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First Quarter Pre-Calculus Assignment Packet Assignment 1.1 (The Wrapping Function) I. Identify the ordered pair that results when the following numbers are wrapped around a unit circle, given a reference value. [Do steps a—d for each problem]. 3 4 1. W (t ) , 2. W (t ) 0.29,0.96 5 5 12 5 3. W (t ) , 4. W (t ) 0.8, 0.6 13 13 a. W (t 2 ) b. W (t ) c. W ( t ) d. W ( t ) 3 1 II. Let W , . Find the coordinates of the following: 6 2 2 7 5 11 1. W 2. W 3. W 4. W 6 6 6 6 2 2 III. Let W , . Find the coordinates of the following: 4 2 2 5 7 3 1. W 2. W 3. W 4. W 4 4 4 4 1 3 IV. Let W , . Find the coordinates of the following: 3 2 2 5 2 4 1. W 2. W 3. W 4. W 3 3 3 3 1 First Quarter Pre-Calculus Assignment Packet V. Identify the ordered pair that results when the following numbers are wrapped around a unit circle. 7 3 1. w 2. w 3. w 6 3 4 11 11 4. w 5. w 6. w 2 6 4 8 5 7. w 7 8. w 9. w 3 6 VI. Find the positive number (greater than 0 but less than 2 ) that wraps onto the following ordered pairs: 2 2 3 1 1 3 2 , 2 1. 2. 2 ,2 3. , 2 2 2 2 4. 0, 1 5. 1,0 6. 2 , 2 VII. Find the negative number (greater than 2 but less than 0) that wraps onto the following ordered pairs: 2 2 3 1 1 3 1. 2 , 2 ,2 2. 3. , 2 2 2 VIII. Given that w t , Z and that 0 and Z 0 , identify the ordered pair that results when the following numbers are wrapped onto a unit circle: 1. w t 2. w t 3. w t 2007 2 First Quarter Pre-Calculus Assignment Packet IX. Evaluate the following without using a calculator: 3 13 1. sec 2. csc 3. sin 3 4 6 7 15 13 4. cos 5. tan 6. cot 3 4 4 23 17 11 7. cos 8. cos 9. sin 6 4 3 19 10. cot 6 Assignment 1.2 (Sine and Cosine Functions with Transformations) I. Remember that when a number line is wrapped around a unit circle, each number t corresponds to the ordered pair (cos t ,sin t ) . [ W (t ) (cos t ,sin t ) ]. 1. Sketch the graph of y cos t over the interval 4 t 4 . 2. What is the amplitude of y cos t ? [Amplitude = ½ (maximum y value – minimum y value)]. 3. What is the period of y cos t ? [How far along the horizontal axis must you go before the complete pattern repeats itself?]. 4. Name 5 key points in one period of the function beginning at t 0 and identify each as a ―maximum‖, a ―minimum‖, or a ―crossing‖. II. Do steps a—d for each of the following functions: 1. y 4sin x 2. y 3cos x 3. y 2cos x 4. y 3sin x a. Find the amplitude b. Find the period c. Sketch the graph showing at least one period of the function. d. List the 5 key points and identify each as a ―maximum‖, a ―minimum‖, or a ―crossing‖. 3 First Quarter Pre-Calculus Assignment Packet III. Do steps a—e for each of the following functions: 1. y sin x 2 2. y cos x 1 3. y 2sin x 3 4. y 3cos x 1 a. Find the amplitude b. Find the period c. Find the vertical shift (include direction) d. Sketch the graph showing at least one period of the function. e. List the 5 key points and identify each as a ―maximum‖, a ―minimum‖, or a ―crossing‖. IV. Do steps a—f for each of the following functions: 1 1. y sin 3x 2. y cos x 2 1 3. y sin(3x) 4. y cos x 2 5. y sin x 6. y cos x 2 4 2 7. y sin 3 x 8. y cos x 6 3 a. Find the amplitude b. Find the period c. Find the vertical shift (include direction) d. Find the phase shift (include direction) e. Sketch the graph showing at least one period of the function. f. List the 5 key points and identify each as a ―maximum‖, a ―minimum‖, or a ―crossing‖. V. Do steps a—f for each of the following functions: 1 1. y sin x 2. y 2sin x 2 2 3. y 3sin 2 x 4. y 2cos x 3 4 2 5. y cos 3x 6. y 2cos 3 x 1 2 12 a. Find the amplitude b. Find the period c. Find the vertical shift (include direction) d. Find the phase shift (include direction) e. Sketch the graph showing at least one period of the function. f. List the 5 key points and identify each as a ―maximum‖, a ―minimum‖, or a ―crossing‖. 4 First Quarter Pre-Calculus Assignment Packet Assignment 1.3 (Interval Notation) I. Intervals can be expressed in the following ways: a. As an inequality or a compound inequality b. In interval notation c. As a graph on a number line Express each of the following intervals in all three ways: 1. 2 x 5 2. x 3 3. 1 x 6 4. (, 2) 5. (3, ) 6. (2, 4] 7. 1 8. -2 3 II. Represent the result of each of the following operations in three ways: a. as a graph on a number line, b. as an inequality or compound inequality, c. in interval notation. 1. [3, 2] [1, 4] 2. [3, 2] [1, 4] 3. (4, 0) (5,1) 4. (4, 0) (5,1) 5. (,3) (1, ) 6. (,3) (1, ) Assignment 1.4 (Secant, Cosecant, Tangent, Cotangent Functions with Transformations) 1. For each function state the vertical stretch factor, the period, the phase shift, the vertical shift, the range, and sketch at least one period of the function. List 5 key points or asymptotes. 1. y 2sec x 2. y 3csc(2 x) 3. y 3sec x 1 4. y 4csc 2 x 4 2. Pre-calculus textbook page 401 #13—16 3. For each function state the vertical stretch factor, the period, the phase shift, the vertical shift, the range, and sketch at least one period of the function. List 3 key points or asymptotes. 1. y 2 tan x 2. y 3tan(2 x) 3. y 3cot x 1 4. y 4cot 2 x 4 5 First Quarter Pre-Calculus Assignment Packet Assignment 1.5 (Sine equations that fit given parameters) I. Write the equation of the Sine curve that satisfies the given conditions: 1. Amplitude = 3 Period = 2 No vertical shift No phase shift 2 2. Amplitude = 4 Period = Vert shift = up 1 No phase shift 3 3. Amplitude = 1 Period = 4 Vert shift = down 2 Phase shift = right II. Write the equation of the Sine curve that satisfies the given conditions: 1. Choose one of the following cities and graph (by hand) the average monthly temperature. Determine the vertical shift, period, phase shift, and amplitude using the graph. Write the equation of the Sine curve that models the average monthly temperature. a. Cleveland, Ohio b. Nome, Alaska c. Memphis, Tennessee d. St. Louis, Missouri 2. Use the sine regression features of your graphing calculator to determine the equation of the Sine curve that models the average monthly temperature of the four cities listed in problem 1. Assignment 1.6 (Applications of periodic functions) 1. A weight attached to a spring bobs up and down 10 times in a minute. You measure the distance between the extreme positions of the bob to be 6 cm. a. What is the frequency (in seconds)? b. What is the period? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell the displacement of the bob (y) at any instant (t). Be sure it matches your sketch. 2. Consider a tuning fork with 66 vibrations per second. (This would give the pitch of C, two octaves below middle C). Suppose its maximum displacement from resting position is 2 mm. a. What is the frequency (in seconds)? b. What is the period of vibration? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell the displacement of the fork (y) at any instant (t). Be sure it matches your sketch. 6 First Quarter Pre-Calculus Assignment Packet 3. Recall the Ferris Wheel example (sun is directly overhead, diameter of Ferris Wheel is 20 feet, it takes 20 seconds to move 1/12 the circumference, where your shadow on the ground will be five feet from its starting position). Consider the SHADOW motion: a. What is the frequency (in seconds)? b. What is the period? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell where the shadow would be (y feet) at any instant (t seconds), assuming you start counting when you are at the bottom of the Ferris wheel. 4. A simple pendulum swings with a bob that makes 30 complete swings in a minute. The horizontal distance between extreme left and extreme right positions is 10 cm. a. What is the frequency (in seconds)? b. What is the period? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell the horizontal displacement of the bob (y) at any instant (t). Be sure it matches your sketch. f. Why is this particular pendulum referred to as a ―seconds pendulum‖? 5. A lead ball is swinging back and forth on a long string. You count 5 complete back and forth swings in a minute and estimate the width of the arc through which the ball swings to be 10 cm. a. What is the frequency (in seconds)? b. What is the period? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell the horizontal displacement of the bob (y) at any instant (t). You begin the time when the ball is at maximum displacement from resting position. 6. At high tide in the Bay of Fundy, the water level is approximately 8 m above the level at low tide, and the time between high tides is 12.4 hours. a. What is the frequency (in hours)? b. What is the period? c. What is the amplitude? d. Sketch a graph of this function. e. Write an equation of a sine curve that will tell the position of the tide (y meters) at any instant (t hours) after high tide. 7 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.7 (Linear Equations) I. For the line AB passing through the given point A with the given slope m, find the following: a. The equation of line AB in point-slope form. b. The equation of line AB in slope-intercept form. c. The equation of line AB in general form. d. The slope of a line perpendicular to line AB. e. The equation (general form) of a line passing through A perpendicular to line AB. f. The equation of a line (general form) passing through (6, -2) parallel to line AB. 1 1. A (3, 7); m = 4 2. A (-2, 4); m = -2 3. A (-4, -3); m = 2 1 4. A (3, -1); m = 0 5. A (0, 0); m = 6. A (4, 0); m is undefined 4 II. Use the definition of rate of change to answer the following: 1. After a person has stopped drinking, his or her blood alcohol level decreases slowly as time passes. We can say that blood alcohol concentration C is a function of time, and write C f (t ) . Write an expression for the rate of change of a person’s blood alcohol concentration over the period from 1 hour to 3 hours after that person has stopped drinking. 2. The population of a city is changing. If P is the population size and t the time in number of years since 1995, we can write P f (t ) . Use this to do the following: a. Use function notation to express the rate of change of P with respect to t for the interval from 1995 to 2000. b. Write down another formula for the rate of change of P with respect to t for the interval from 2000 to 2005. c. The population was 650,000 in 1995 and had grown to 700,000 by 1997. Assuming a linear model, sketch a graph of the function P f (t ) and find an equation to describe the model. d. Calculate the rate of change of P with respect to t over the intervals 1995 to 2000 and 2000 to 2005. (Use the graph or the equation to help you.) What do you notice? Why does this make sense? 3. An environmental action group sells screen-printed T-shirts to raise funds. If they set the price at $20 each, they sell an average of 60 shirts a week. They have experimented with changing the price and have found that, on the average, for each extra $5 they charge, they sell 10 fewer T-shirts. a. Sketch a graph showing the average number of T-shirts sold as a function of the price, and find a formula (equation) for this function. b. If they increase the price from $20 to $25, how does the number sold change? What is the average rate of change of sales with respect to price? c. If they increase the price from $25 to $30, what is the average rate of change of sales with respect to price? 8 First Quarter Pre-Calculus Assignment Packet 4. Refer to in-class examples and numbers 1-3 above to answer the following questions: a. Does the rate of change of a linear function depend on the interval over which it is measured? b. How can you tell the rate of change of a linear function from its graph? c. If a linear function is defined by the formula y mx b , what is the rate of change of y with respect to x? d. If a linear function is defined by the formula ax by c 0 , what is the rate of change of y with respect to x? (Hint: solve for y). e. What is the significance of a positive or negative rate of change? III. For the line AB passing through the given points A and B, find: a. The length of the line segment AB b. The midpoint of line segment AB c. The slope of line segment AB d. The equation of line AB in point-slope form e. The equation of line AB in slope-intercept form f. The equation of line AB in general form g. The slope of a line perpendicular to line AB h. The equation of a line in general form passing through B perpendicular to line AB i. The equation of the perpendicular bisector of line segment AB in slope-intercept form j. The equation of the line passing through (5, -1) parallel to line AB in general form. 1. A: (-3, 2); B: (1, -1) 2. A: (3, 5); B: (6, 5) IV. Marion is located 27 miles north and 5 miles west of Columbus. Ashland is located 45 miles north and 28 miles east of Columbus. a. Find the distance between Marion and Ashland. b. The point located exactly half way between Marion and Ashland is located where in reference to Columbus? c. What is the slope of the line connection Marion and Ashland? d. What does this number (the slope) mean in the context of the problem? 9 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.8 (Linear Equations: Line of Best Fit) The table gives net sales y (in millions of dollars) for Proctor & Gamble Company for the years 1980 through 2003 (with some years excluded) where t is time in years with t=0 corresponding to 1980. (Source: 1992 and 2003 Annual Reports for Proctor & Gamble Company). Year t y a. Plot the points on your graphing calculator. Make a sketch of the screen (it 1980 0 10,772 does not have to be perfect). Find the equation of the line of best fit using a linear model. 1981 1 11,416 1982 2 11,994 b. Explain the significance of the y-intercept and the slope in the context of the 1983 3 12,452 data. 1984 4 12,946 1985 5 13,552 c. Use the model to predict net sales for the year 2005. 1986 6 15,439 1987 7 17,000 1988 8 19,336 1989 9 21,398 1990 10 24,081 1991 11 27,026 1992 12 29,362 1999 19 38,125 2000 20 39,951 2001 21 39,244 2002 22 40,238 2003 23 43,377 10 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.9 (Zeros of Polynomial Functions by Factoring, Quadratic Formula, Synthetic Division) I. Find all zeros (simplest radical form if necessary) for the following functions: 1. f ( x) 6 x 12 2. f ( x) 2 x 10 3. f ( x) 3x 10 4. f ( x) x 2 10 x 24 5. f ( x) 3x 2 x 10 6. f ( x) x 2 5 x 6 7. f ( x) x 2 9 8. f ( x ) x 2 7 9. f ( x) 2 x 2 x 2 10. f ( x) x 2 5 x 7 11. f ( x) 5 x 2 10 12. f ( x) x 4 16 x 2 II. Use polynomial long division or synthetic division on the following problems to simplify. x2 4 x4 1 1. 5. x2 x2 1 x3 8 x4 1 2. 6. 2 x2 x 1 x3 8 5 x5 3x 2 2 x 7 3. 7. x2 x2 4 3x3 2 x 2 x 1 x3 x 2 x 1 4. 8. x2 x2 2 III. For each function do the following: a. Write all the potential rational zeros of the function b. Find all the zeros of the function 1. f ( x) x3 4 x 2 x 6 2. f ( x) 6 x 3 13 x 2 x 2 3. f ( x) x 4 2 x 2 3x 2 4. f ( x) x3 5 x 2 2 x 12 11 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.10 (Zeros of Polynomial Functions using a graphing calculator) I. Find all zeros of the following polynomial functions to the nearest hundredth. 1. f ( x) 24 x 3 22 x 2 x 2 2. f ( x) 2 x3 10 x 2 7 x 15 3. f ( x) 2 x 3 5 x 2 6 x 8 4. f ( x) 2 x3 10 x 2 7 x 17 5. f ( x) x 4 3x 3 2 x 2 5 x 6 6. f ( x) x 4 3x 3 2 x 2 5 x 6 7. f ( x) x 4 3x 3 2 x 2 5 x 2 ASSIGNMENT 1.11 (Limits of Functions) I. Use the graphs to find the limits of the given functions. 1. For the function f graphed below, find 2. For the function F graphed below, find a. lim f ( x) b. lim f ( x) a. lim F ( x) b. lim F ( x) x3 x3 x2 x2 c. lim f ( x) d. f (3) c. lim F ( x) d. F (2) x3 x2 e. lim f ( x) x f. lim f ( x) . x e. lim F ( x) x f. lim F ( x) . x 8 6 6 4 4 2 -10 -5 5 10 2 -2 -4 -5 -6 -8 Graph of f Graph of F 12 First Quarter Pre-Calculus Assignment Packet 3. For the function graphed below, find 4. For the function G graphed below, find a. lim ( x) b. lim ( x) a. lim G( x) b. lim G( x) x2 x2 x0 x0 c. lim ( x) x2 d. (2) c. lim G( x) x0 d. G (0) e. lim ( x) x f. lim ( x) . x e. lim G( x) x f. lim G( x) . x 8 4 6 2 4 2 -10 -5 5 10 -2 -10 -5 5 10 -4 -2 -6 -4 -8 -6 -10 -8 Graph of φ Graph of G 5. Consider the function g in the following graph. 8 a. For what values of x0 does lim g ( x) NOT exist? 6 x x0 4 b. For what values of x0 does lim g ( x) exist? 2 x x0 -10 -5 5 10 -2 -4 -6 -8 Graph of g 13 First Quarter Pre-Calculus Assignment Packet Section II. Evaluate the following limits algebraically: 1. lim 2 2. lim 2 3. lim 10 x 6 x 7 x4 4. lim 3x 5. lim 2 x 5 6. lim5x 6 x 4 x 2 x 0 7. lim x 2 8. lim x 2 3x 9. lim x 6 2 x 3 x 2 x 5 10. lim x3 x 2 3 11. lim x2 2 x 3 12. lim 11 x 2 x 3 x 2 x 5 13. lim 9 x 3x x 2 3 14. lim 3x 8 x x 15. lim x x 2 x 16. lim 12 5x 4 3 x x 17. lim x 2 x 1 3 2 18. lim 11 x x 19. lim 3 20. lim9 x x3 21. lim 2 x 4 x x 6 x 2 14 First Quarter Pre-Calculus Assignment Packet III. Evaluate the following limits (feel free to use graphs from previous notes). YOUR ANSWER MAY BE: A Real Number DNE= the limit Does Not Exist = the limit tends to positive infinity - = the limit tends to negative infinity x x x 1 1. lim e 2. lim e 3. lim x x x 2 x 1 4. lim 5. lim csc x 6. lim csc x x 2 x x 7. lim sec x 8. lim tan x 9. lim cot x x x x 2 10. Does the lim cot x = lim cot x ? Does the lim cot x exist? x x x 1 11. lim sin x 12. lim 13. lim x 2 x x 0 x 2 x 0 1 1 1 14. Does the lim 2 = lim 2 ? Does the lim exist? x 0 x x 0 x x 0 x2 15. lim ln x 16. lim ln x x 1 x 0 15 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.12 (Average and Instantaneous Rates of Change) I. For the following functions find: a. The numerical value for the average rate of change in f ( x) between x1 0 and x2 2 . b. The general expression for the average rate of change of f ( x) between x1 x and x2 x h . 1. f ( x) 4 x 2 2. f ( x) 2 6 x II. For the function y x 3 , find: a. The numerical value for the average rate of change of y between x1 0 and x2 2 . b. The general expression for the average rate of change of y between x1 x and x2 x h . III. For the following functions find: a. The general expression for the slope of the secant to the graph of f ( x) between the points x, f ( x) and x h, f ( x h) . b. The numerical value for the slope of the secant to the graph of f ( x) between 2, f (2) and 3, f (3) . NOTE: For #1 just use your result from letter a and substitution. For #2 use f (b) f (a) msec where a 2 and b 3 . For # 3 use both methods. ba c. The numerical value for the instantaneous rate of change of f ( x) with respect to x at the point where x 3 . d. The equation of the line tangent to the graph of f ( x) at the point where x 3 . e. The equation of the line perpendicular to the graph of f ( x) at the point 3, f (3) . 1. f ( x) 3x 2 2. f ( x) 9 x 5 3. f ( x) x 2 3x 2 16 First Quarter Pre-Calculus Assignment Packet IV. The graph below shows the squirrel population in a certain wilderness area. The population increases as the squirrels reproduce, but then decreases sharply as predators move into the area. a. What is the average growth rate from day 10 to day 50? (remember units!) What does this value mean in the context of the problem? b. During what approximate time period, beginning at day 0, is the average growth rate of the squirrel population positive? Hint: remember definition of average rate of change. c. During what approximate time period, beginning at day 0, is the average growth rate of the squirrel population zero? d. During what approximate time period is the instantaneous growth rate of the squirrel population positive? e. Approximate the instantaneous growth rate of the squirrel population on day 30. Hint: sketch a tangent line and determine its slope. f. Approximate the instantaneous growth rate of the squirrel population on day 100. 17 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.13 (Derivatives and Derivative Notation) I. For the following functions find f '( x) using the definition of the derivative. 1. f ( x) 4 x 2 2. f ( x) 2 x 6 x 2 3. f ( x) x3 2 4. f ( x) x3 4 x II. Find the equation of the line tangent to f ( x) x3 4 x at the point where x 2 . (Hint: Use results from section I # 4). III. 1. Let y 4 x 2 2 . dy dy a. Find b. Find dx dx x 1 2. Find f (t ) if f (t ) 4t 2 t . dA 3. Find if A 3 2 . d IV. Use the power rule to find the required derivative for each of the following: dy dy 1. for y 4 x7 2. for y 3x8 2x 1 dx dx dy dy 1 3. for y 3 4. for y ( x7 2 x 9) dx dx 3 5. Find d dt 16t 2 4 6. Find V (r ) for V (r ) r 3 3 d2y d2y 7. Find for 7 x3 5 x 2 8. Find for 12 x2 2 x 3 dx 2 dx 2 9. Find f (2) for f ( x) x 4 2 x 3 x 5 d2y 10. Find for y 6x5 4x2 dx 2 x 1 18 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.14 (Position-Time Graphs) 1. Some students were asked to move according to the directions below. Each student moved for 40 seconds. Draw sketches to show what the resulting position-time graphs should look like. a. Run as fast as you can for 10 seconds, stop for 10 seconds, and then walk for the rest of the time. b. Start at the far end of the room and then walk back. c. Walk for 10 seconds, run for 10 seconds, and then turn around and walk slowly back towards the starting point. d. Walk for 4 seconds, stand still for 4 seconds, and keep repeating this pattern until the time is up. e. Walk for 20 seconds, turn and walk back towards your starting point for 10 seconds, then turn around and walk away from the starting point again. 2. A class of precalculus students was required to turn in a sketch of the position-time graph they made while doing an activity involving straight line motion and a motion detector. Determine which of the groups actually did the activity and which groups must have faked their results. Give a description of the motion produced for each graph and tell why the motion is or is not possible! Group A: Group B: d d d d t t d d Group C: Group D: d d d d t t d d 19 First Quarter Pre-Calculus Assignment Packet Group E Group F d d d d t d Group G Group H t d d d d d t t d d 3. The figure below shows the position-time function for a certain particle moving on a straight line (“one-dimensional” motion): d d t0 t1 t2 a. Is the particle moving faster at time t0 or at time t1 ? Explain. b. At the origin, the tangent line is horizontal. What does this tell us about the initial velocity of the particle? c. Is the particle speeding up or slowing down in the interval t0 , t1 ? Explain. d. Is the particle speeding up or slowing down in the interval t1 , t2 ? Explain. 20 First Quarter Pre-Calculus Assignment Packet Assignment 1.15 (Vertical Motion Applications) I. Note: neglect air resistance in the following problems 1. A construction worker at the top of Cleveland’s Terminal Tower, 708 feet above the ground, had to sneeze and accidently dropped his wrench. a. Write the equation describing the vertical position of the wrench as a function of time. b. Write the equation describing the instantaneous velocity of the wrench as a function of time. c. Find the velocity of the wrench just before it hits the ground. d. How much time does it take for the wrench to reach the ground? 2. A toy rocket is launched straight up in the air from a platform 10 feet above the ground. The initial velocity of the rocket is 170 feet per second. a. Write the equation describing the vertical position of the rocket as a function of time. b. Write the equation describing the instantaneous velocity of the rocket as a function of time. c. Find the average velocity of the rocket from launch to t 3sec . d. How much time does it take for the rocket to reach the ground? 3. Galileo is reported to have dropped two iron spheres from an upper balcony of the Leaning Tower of Pisa, approximately 49 meters above the ground. a. Write the equation describing the vertical position of the iron spheres as a function of time. b. Write the equation describing the instantaneous velocity of the iron spheres as a function of time. c. Find the average velocity of the spheres from the time they are released to t 2sec . d. Find the instantaneous velocity of the at time t 2sec . e. How much time does it take for the spheres to reach the ground? 4. Suppose a stone is hurled downward from a bridge that is 20 meters above the Cuyahoga River. If the initial velocity is 4 meters per second, determine how fast the stone is traveling exactly one second later. 5. Suppose a stone is thrown upward from a much lower bridge that is 20 feet above the same river. If the initial velocity is 52 feet per second, determine how fast the stone is traveling exactly one second later, three seconds later, and 5 seconds later. 21 First Quarter Pre-Calculus Assignment Packet II. Solve all problems on your own notebook paper. 1. A rock is thrown straight up from level ground. The distance (in ft) the rock is above the ground (the position function) is h(t ) 3 48t 16t 2 at any time t (in sec). Find The following: a) h(t ) b) h(0) c) The initial velocity of the rock. d) The time at which the instantaneous velocity of the rock is zero. e) The maximum height of the rock. f) The time the rock reaches the ground again (to the nearest hundredth). g) The velocity with which the rock hits the ground (to the nearest hundredth). h) Can you think of a reason why the answer to (g) is not exactly 48 ft/sec? i) The average velocity between 1.5 and 3 seconds. j) The instantaneous velocity at 2 seconds. 2. A flare is launched from a cruise ship deck, 75 feet above the ocean, with an initial vertical velocity of 130 ft per second. Find the following: a) The position function h(t ) where t is time from launch (in seconds) and h is the position of the flare (in feet) above the ocean. b) The instantaneous velocity function h(t ) . c) The time at which the instantaneous velocity of the flare is zero. d) The maximum height of the flare. e) The time the flare reaches the ocean surface. f) The velocity with which the flare hits the ocean surface. 3. An arrow is shot vertically upward with an initial velocity of 50 meters per second from a height of 2 meters. Find the following: a) The position function h(t ) where t is time in the air (in seconds) and h is the position of the arrow (in meters) above the ground. b) When does the arrow reach its maximum height? c) What is the arrow’s maximum height? d) When does the arrow hit the ground? ASSIGNMENT 1.16 (Intermediate Value Theorem) I. What values do we know that f(x) MUST take on the given interval (a, b)? HINT: Use the Intermediate Value Theorem and write your answers in interval notation. 1. f ( x) x 2 3x 7 on (-2,1) 2. f ( x) x 2 x 3 3 on [-1,3) 3. f ( x) x x 1 3 on (1,2] 4. f ( x) x 2 x 5 4 on (-3,-2) 5. f ( x) x 10 5 on [2,4] 22 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.17 (Transformations and Symmetry) I. Let the graph of y=f(x) pass through (-2, 6), (0,3), and (4,11). Identify 3 points on the graph of the following functions. 1. y f ( x) 2. y f ( x) 3. y 6 f ( x) 4. y f ( x 1) 5. y f (4 x) 6. y f ( x) 2 7. y 3 f ( x 1) 1 1 8. y f x 1 1 3 II. For the following functions g(x) a) Find g(-x) b) Identify the function as even, odd, or neither c) Are there any symmetries for g(x)? 1. g ( x) x 2 x 2 2. g ( x) x 4 6 x 2 8 3. g ( x) 3x 3 2 x III. The graph of y=f(x) is pictured below with a table of 5 point of f(x). X Y 6 -3 0 fx = x+3x-2 x-5 18 -1 2 4 2 0 5 0 2 6 2 -5 5 -2 For the following functions, identify 5 points on the graph and sketch the graph. 1. y 2 f ( x) 2. y f (2 x) 3. y f ( x 1) 2 1 4. y f ( x 1) 2 23 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.18 (Number Line + - Graphs) Using the given graph: 2 -5 5 -2 -4 1. Determine the equation of f(x) using the identified points and the TI-83 regression capabilities. 2. Make a number line graph of the sign of f(x). 3. Determine the interval(s) where f(x) is above the axis by examining the graph. 4. Compare your answer for #3 with your line graph in number 2. What do you notice? 5. Find f ’ (x). 6. Make a number line graph of the sign of f ’ (x). 7. Determine the interval(s) where f(x) is increasing by examining the graph. 8. Determine the relative maximum of f(x) by examining the graph. 9. Determine the relative minimum of f(x) by examining the graph. 10. Compare your answers for #7-9 with your number line graph for #6. What do you notice? 11. Find f ‘‘ (x). 12. Make a number line graph of the sign of f ‘‘ (x). 13. Determine where f(x) changes concavity by examining the graph. 14. Determine the interval(s) where f(x) is concave up by examining the graph. 15. Compare your answers for #13 and 14 with your number line graph for #12. What do you notice? 24 First Quarter Pre-Calculus Assignment Packet ASSIGNMENT 1.19 (Number Line + - Graphs Continued) Draw a possible graph of f(x) using the given number line graphs of f(x), f ‘ (x), f ‘‘ (x). 1. Sign of f(x) + 0 - 0 + 0 - -1 1 3 Sign of f ‗ (x) - 0 + 0 - 0 2 Sign of f ‗‗ (x) + 0 - 3/2 2. Sign of f(x) + 0 - 0 + 0 - 0 + -3 -1 2 4 Sign of f ‗ (x) - 0 + 0 - 0 + -2 1 3 Sign of f ‗‗ (x) + 0 - 0 + 0 3/2 3. Sign of f(x) - 0 - -2 Sign of f ‗ (x) + 0 - 0 + 0 - -2 0 1 Sign of f ‗‗ (x) - 0 + 0 - -1 1 25