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Q1:Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean Question 1 Question 3 Which of the following polynomials are written with their Indicate the degree of the polynomial given below. terms in the proper order? 1. 4x-1 2. x4 - 4x2 3. 2x+5x2 + 9×4 16x+4-x2 Question 4 Indicate the degree of the polynomial given below. Question 2 Which of the following polynomials are written with their terms in the proper order? 7-5x2 -12x -- 1. 3x2 -> + 5x4 + 10 2. 4x3 - x2 +9x-7 3. 5x3 -x+12 Question 5 indicate the degree of the polynomial given below. 9+5x4 http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 3 Math-Help-Services.com 8 Question 6 Which of the following polynomials are written with their terms in the proper order? Question 8 Which of the following polynomials are written with their terms in the proper order? 1. x - x4 -x3 2. 7x-4x2 -2x4 3. 5x+ +9x2 - 8 1. x2 - 4 2. 5x2 - x-15x4 3. 81-x4 Question 7 Question 9 Which of the following polynomials are written with their terms in the proper order? Indicate the degree of the polynomial given below. 6x-5x3 http://math-help-services.org 20/04/2018 09:42:46 Page 2 of 3 1. 6x5 -3x4 -x3 2. 2x4 +7x- 422 - x3 3. 3x2 + 7x4 -12x6 9 Math-Help-Services.com Question 10 Which of the following polynomials are written with their terms in the proper order? 1. x3 -7x2 + 12x 2. x2 - 19+ 11x3 3. x+1 http://math-help-services.org 20/04/2018 09:42:46 Page 3 of 3 10 Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 1 and 2 only Question 2 2 and 3 only Question 3 2 Question 4 2 Question 5 4 Question 6 3 only Question 7 1 only Question 8 1 only Question 9 3 Question 10 1 and 3 only - http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 1See Answer
Q2:Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean Question 1 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 4 Identify the variable and the constant in the polynomial given below. 2-x2 +3x2 - 4x+2x +6+ 4x2 = ? 9x3 +5x2 - 5 Question 2 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 5 Combine the like terms of the polynomial given below and. identify the leading coefficient of the resulting polynomial. 2x2 -2 -3+2x+6+4x2 -10-2 = ? Question 3 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 6 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. 3x2 +5x+6-2x2 +2x-1 =? 2×3 -3x-2x3 - 1+3x+52 - 1 =? 9x3 -x-2x3 - 1+3x+5x2 - 5 =? http:/math-help-services.org 20/04/2018 09:39:56 Page 1 of 2 Math-Help-Services.com Question 7 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 9 Identify the variable and the constant in the polynomial given below. 4-3x2 +x+5-4x+ 7x2 = ? ₹ -4x+5 Question 8 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 10 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. ≥ +3x+2x3 +4+23 - 2x2 -3x+6 =? 2x3 +5% +2x-62 -1 =? http://math-help-services.org 20/04/2018 09:39:57 Page 2 of 2 Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 6x2 - 2x+8 leading coefficient is 6 Question 2 7x3 + 5x2 + 2x-6 leading coefficient is 7 Question 3 5x2 -2 leading coefficient is 5 Question 4 variable is x constant is - 5 Question 5 -2x3 +6x2 +2x-7 leading coefficient is - 2 Question 6 x +7x+5 leading coefficient is 1 Question 7 4x2 - 3x+ 9 leading coefficient is 4 Question 8 4x3 - 2 + 10 leading coefficient is 4 Question 9 variable is x constant is 5 Question 10 2x3 - x2 +2x-1 leading coefficient is 2 http:/math-help-services.org 20/04/2018 09:39:57 Page 1 of 1 3See Answer
Q3:Problem 3 Easy one here. Lets find the integral of f(x) = (sin (2x) + cos(2x))2 from 0 to 2n. a) Use Trapezoidal rule on excel to do this numerically with step size of 0.1. b) Optional .. try this by hand .. symbolically ... (you did take calc !! ) c) Do this both numerically and symbolically with MATLAB using built in functions. Like "trapz" (this is 4 lines of code !! ) d) Find the derivative at x= 3 i. Numerically with excel 1. Plot derivative as a function of deta x used ii. Numerically with MATLAB using the built in function iii. Symbolically with MATLAB iv. by hand (calculus)See Answer
Q4:5. Consider the function f : Z+ -> Z+ defined by f(n) = 2n 3n if n is even. Let S = {1, 2, 3, 4,5,6}. Determine the following. (a) f(S) = (b) f-1(S)= (c) f-1(f(S) = (d) f(f-1(S)= if n is odd,See Answer
Q5:Question 2 Consider the gambler's ruin with draws: Alice starts with fa and Bob with £(m - a), and at each time step Alice wins £1 off Bob with probability p, loses £1 to Bob with probability q, and no money is exchanged with probability s, where p +q + 8 =1. We consider the case where Bob and Alice are equally matched, so p= q and s = 1- 2p. (We assume 0<p <1/2.) Let ry be Alice's ruin probability from the point she has Li. (a) By conditioning on the first step, explain why pri+1-(1-s)r.+pr ;- 1 = 0, and give appropriate boundary conditions. [2] (b) Solve this linear difference equation to find an expression for ry- [2] Let d, be the expected duration of the game from the point Alice has &i. (c) Explain why pd141 - (1-8)d, + pd1-1 =- 1, and give appropriate boundary conditions. [2] (d) Solve this linear difference equation to find an expression for d .. [2] (e) Compare your answer to parts (b) and (d) with those for the standard gambler's ruin problem with p = 1/2, and give reasons for the similarities or differences. [2]See Answer
Q6:Question 2 (25 Marks) Use the Lagrange multiplier method to find the dimensions of the box (Fig 1) with the lowest surface area. The box has top face missing and its volume must be at least 10 + N cm3. Fig. 1. A rectangular box with a top face missingSee Answer
Q7:13. Suppose that the joint probability density of two random variables X1 and X2 is given by: f(X1,X2)= (6e-2X1-3x2 for x1 > 0 and x2 > 0, 0 otherwise Find the following probabilities. Show all steps. a) P(1≤X1≤2,2≤X2≤3) b) P(X1<2, X2>2). c) Find the marginal density fi(x1).See Answer
Q8:12. Let X and Y have joint pdf: f(x,y) =12x(1-x)y, 0<x<1, and 0<y< 1 a) Find the marginal densities of X and Y. b) Determine the conditional pdf f1 (x|y). c) Find E(X) and E(Y). d) Find P(X ≤ 0.5|Y = 0.5). e) Determine if X and Y are independent.See Answer
Q9:6. Suppose that the annual amount of rainfall (in million tons) accumulated in a lake follows a gamma distribution with a = 3 and ß = 5. a) Find the expected annual rainfall accumulated in this lake. b) Find the standard deviation of the annual amount of rainfall in this lake. c) Find P(X < 4), use R. d) Find P(2<X <5, use R.See Answer
Q10:Problem 4. The following data are the temperatures (in degree Fahrenheit) of effluent at discharge from a sewage treatment facility on consecutive days: 43 47 51 48 52 50 46 49 45 52 46 51 44 49 46 51 59 45 44 50 48 50 49 50 Using the Method of Moments, fit normal, lognormal distribution, 2-parameter exponential, and uniform distributions to the data and provide the following for each distribution: a. Parameters of the distribution. b. Using the fitted distribution, find the probability that temperature exceeds 58 °F? 1See Answer
Q11:Problem 1. Exercise 4.10.8 in Applied Statistics & Probability for Engineers, 7th Ed. An article in Journal of Hydrology titled "Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data" (2006, Vol. 323(1), pp. 325-334) considered a lognormal distribution model to estimate water retention curves for a range of soil textures. The particle-size distribution (in centimeters) was modeled as a lognormal random variable X with 0 = -3.8 and w = 0.7. Determine the following: a. P(X < 0.02) b. Value for x such that P(X ≤ x) = 0.95 c. Mean and variance of XSee Answer
Q12:Problem 3. A nuclear facility in a coastal region is built to withstand ocean wave forces. Suppose the annual maximum wave height of the ocean waves (above the sea level) is a Gamma distributed random variable with mean height of 4.0 meters and coefficient of variation of 0.8. a. What is the probability that the wave height will exceed 6 meters in any year? b. To construct the nuclear facility in a safe manner, an engineer is assigned to compute a design wave height (above sea level) that will not be exceeded by ocean waves over a 10- year period with a probability of 95%, i.e., 10-yr reliability = 0.95. Assuming that wave height exceedances between years are statistically independent, what should be the height of the design wave above the sea level?See Answer
Q13:The distance between major cracks in a highway follows a [1-parameter] exponential distribution with a mean of 5 miles. a. What is the probability that there are no major cracks in a 10-mile stretch of the highway? b. What is the standard deviation of the distance between major cracks? c. What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection? d. Given that there are no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?See Answer
Q14:7. A printer company claims that the mean warm-up time of a certain brand of printer is at most 10 seconds. An engineer of another company is conducting a statistical test to show this is an underestimate, and the mean warm-up time is more than 10 seconds. a) State the testing hypothesis. b) The test yielded a p-value of 0.035. What would be the decision of the test if α = 0.05? c) Suppose a further study establishes that the true mean warm-up time is 9 seconds. Did the engineer make the correct decision in part (b)? If not, what type of error did he/she make? 8. The incubation period of respiratory infection is known to have a normal distribution with a mean of 8 days and a standard deviation of 3 days. Suppose a group of researchers claimed that the true mean incubation period is shorter than 8 days. A test is conducted using a random sample of 20 patients. a) Formulate hypotheses for this test. b) Consider a rejection region (X ≤6}. Suppose the test failed to reject Ho. If the true mean incubation period is 5 days, what is the Type II error probability of the test using σ =3?See Answer
Q15:5. (20 pts) A conjecture is a statement which is probably true but has not been proved or disproved. In this problem, we will "discover" a conjecture. a) (5 pts) Start with any positive integer. If it is even, divide by 2. If it is odd, multiply by 3 and add 1. So, we have a new integer, use the same rule to generate the next integer. Repeat and record the sequence of integers until it is natural to stop. b) (5 pts) Repeat part (a) with another initial positive integer. c) (10 pts) Write down a conjecture based on your observation in parts (a) and (b). Feel free to do more experimentation.See Answer
Q16:Question 4. Let R > 0 be a real number and let CR denote the semicircular arc of radius R in the upper half of the complex plane centred at the origin from R to -R. -R C R Use lemma 3.2.9 to prove that SCH 1 z² +4 dz 0 as R→ ∞. [5 marks] Question 5. (a) Show that there does not exist a holomorphic function F: C\{0} → C such that F' (z) = 1/z for all non-zero z = C. (b) Deduce that there does not exist a holomorphic function L: C\{0} → C such that exp(L(z)) = z for all non-zero z Є C. [5 marks]See Answer
Q17:1. Let A be the area between the x-axis, the curve y=x+, and x = 2 and x 5. Find a such that the line xa divides this area into two regions of equal area. 2. Find the volume of the solid of revolution when the area between y x and y =x³ between x=0 and x = 1 is rotated around the x-axis.See Answer
Q18: Finding Tangent Vectors and Lengths In Exercises 1-8, find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. 7. r(t)=(t cost)i +(t sin t)j + (2√2/3) +³/2k, 0 ≤1≤ π Instructions Include at least one relevant graph and while you may utilize Maple to work the problem, clearly identify and describe the method and steps to evaluate the problem as if one were using pencil and paper. include the problem statement, clearly labeled mathematical work, clearly labeled and relevant plots, and explanations for all (mathematical work, plots, and conclusions). include a paragraph sharing what you have learned and any difficulties you found at bottom of fileSee Answer

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