t 4 ) 2 (3rs 5 t 2 ) 5. ( (2a4 b 2 ) 3 (4a 9 b 5 ) 2) (5a 2b) 2 6. ( (3m3 n 4 ) 4 (6m 8 n 12 ) 2) (c 4) 3
17 Problem Set #11 Logarithms ***Calculators Allowed*** Solve the following equations: 1. log x 16 = 4 6. log(2x + 3) = log (12x 1) 2. log x 125 = x = x+2 = x = x 3 = log 2 (r + 3) + log 2 (r) = log log(7x + 3) = log (2x + 23) 10. log 4 (r + 5) log 4 (r) = log 4 10
18 REVIEW PROBLEM SET ANSWER KEYS Problem Set #1 Slopes undef x 17 3 x 9. y = 1 2 x y = 1 3x y = 2(x + 2) y = 3ln(x 5) + 2 Problem Set #4 Piecewise Functions 1. See graph Problem Set #2 Eqns. of Lines 1. y 3 = 2(x 2) or y = 2x 1 2. y 4 = 1 2 (x + 2) or y = 1 2 x y + 5 = 2(x + 4) or y = 2x y + 5 = 5 3 (x 3) or or y = 5 3 x 2. See graph 5. y = 2 6. x = 1 7. y = 2x y 2 = 3 2 (x + 3) or y = 3 2 x y = 3 5 x 10. y = See graph Problem Set #3 Functions/Graphing 1. a) TRUE b) Each x-value corresponds to only one y-value. 2. a) FALSE b) Does not pass vertical line test; more than one y-value for each x-value 3. a) TRUE b) Each x-value corresponds to only one y-value 4. a) FALSE b) Does not pass vertical line test; more than one y-value for each x-value x 2 + 5x See graph 5. a) 44 b) 11 c) 11 d) 7 e) a) 13 b) -20 c) -4 d) -2
19 Problem Set #5 Function Composition x x x 2 1 3x 8. 2x x x x x+1 4x 2 Problem Set #6 Function Roots 1. x = 2 & x = 4 2. x = 8 & x = 4 3. t = 0 & t = 2 & t = 9 4. x = 2 & x = t = 0 6. no real roots 7. x = 2 & x = x = 2 & x = 6 9. no real roots 10. x = 4± x = 0, x = 7 5 & x = x = 3±2 3 3 and x = 0 Problem Set #7 Domain & Range 1. Domain: (, 1) [4, ) Range: R 2. Domain: R Range: (, 3] 3. Domain: [ 5,5] Range: [ 2,2] 4. Domain: R Range: y = 3 5. Domain: (, 3] ( 2, ) Range: (, 3] 6. Domain: (, 2] (3, ) Range: y = 2 and ( 1, ) 7. Domain: R Range: R 8. Domain: R Range: [ 36, ) 9. Domain: R Range: (, 8] 10. Domain: [ 5, ) Range: [ 2, ) 11. Domain: [ 7, ) Range: (, 5] 12. Domain: ( 3, ) 13. Domain: (3, ) Range: R 14. Domain: ( 2, ) Range: R 15. Domain: R Range: R 16. Domain: R Range: R 17. Domain: R x 2 x 4 Range: (, 0) (0, ) 18. Domain: R x 1 2 x 3 Problem Set #8 Inverses 1. No 2. Yes 3. Yes 4. No 5. h 1 (x) = ( x 2 4 )3 6. k 1 (t) = 11 t 5 7. m 1 (x) = x g 1 5 (z) = z 2 + 3
20 Problem Set #9 Trigonometry undef undef π π π π π π π π 3 Problem Set #10 Exponents 1. 3m7 k f17 2e x 4 + 4x 3 16x x y y y 3 13y 2 28y a18 b n16 81m y x 9 z 12 m 4 16k 8 p x 16 y 10 z r 11 st a 2 20ab + 4b c 3 12c c + 64 Problem Set #11 Logarithms 1. x = 2 2. x = 5 3. x = or x = x = 4 6. x = x = x = or r = r = π 2
21 FUNCTIONS Topic: How can the direction of a function be determined by its graph, table, or equation? Explanation: Problems: 1) Graph the following: a) Y= x 2-1 b) Y= (x-1) 2 2) Find the intercepts of y= 3x 2 +13x-10
22 SLOPE Topic: How do you find the slope and what does it mean for a line? What does it mean in a nonlinear situation? Explanation: Problems 1) Find the following equations of each line: a) Containing (1,3) and (5,-7) b) Perpendicular to y-3=4(x+2) containing (-1,1) Average Weight of Adult Males in US Census Year Weight (lbs) 2) Use the table to complete the following: a) Graph b) Find a linear regression equation for a line of best fit c) Use equation to predict average weight in 2010
23 EVEN/ODD FUNCTIONS Topic: How do you determine from a graph, equation, and table whether a function is even an odd? Explanation: Problems: 1) Are the following even, odd, or neither? Explain why. a) y=(x 2 )+1 b) y=1-cosx 2) A portion of a function on [-3,3] is shown. Complete the graph assuming the function is a) Even (solid line) b) Odd (dotted line)
24 Topic: What does the mnemonic Pert mean? EXPONENTIAL FUNCTIONS Explanation: Problem: 1) The population of Knoxville is now 500,000 and is increasing at a rate of 3.75% each year. How long will it take for the population to reach 1 million?
25 Topic: In what order are transformations applied? TRANSFORMATIONS Explanation: Problem: 1) The graph of f is shown. Draw the graph of each real function. a) Y= f(-x) b) Y=-f(x) c) Y= -2f(x+1)+1 d) Y=3f(x-2)-2
26 Topic: What are the steps to solving word problems? SOLVING WORD PROBLEMS Explanation: Problem: Given a sheet 14x8 in. Cut out squares as shown to make a box with a lid that has maximum volume. Show your answer for each step. (B folds over A) 14 inches A B 8 inches
27 INVERSES Topic: How do you take the inverse of a function? Is the inverse of a function a function? Explanation: Problems: Find f -1 for each function. Show that f(f -1 x)=x. Sketch the graph of f and f -1 on the same set of axis. a) f(x)=2-3x b) f(x)= (x+2 )2, x> -2
28 GRAPHING INVERSE TRIG FUNCTIONS Topic: How does graphing a trig function relate to graphing inverse trig functions? Explanation: Problems: 1) Solve for x a) Cos(x) = b) Tan -1(x) = 5π 6 2) Find cos 2 π 6 a) By finding cos π and then squaring 6 b) Double-angle formula
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AP Calculus BC Summer Assignment
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