MTH 99 Chapter Notes - Chapter 2: Ferrari P, Piecewise, Floor And Ceiling Functions

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plummoose320

21 Nov 2017

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When functions are increasing, decreasing, or constant from its own graph. When it"s on an open interval, i, if, for any possible choice of both x1 and x2 in i when noting that x1 < x2, which therefore notes that f (x1) < f (x2) When it"s on an open interval, i, if, for any possible choice of both x1 and x2 in i when noting that x1 < x2, which therefore notes that f (x1) > f (x2) When it"s on an open interval, i, if, for any possible choice of just x in i when the values in f (x) are all equal. What"s the di erence between a maximum and minimum: maximum. The maximum would be the largest y-value for any function: minimum. The minimum would be the smallest y-value for any function. What"s the di erence between a local minimum and an absolute mini- mum: local minimum.

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Q1. Determine, without graphing, whether the quadratic function f(x) = x2 + 2x - 6 has a maximum value or a minimum value and then find that value. a. minimum; -7 b. minimum; -1 c. maximum; -1 d. maximum; -7 Q2. Find (f - g)(4) when f(x) = 5x2 + 6 and g(x) = x + 2. a. 80 b. 88 c. 84 d. -90 Q3. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q4. Find the average rate of change for the function f(x) = -3x2 - x from the values 5 to 6. a. 1/2 b. -2 c. -34 d. -1/6 Q5. Find the vertex and axis of symmetry of the graph of the function f(x) = 3x2 + 36x. a. (-6, -108); x = -6 b. (-6, 0); x = -6 c. (6, -108); x =6 d. (6, 0); x = 6 Q6. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q7. Use the graph to find the intervals on which it is increasing, decreasing, or constant. a. Increasing on (-3, -2) and (2, 4); decreasing on (-1, 1); constant on (-2, -1) and (1, 2) b. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1); constant on (-2, -1) and (1, 2) c. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1) d. Decreasing on (-3, -1) and (1, 4); increasing on (-2, 1) Q8. Graph the function F(x) = -5. State whether it is increasing, decreasing, or constant. a. b. c. d. Q9. Determine algebraically whether f(x) = -2x3 is even, odd, or neither. a. even b. odd c. neither Q10. Find the average rate of change for the function f(x) = 3/(x - 2) from the values 4 to 7. a. 7 b. 1/3 c. -3/10 d. 2 Q11. For the graph of the function y = f(x), find the absolute maximum and the absolute minimum, if it exists. a. Absolute maximum: f(4) = 7; Absolute minimum: f(1) = 2 b. Absolute maximum: f(4) = 7; Absolute minimum: none c. Absolute maximum: none; Absolute minimum: none d. Absolute maximum: none; Absolute minimum: f(1) = 2 Q12. The graph of a function f is given. Find the numbers, if any, at which f has a local minimum. What are the local minima? a. f has a local minimum at x = -2; the local minimum is 0 b. f has a local minimum at x = 0; the local minimum is 3 c. f has a local minimum at x = -2 and 2; the local minimum is 0 d. f has no local minimum Q13. Graph the function h(x) = -2x + 3. State whether it is increasing, decreasing, or constant. a. b. c. d. Q14. Find -f(x) when f(x) = -2x2 + 5x + 2. a. 2x2 - 5x + 2 b. -2x2 - 5x - 2 c. -2x2 - 5x + 2 d. 2x2 - 5x - 2 Q15. Determine algebraically whether f(x) = 1/x2 is even, odd, or neither. a. even b. odd c. neither Q16. Match the graph to one of the listed functions. a. f(x) = x2 - 8 b. f(x) = -x2 - 8x c. f(x) = x2 - 8x d. f(x) = -x2 - 8 Q17. In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $2.90 as soon as you get in the taxi, to which a charge of $1.70 per mile is added. Find an equation that can be used to determine the cost, C(x), of an x-mile taxi ride. a. C(x) = 1.70 + 2.90x b. C(x) = 3.10x c. C(x) = 4.60x d. C(x) = 2.90 + 1.70x Q18. Graph the function f(x) = -x2 + 3 by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). a. b. c. d. Q19. Graph the function f(x) = x2 + 8x + 7 using its vertex, axis of symmetry, and intercepts. a. b. c. d. Q20. Determine where the function f(x) = -x2 + 8x - 7 is increasing and where it is decreasing. a. increasing on (4, â) and decreasing on (-â, 4) b. increasing on (-â, 4) and decreasing on (4, â) c. increasing on (9, â) and decreasing on (-â, 9) d. increasing on (-â, 9) and decreasing on (9, â) Q21. Find the domain of the rational function f(x) = (x + 9)/(x2 - 4x). a. {x|x â -2, x â 2} b. {x|x â -2, x â 2, x â -9} c. all real numbers d. {x|x â 0, x â 4} Q22. For the polynomial f(x) = 4(x - 5)(x - 6)3, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 5, multiplicity 1, touches x-axis; 6, multiplicity 3 b. -5, multiplicity 1, touches x-axis; -6, multiplicity 3 c. 5, multiplicity 1, crosses x-axis; 6, multiplicity 3, crosses x-axis d. -5, multiplicity 1, crosses x-axis; -6, multiplicity 3, crosses x-axis Q23. Find all zeros of the function f(x) = 2x4 + 3x3 + 16x2 + 27x - 18 and write the polynomial as a product of linear factors. a. f(x) = (2x - 1)(x + 2)(x + 3)(x - 3) b. f(x) = (2x + 1)(x - 2)(x + 3)(x - 3) c. f(x) = (2x - 1)(x + 2)(x + 3i)(x - 3i) d. f(x) = (2x + 1)(x - 2)(x + 3i)(x - 3i) Q24. For the polynomial f(x) = (1/5)x(x2 - 5), list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 0, multiplicity 1, touches x-axis; â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis b. 0, multiplicity 1, crosses x-axis; â5, multiplicity 1, crosses x-axis; -â5, multiplicity 1, crosses x-axis c. â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis d. 0, multiplicity 1 Q25. List the potential rational zeros of the polynomial function f(x) = x5 - 6x2 + 5x + 15. Do not find the zeros. a. Â± 1, Â± 1/5, Â± 1/3 Â± 1/15 b. Â± 1, Â± 5, Â± 3, Â± 15 c. Â± 1, Â± 5, Â± 3 d. Â± 1, Â± 1/5, Â± 1/3, Â± 1/15, Â± 5, Â± 3,Â± 15 Q26. The function f(x) = x4 - 5x2 - 36 has the zero -2i. Find the remaining zeros of the function. a. 2i, 6, -6 b. 2i, 3i, -3i c. 2i, 3, -3 d. 2i, 6i, -6i Q27. Use transformations of the graph of y = x4 to graph the function f(x) = 5x4. a. b. c. d. Q28. State whether the function f(x) = x(x - 9) is a polynomial function or not. If it is, give its degree. If it is not, tell why not. a. Yes; degree 2 b. No; it is a product c. Yes; degree 0 d. Yes; degree 1 Q29. Solve the inequality algebraically. Express the solution in interval notation. (x - 2)2(x + 9) < 0 a. (-â, -9) or (9, â) b. (-â, -9] c. (-â, -9) d. (-9, â) Q30. Use the Intermediate Value Theorem to determine whether the polynomial function f(x) = -4x4 - 9x2 + 4; has a zero in the interval [-1, 0]. a. f(-1) = -9 and f(0) = -4; no b. f(-1) = 9 and f(0) = 5; no c. f(-1) = 9 and f(0) = -4; yes d. f(-1) = -9 and f(0) = 4; yes Q31. Use the Factor Theorem to determine whether x + 5 is a factor of f(x) = 3x3 + 13x2 - 9x + 5. a. Yes b. No Q32. Find a bound on the real zeros of the polynomial function f(x) = x4 - 8x2 - 9. a. -17 and 17 b. -9 and 9 c. -18 and 18 d. -10 and 10 Q33. Find the x-intercepts of the graph of the function f(x) = (x - 3)(2x + 7)/(x2 + 9x - 8). a. (3, 0), (-7, 0) b. (-3, 0), (7/2, 0) c. (3, 0), (-7/2, 0) d. none Q34. Solve the inequality algebraically. Express the solution in interval notation. (9x - 5)/(x + 2) â¤ 8 a. (-2, 13] b. (-2, 21) c. (-2, 21] d. (-2, 13) Q35. A polynomial f(x) of degree 3 whose coefficients are real numbers has the zeros -4 and 4 - 5i. Find the remaining zeros of f. a. 4, -4 + 5i b. 4, 4 + 5i c. 4 + 5i d. -4 + 5i Q36. Find the intercepts of the function f(x) = x3 + 3x2 - 4x - 12. a. x-intercepts: -2, 2, 3; y-intercept: -12 b. x-intercepts: -3, -2, 2; y-intercept: -12 c. x-intercept: -3; y-intercept: -12 d. x-intercept: -2; y-intercept: -12 Q37. Find the real solutions of the equation 3x3 - x2 + 3x - 1 = 0. a. {-3, 1/3, -1} b. {1/3} c. {1/3, -1} d. {-3, -1/3, -1} Q38. Solve the inequality algebraically. Express the solution in interval notation. x3 â¥ 27 a. (-â, 3] b. (-â, -3] âª [3, â) c. [-3, 3] d. [3, â) Q39. Find the intercepts of the function f(x) = x2(x - 1)(x - 6). a. x-intercepts: 0, -1, -6; y-intercept: 0 b. x-intercepts: 0, 1, 6; y-intercept: 6 c. x-intercepts: 0, 1, 6; y-intercept: 0 d. x-intercepts: 0, -1, -6; y-intercept: 6 Q40. Find the domain of the rational function R(x) = (-3x2)/(x2 + 2x - 15). a. {x|x â 5, 3} b. {x|x â 5, -3} c. {x|x â - 15, 1} d. {x|x â -5, 3}

lemongiraffe898

you must use matlab or octave program. others are not allowed T.T. please use only above the two program to get solution.

Problem 3: The Inverse Transformation Method A general method for simulating a RV having a continuous distribution called the inverse transformation method is based on the following result (see also Section 10.2 of the textbook and Quiz 2) Proposition 2.1. The Inverse Transformation Method). L UN U(0,1). For any continuous et distribution function F, X F-1(U) has distribution function F. (F-1(ar) is defined to equal that value y for which F(y) r.) In order to simulate X n Exp(1), for example, generate an U U(0,1) and let X log(1 -U) (see Quiz 2(c)). In fact, X log(U) is also Exp(1), since 1 U (0,1). Since X Exp(A) if X Exp(1) (why?), it follows that log(U)A has an Exp(1) distribution. This can be easily illustrated in Octave xi -log Grand (100,1)); 100 random samples from Exp (1) x2 -log (rand (100, 1))/2; 100 random samples from Exp(2) x3 -log Grand (100, 1))*2; 100 random samples from Exp(0.5) [mean (x1) mean(x2) mean (x3)] check their mean values In addition, Y X Gamma(n, A) if X Xn are i.i.d. Exp(A). Therefore, if Un are i.i.d. U(0, 1), log (Ui) 1 log Ui Gamma(n, A) (1) i-1 i-1 (a) Write an Octave code to generate 100 random samples from Gamma(10, A) by using the inverse transformation method (1). Let A 0.5, 1, 2. Plot histograms for the Gamma random samples. Octave also has a built-in gamma RV generator gamrnd (google it for reference!). Generate 100 random samples of Gamma(10, A) using gamrnd, make histograms, and compare two results. (b) As a function of U U(0, 1), create a new RV W with CDF F(w) 1 e-ti (w 0). (This is Quiz 2 (b).) Write an Octave code to generate 100 random samples of W F using the inverse transformation method (c) Consider a method for generating Weibull(a,B) RVs that have the following CDF F(z) exp(-ar 0 z oo, a Note that W in (b) is in fact Weibull(1,2). Write an Octave function myweibrnd.m that will generate n random samples for given parameters a and B. It should have the following format function CRJ myweibrnd n, alpha, beta) this function receives sample size n and parameters alpha and beta as input arguments and return R Ca vector of n random samples) (put your code here) Using your myweibrnd function, generate 1000 simulated values when (i) (a, B) (1,2) and (ii) (a,B) (2,1), respectively. Plot histograms and estimate their mean and variance

Show transcribed image text Problem 3: The Inverse Transformation Method A general method for simulating a RV having a continuous distribution called the inverse transformation method is based on the following result (see also Section 10.2 of the textbook and Quiz 2) Proposition 2.1. The Inverse Transformation Method). L UN U(0,1). For any continuous et distribution function F, X F-1(U) has distribution function F. (F-1(ar) is defined to equal that value y for which F(y) r.) In order to simulate X n Exp(1), for example, generate an U U(0,1) and let X log(1 -U) (see Quiz 2(c)). In fact, X log(U) is also Exp(1), since 1 U (0,1). Since X Exp(A) if X Exp(1) (why?), it follows that log(U)A has an Exp(1) distribution. This can be easily illustrated in Octave xi -log Grand (100,1)); 100 random samples from Exp (1) x2 -log (rand (100, 1))/2; 100 random samples from Exp(2) x3 -log Grand (100, 1))*2; 100 random samples from Exp(0.5) [mean (x1) mean(x2) mean (x3)] check their mean values In addition, Y X Gamma(n, A) if X Xn are i.i.d. Exp(A). Therefore, if Un are i.i.d. U(0, 1), log (Ui) 1 log Ui Gamma(n, A) (1) i-1 i-1 (a) Write an Octave code to generate 100 random samples from Gamma(10, A) by using the inverse transformation method (1). Let A 0.5, 1, 2. Plot histograms for the Gamma random samples. Octave also has a built-in gamma RV generator gamrnd (google it for reference!). Generate 100 random samples of Gamma(10, A) using gamrnd, make histograms, and compare two results. (b) As a function of U U(0, 1), create a new RV W with CDF F(w) 1 e-ti (w 0). (This is Quiz 2 (b).) Write an Octave code to generate 100 random samples of W F using the inverse transformation method (c) Consider a method for generating Weibull(a,B) RVs that have the following CDF F(z) exp(-ar 0 z oo, a Note that W in (b) is in fact Weibull(1,2). Write an Octave function myweibrnd.m that will generate n random samples for given parameters a and B. It should have the following format function CRJ myweibrnd n, alpha, beta) this function receives sample size n and parameters alpha and beta as input arguments and return R Ca vector of n random samples) (put your code here) Using your myweibrnd function, generate 1000 simulated values when (i) (a, B) (1,2) and (ii) (a,B) (2,1), respectively. Plot histograms and estimate their mean and variance

burgundywhale451

Help with 38,42,47,55 please

Mnct 60. fx)-3x3x 61, f(x) = x.(x-z? Applications of Differentiation 212 CHAPTER 3 Find all critial hunber 35. 40, g(Î¸)-48-tane 39. F(x)-x""(x-4): 41. f)-2 cos8 + sino 63. Between 0Â°C and 30Â°C, the volume V Gn of I kg of water at a temperature T is 43-44 A formala for the derivartive of a function f'is given. How many critical numbers does f have? V-999.87-0.06426T0.00850437 Find the temperature at which water has its m APTE - 6x + 10 64. An object with weight W is dragged along a h attached to the o by a force acting along a rope rope makes an angle Î¸ with the plane, then the 45-56 Find the absolute maximum and absolute minimurn values the force is Î¼ sin Î¸ + cos Î¸ of f on the given interval. 45. fx)-12+4x -. [0,S] 46. fix)-5+54x-2x. (0,4) å¸f(x)-2x,-3x1-12x + 1, [-23] where Î¼ is a positive constant called the coefficie and where 0 Î¸ Ï/2. Show that F is minimize 65. The water level, measured in feet above mean sea l Lake Lanier in Georgia, USA, during 2012 can be im the function L()-0.01441,, 50. f)- 4). [-2,3] 51.)(02.4) 52. x)-+T -0.4177,2 + 2703t + 1060. where t is measured in months since January 1,201 when the water level was highest during 2012. 66. On May 7, 1992, the space shuttle Endeavour was laun on mission STS 49, the purpose of which was to install perigee kick motor in an Intelsat communications satellit The table gives the velocity data for the shuttle between l and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubi S4. ft)- 0.2 pol ymomial that best models the velocity of the shuttle interval r e [o, 125). Then graph this polynom acceleration of the shuttle and usew ss fc)-2 cos+sin 2t, [o, -/2 56. f(t)-' + cot('/2), [Ï/4.7v/4] 57. If a and b are positive bumbers, find the maximum value 58. Use a graph ximum and minimum values of the to estimate the critical numbers of fx)-II+5xcorect i e decil place 59-62 (a) Use a graph to estimate the absolute (b) Use calculus to find the exact 0 maximurm and 0 185 319 447 values of the function to two decimal places. minimum maxirsum and minimum values 20 32

MTH 99 Chapter Notes - Chapter 2: Ferrari P, Piecewise, Floor And Ceiling Functions

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