Rectilinear Motion PDF

UcyBeldner 91 Rectilinear Motion 1 The velocity of a particle moving along a line on the interval [0,g] is represented by the graph to the right. (a) State the interval(s) in which...

UcyBeldner 91 Rectilinear Motion 1 The velocity of a particle moving along a line on the interval [0,g] is represented by the graph to the right. (a) State the interval(s) in which the particle is moving in a positive direction. ooo (b) State the interval(s) in which the acceleration of the particle is positive. (c) State the interval(s) in which the particle is speeding up. of (d) State the time(s) when the particle is not moving. Fi (e) State the time(s) when the particle has zero acceleration. Is the particle stopped at each of these times? F yes t (f) State the interval(s) is which the particle is speeding up in a negative direction. oo Pre-Calculus Honors Rectilinear Motion 2 Based on the velocity function shown below of a hockey player practicing skating along a straight line, answer the following questions: (a) On what interval(s) is the hockey player moving forward? Explain. (b) On what interval(s) is the player stopped? Explain. (c) On what interval(s) is the player moving forward at an increasing rate? Explain. (d) On what interval(s) is the player moving forward at a decreasing rate? Explain. (e) When is the player speeding up? Explain. (f) Sketch a graph of the position of the player as a function of time. Assume that the players position at time t  0 , is at 0. (Do your best with this, try to make everything curve correctly) Pre-Calculus Honors UcyBeldner 91 Rectilinear Motion 1 The velocity of a particle moving along a line on the interval [0,g] is represented by the graph to the right. (a) State the interval(s) in which the particle is moving in a positive direction.velocity is Pos offloat 0,5 D 9 (b) State the interval(s) in which the acceleration of the particle is positive. velocity is odor F9 (c) State the interval(s) in which the particle is speeding up. C O b C speeding up in C direction fondness 8 speeding up in positivedirection (d) State the time(s) when the particle is not moving. d Vel o t b lead (e) State the time(s) when the particle has zero acceleration. Is the particle stopped at each of these times? vet not for t a ctoe F (f) State the interval(s) is which the particle is speeding up in a negative direction. A B onC B GoE f Pre-Calculus Honors Rectilinear Motion 2 Based on the velocity function shown below of a hockey player practicing skating along a straight line, answer the following questions: (a) On what interval(s) is the hockey player moving forward? Explain. 0,2 S 12f because the velocity is positive (b) On what interval(s) is the player stopped? Explain. 7 2 4,5 7 12 when V O the player is stopped rosacea (c) On what interval(s) is the player moving forward at an increasing rate? Explain. 0,1 5,6 the velocity as positive C more and increasing hot as a cop is getting in (d) On what interval(s) is the player moving forward at a decreasing rate? Explain. 12 7,8 told is 11,12 velocity is positive but getting smaller (e) When is the player speeding up? Explain. 0.1 II2 3 3,4 5 6 getting farther from negatives (f) Sketch a graph of the position of the player as a function of time. Assume that the players position at time t  0 , is at 0. (Do your best with this, try to make everything curve correctly) Pre-Calculus Honors Name: Date:9 13 Lucy Beldner Runner A runs on a straight track for 0  t  10 seconds. The graph, which consists of two line segments, shows the velocity, in meters per second of Runner A. a) Find the acceleration at time t  2 seconds. Show your work. Indicate units of measure. 1 oooh 35 s b) Find the velocity at time t  2 seconds. Show your work. Indicate units of measure. V 6 SI t axt ammam 13.2 875 c) Is the runner speeding up or slowing down at time t  6 seconds. Justify your answer. constant Mktg neither at 6 the runner is moving at a soooooooooooo 2) The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a continuous and strictly increasing function R or time t. The graph of R and a table of selected values of R(t), for the time interval 0  t  90 minutes, are shown below. t R(t) (minutes) (gallons per minute) Use the data from the table to find the average rate 0 20 of change from t  50 to t  70. Indicate units of 30 30 measure. Explain the meaning in context of the 40 40 problem. 50 55 70 65 o soso.tooo 90 70 09m rate inc by is 91m between so 70 3) A car travels on a straight track. During the time interval 0  t  60 seconds, the car’s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. The table below shows selected values of these functions. Use the table to answer the following questions. t (sec) 0 15 25 30 35 50 60 v (t ) (ft / sec) -20 -30 -20 -14 -10 0 10 a (t ) (ft / sec2 ) 1 5 2 1 2 4 2 a) At what time(s) does the car turn around? Justify your answer. changes signs 7 50 Vtooooo b) Explain why the acceleration from t  0 to t  15 must be negative at least once. a e've S sooooooang reased c) Is the car speeding up or slowing down at t  15 , Justify your answer. Slowing down going up before neg vel Name: _____________________________________ Date: _________________ 1) Which of the following represents a function that is increasing at a decreasing rate over the entire interval shown? Justify your answer. 2) Tamara is running a race in a straight line. At time t  0 , she crosses the finish line. Her distance from the finish line is a function of the time since she crossed it as shown by the graph to the right. Answer the following based on the graph: a) On what interval is Tamara’s velocity positive? Justify your answer. b) On what time interval is Tamara running at a decreasing rate? Justify your answer. c) Is Tamara speeding up or slowing down on the time interval 5  t  10 ? Justify your answer. 3) A bug begins to crawl up a vertical wire at time t  0. The velocity of the bug a time t, 0  t  8 , is given by the function shown to the right. a) At what value(s) of t does the bug change direction? Justify your answer. b) On what interval of time does the bug move at a constant rate? Justify your answer. 4) The graph of the velocity v (t ) , in ft/sec, of a car traveling north on a straight road, for 0  t  50 , is shown at the right. a) On what time interval is the car slowing down? Justify your answer b) Does the car ever go south? Justify your answer. A metal wire, of length 8 centimeters is heated at one end. The table below gives selected values of the temperature T (x ) , in degrees Celsius, of the wire x cm from the heated end. a. Find the rate of change of the temperature of the wire with respect to its’ distance over the interval x  6 to x  8. Indicate units of measure. b. Explain the meaning, using correct units, of the answer you found in part a. c. Evaluate T (8) T (0) and indicate units of measure. d. Explain the meaning, using correct units, the answer you found in part c. Name: _________________________________________ Date: _____________________ Practice with Limits Pre Calculus Sketch a single function that satisfies each of the following properties. f 2   1 f 2   4 lim f (x )  4 x 2 lim f (x )  4 x 2 lim f (x )  1 x 2 lim f (x )  1 x 2 a. Find the value of lim f (x ) a. Find the value of lim f (x ) x 2 x 2 b. Is the function continuous at x = 2? b. Is the function continuous at x = 2? f 2   4 f 2   4 lim f (x )  1 lim f (x )  4 x 2 x 2 lim f (x )  1 lim f (x )  DNE x 2 x 2 a. Find the value of lim f (x ) a. Find the value of lim f (x ). x 2 x 2 b. Is the function continuous at x = 2? b. Is the function continuous at x = 2?  f 2   4 f 2  DNE lim f (x )  1 lim f (x )  1 x 2 x 2 lim f (x ) exists x 2 a. Find the value of lim f (x ) a. Find the value of lim f (x ). x 2 x 2 Find the value of lim f (x ). x 2 b. Is the function continuous at x = 2? b. Is the function continuous at x = 2? f 2   4 f 2   4 lim f (x )  1 lim f (x )  4 x 2 x 2 a. Find the value of lim f (x ) a. Find the value of lim f (x ). x 2 x 2 Find the value of lim f (x ). Find the value of lim f (x ). x 2 x 2 b. Is the function continuous at x = 2? b. Is the function continuous at x = 2? Continuity e x  2, x 0  1 Is the function f (x )  2, x 0 continuous at x  0 ? cos x , x 0  3 What value of k would make the following function continuous at x  e ?  4ln(x )  5, 2 x e f (x )   k e  6,  x e Continuity 4 Find the value of p so that the limit at x approaches -1 exists 4x  7, x  2  f (x )  q  3, x  2 x 2  px  1, x  2  Find the value of q so that the function is continuous at x  1. 5 Find the values of j and k so that g (x ) is continuous at x  2.  x  3  4, x 2  g (x )   j , x 2 kx 2 , x 2  Continuity 6 Answer the following questions based on the function below: x  4, x  3  6 3  x  0 f (x )   2 x  6, 0 x  4 3log x x 4  2 lim f (x ) x 3 lim f (x ) x 0 lim f (x ) x 4 Is the function continuous at x = 4? Name: Date: 1) Find each of the following limits: lim 4x 3  2x 2  10x  6 a) x  lim 4x 3  2x 2  10x  6 b) x  lim  4x 6  3x 3  2x 2  12 c) x  lim  4x 6  3x 3  2x 2  12 d) x  lim 2  3x 4  6x 5 e) x  lim 2  3x 4  6x 5 f) x  2) Answer the following about the function f (x )  13x 5  9x 3  11x 2  3 a) What is the minimum number of x – intercepts that the function could have? (I don’t want to know how many it does have) Explain b) What is the maximum number of x – intercepts that the function could have? Explain. lim f (x ) c) Find x  lim f (x ) d) Find x  Name: Date: For each of the following functions: a) Draw a sketch of the function. (you may need to show some “extra” work) b) State the domain of the function. c) State the range of the function. d) Find the end behaviors of the function. 1 f (x )  x 4 g (x )  4x 7 1 h (x )  x 5 m (x )  ln(x  4) p (x )  3e  x 10) True or false: A graph of a function and its inverse can never intersect. Explain your reasoning. (3 pts) False if a coordinate has the same x y value the function inverse will intersect 3x  5 11) Determine the inverse of the y  (3pts) 6x  11 64 11 6y.in X YE Kahaani 4 6 4 11 311  LUCY Beldne 5) Let f ( x ) = ln ( x − 5 ) + 4 Iiiii a) Sketch the graph of f. Hawes 1 50 b) State the domain and range of f c) Find the coordinates of the x intercept 5O d) Find the coordinates of the y intercept DNE all y values 0 e) Find the inverse to f Y In x 5 4 ohhh 6) Let g ( x ) = e x +1 + 7 a) Sketch the graph of g. T.es b) State the domain and range of g. c) Find the coordinates of the x intercept DNE d) Find the coordinates of the y intercept ye 7 8 en e) Find the inverse to g sad in x 7 in e g x In x 7 1 Name: Date: 6x  x 2 6x  x 2 lim lim x  x 3  36x x 6 x 3  36x 6x  x 2 6x  x 2 lim lim x 6 x 3  36x x 0 x 3  36x 5  7x  4 lim x 3 x2  9 x 4x  3  lim x 22 5x x 3 x  4x  3 x 2  6x  8 Determine if f x    x 2  8x  16 has a removable discontinuity, vertical asymptote, or jump discontinuity at x  4. Justify your answer. Name: ____________________________________ Date: _________________ Find the exact value of each of the following:  3 tan  cos csc 2 2 3 7  cot sec sin 4 6 3  If f (x)  csc(2x)  tan x cot x  ln(e x ) find the exact value of f    3 For what value of x , 0  x  2 is sin x cot x  sin x  0 Find:  tan lim tan x lim tan x 2 x   x   2 2 3 sec lim sec x lim sec x 2 x 3  x 3  2 2 5) Solve the following equation for x , 0  x  2 : 2cot x sin x  2 cot x  0 6) Given the function f(x)= 5cos(x)sin(x)-cos(x). Find the zeros of f(x) on the domain [0, 2π] Name: Lucy _____________________________________ Beldner Date: 10123 _________________ 1) For each rational function below, find:  The Domain  Critical valueszeros  The horizontal asymptote(s).  Make a # line to determine the sign of the functions 1 a) f (x )  x 4 2 x2 x 2 Domain all realnumbersexcept 2 2 rotate 0 off 1f x 15 x O x 2 b) f (x )  x2 4 x 2 x 2 ementa is oak is  x 3 2) Let f (x ) . Find each of the following: x2 1 f (3) f (1) f ( 1) FCI 80 ooo f i see undefined fc.ge y sad f undefined1mtFCD lim f (x ) x  E2 0 erodes ooo lim f (x ) x 1 time ii rest to lim f (x ) x 1 pigfasting es if rode lim f (x ) node has lim f (x ) lim f (x ) x  YIoflxs E O x 1 glimpses re x 1 ftp.eflxtl 3) Determine each of the following limits: x3 x a) lim x  5  x  2x 3 116 2xs (x  2)(x  3) b) lim off x  (2x  3)2 ax II x5  x2 1 c) lim x  x7 O x3 x d) lim X O x  5  x 3 e) lim x4 x x  5  x time x i  Name: Date: Lucy Beldner 10 31 - Find all information and sketch the following functions 3x 4  27 x 2 Iii f (x )  x 2  9x  18 ii x 2  5x 5 3 6 g (x )  3 all real Domain x  8x  15x 2 cus 1 s.si 2x 2  10x  12 h(x )  2 x  7 x  12 removabedc xe 3iim.il 15 3131 16,2 54 end behavior 3 3 ha g Ears 8 6 Wmgag yd  Elites Domain all real S 0 3 S cv it nooooge Iii PV o e P I domain all real 4 3 cus 9 end 23ha 4 2 time removable dc sat iii i I I 6) – 7) Find the inverses of the functions given below and check your work using the composition of inverses: (4 pts each) 6) g (x )  f (x 3  5) 5 Tests fluftif f 5 s x F s g gas Sx F Y s jj9 offdffkf F sx FFC s 7) f (x )  3(x 2  7) pepgy.gg are E 1 gfyfcr.ca gpgrss 8) Given the function g(x) = (h(x  2))2 find g-1(x). (3 pts) h 2T x 9 x 2 X h 9) True or False: If f(x) has a horizontal asymptote at y = 4, then the inverse of f(x) has a vertical asymptote at x = 4. Explain your reasoning. (3 pts) the x and y values are being switched 10) True or false: A graph of a function and its inverse can never intersect. Explain your reasoning. (3 pts) False if a coordinate has the same x y value the function inverse will intersect 3x  5 11) Determine the inverse of the y  (3pts) 6x  11 64 11 6y.in X YE Goto 4 6 4 11 311 3 DNE  Domain of Ln E m in 2 x x2 2x 3 I ftp.t Y  161 1 160 Fiftyte 16 32 e a a x 64 2  2 4105 t 3Csc tan E  312 3 4 x unfix n x 4 xy 21 2 16 i va x 4 In top or bottom  Y e Lu f x in x 4 2 fa.ee in11 xth O n 4 2 2 e y  3 213 3 2A 22 753 3 03 2 2 2 12 3 2 257 k ex 2 2Gt k ez 2 2 tiffiti 2EE.EE k 24 3t3K I5 0