1. (a) Use integration by parts to solve (b) For integer n 20 and constant b>0...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
1. (a) Use integration by parts to solve (b) For integer n 20 and constant b>0 define In (b) = So √2²+1 d 2² 2. Solve log(1 +2²) dr. i. Use integration by parts to show that I, (b) with n 2 1 satisfies the recurrence relation In(b) ii. Solve the Io(b) integral and then use the recurrence rela- tion to determine 13(b). B√√2b + 1 2n + 1 ¹+3³-4 2³+ 4z 3. Consider the double integral dr. 11 2n+1 dr. -In-1(b). [[* gedy dr, (a) Draw the domain of the double integral. (b) Solve the double integral. 4. Consider the parametric curve a(t) = (r.y)= (-2 sint, sin(2t)) for time t = [0,2]. (a) Find the values of t where dr/dt = 0. Then find the values of t where dy/dt = 0. What is the meaning of these zero derivatives? (b) Plot a(t) on the ry-plane. In your answer you must also explain how you created the plot. (e) Calculate the velocity and acceleration of this curve. Then calculate the curvature at f = 0, 1 = x/4 and t = x/2. Explain your curvature results by referring to your plot in Question 4(b). (d) Calculate the area bound by the curve for all 1 € [0.2m]. (Note: treat all area bounded by the curve as positive area, so you should not obtain a zero or negative value.) 5. The shape of a hill is described by = f(r.y)=²(1-6)² cos² y for x = [0,6] and y = -x/2.7/2]. Figure 1 plots == f(x,y) z=f(x,y) 100 50 ON */4 Figure 1: The hill of Question 5 modelled by = f(z. y) = x²(x - 6) cos² y for 2 € [0,6] and y € [-/2,=/2). (a) Where on the hill is the steepest slope in the direction? (b) Where on the hill is the steepest slope in the y direction? How steep is this slope? (e) Use partial derivatives to find the location of the maximum and minimum heights of the hill. (The maximum and mini- mum should be clear from Figure 1, but confirm them with partial derivatives.) 6. Consider f(x) = e cos z over the domainz € [-*.*] (a) Construct the Taylor polynomial of order three pa(r) about To=0. (b) Write a formula for the Lagrange Remainder term Rs(r) for Taylor polynomial pa(z) about zo=0. Your solution should be simplified as much as possible, and it should be in terms of an unknown constant c. Make sure you define the domain of c. (e) For the domain € [0, ], what is the upper bound of the absolute value of the Lagrange remainder term [R₂(e) at any r? In other words, what is the worst-case scenario for the Lagrange remainder term? (d) Calculate function f(x), Taylor polynomial ps(r), error f(x)- ps(r) and the upper bound of Ra(z)) at r=0.0.5, 1.0, π. For all four values of r, is the error of the Taylor polynomial within the expected range? Comment on how the error changes as 2-ro increases. 1. (a) Use integration by parts to solve (b) For integer n 20 and constant b>0 define In (b) = So √2²+1 d 2² 2. Solve log(1 +2²) dr. i. Use integration by parts to show that I, (b) with n 2 1 satisfies the recurrence relation In(b) ii. Solve the Io(b) integral and then use the recurrence rela- tion to determine 13(b). B√√2b + 1 2n + 1 ¹+3³-4 2³+ 4z 3. Consider the double integral dr. 11 2n+1 dr. -In-1(b). [[* gedy dr, (a) Draw the domain of the double integral. (b) Solve the double integral. 4. Consider the parametric curve a(t) = (r.y)= (-2 sint, sin(2t)) for time t = [0,2]. (a) Find the values of t where dr/dt = 0. Then find the values of t where dy/dt = 0. What is the meaning of these zero derivatives? (b) Plot a(t) on the ry-plane. In your answer you must also explain how you created the plot. (e) Calculate the velocity and acceleration of this curve. Then calculate the curvature at f = 0, 1 = x/4 and t = x/2. Explain your curvature results by referring to your plot in Question 4(b). (d) Calculate the area bound by the curve for all 1 € [0.2m]. (Note: treat all area bounded by the curve as positive area, so you should not obtain a zero or negative value.) 5. The shape of a hill is described by = f(r.y)=²(1-6)² cos² y for x = [0,6] and y = -x/2.7/2]. Figure 1 plots == f(x,y) z=f(x,y) 100 50 ON */4 Figure 1: The hill of Question 5 modelled by = f(z. y) = x²(x - 6) cos² y for 2 € [0,6] and y € [-/2,=/2). (a) Where on the hill is the steepest slope in the direction? (b) Where on the hill is the steepest slope in the y direction? How steep is this slope? (e) Use partial derivatives to find the location of the maximum and minimum heights of the hill. (The maximum and mini- mum should be clear from Figure 1, but confirm them with partial derivatives.) 6. Consider f(x) = e cos z over the domainz € [-*.*] (a) Construct the Taylor polynomial of order three pa(r) about To=0. (b) Write a formula for the Lagrange Remainder term Rs(r) for Taylor polynomial pa(z) about zo=0. Your solution should be simplified as much as possible, and it should be in terms of an unknown constant c. Make sure you define the domain of c. (e) For the domain € [0, ], what is the upper bound of the absolute value of the Lagrange remainder term [R₂(e) at any r? In other words, what is the worst-case scenario for the Lagrange remainder term? (d) Calculate function f(x), Taylor polynomial ps(r), error f(x)- ps(r) and the upper bound of Ra(z)) at r=0.0.5, 1.0, π. For all four values of r, is the error of the Taylor polynomial within the expected range? Comment on how the error changes as 2-ro increases.
Expert Answer:
Answer rating: 100% (QA)
Sure Ill go through each of the questions one by one and provide the solutions 1 a To solve the integral 0 to 1 log1 x2 dx using integration by parts we can choose u log1 x2 and dv dx Then du 2x 1 x2 ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
Students also viewed these accounting questions
-
If gold is selling for $1,540 per ounce in the United States, using the currency exchange rates in this table, what should gold's price be in Australian dollars according to the concept of purchasing...
-
Use integration by parts to show that, for all x > 0, sin ! dt
-
test-https://drive.google.com/drive/u/0/mobile/folders/1vfAYOR-ic5VzgRQ-1QozGmBnouRo7ndI?usp=sharingtrain-https://drive.google.com/drive/u/0/mobile/folders/1DmJ0YMIfYD-0CLmWdZC4f1flddkaiYMh?usp=sharin...
-
In determining an employee's net pay, which of the following taxes would be deducted? a. FUTA taxes b. SUTA taxes c. FICA taxes d. All of these choices are correct.
-
Airport Connection provides shuttle service between four hotels near a medical center and an international airport. Airport Connection uses two 10-passenger vans to offer 12 round trips per day. A...
-
Companies often budget selling expenses and general and administrative expenses (SGA) as a percentage of expected sales. Required: 1. For both Apple and Google, list the prior three years sales (in...
-
For a linear viscoelastic material, the creep response under a constant stress is followed by a "recovery response" after the stress is removed at some time, \(t_{0}\). Using the Boltzmann...
-
Why has IDEO been so successful? What is the most difficult challenge it faces in conducting its research and designing its products?
-
The costs per equivalent unit of direct materials and conversion in the Rolling Department of Jabari Steel Company are $1.00 and $2.15, respectively. The equivalent units to be assigned costs are as...
-
Match the four functions of payroll activities with their related internal controls: 1. Hiring employees. 2. Timekeeping. 3. Preparing the payroll. 4. Paying the payroll. ________ a. An independent...
-
On January 1, Britta sends Jeff an offer via U.S. mail, which is received by Jeff on January 4. On January 3, Britta sends a revocation of the offer (again via U.S. mail), which was received by Jeff...
-
The right-hand side of formula (3.1.5) reads, on the set \(y \geq 0, y-x \geq 0\), \[\frac{\mathbb{P}\left(T_{y-x} \in d t ight)}{d t} d x d y=\frac{2 y-x}{t} p_{t}(2 y-x) d x d y\] Check simply that...
-
Let \(b\) and \(\theta\) be continuous deterministic functions. Prove that the process \(Y_{t}=\int_{0}^{t} b(u) d u+\int_{0}^{t} \theta(u) d W_{u}\) is a Gaussian process, with mean...
-
Prove that \[\begin{aligned}C_{E}(x, K ; r, \delta ; T-t) & =P_{E}^{*}\left(K e^{-\mu(T-t)}, x e^{\mu(T-t)} ; T-t ight) \\& =e^{-\mu(T-t)} P_{E}^{*}\left(K, x e^{2 \mu(T-t)} ; T-t...
-
Prove Theorem 1.4.1.2, i.e., if \(X\) is continuous, \(X_{t}\) and \(X_{t}^{2}-t\) are martingales, then \(X\) is a BM. Theorem 1.4.1.2: The process \(X\) is an \(\mathbf{F}\)-Brownian motion if and...
-
Prove that the two martingales \(N\) and \(\widetilde{N}\), defined in Exercise 1.5.2.2 are not orthogonal although as r.v's, for fixed \(t, N_{t}\) and \(\widetilde{N}_{t}\) are orthogonal in...
-
Consider the two following investments which you will maintain for ten years. What is the lowest interest rate which Chase would have to offer you in order for you to invest your money with Chase...
-
Repeat Exercise 16.6 using the t-test of the coefficient of correlation. Is this result identical to the one you produced in Exercise 16.6?
-
Deborah purchases a new $30,000 car in 2012 to use exclusively in her business. If Deborah does not elect to expense or take bonus depreciation in 2012 and holds the car until it is fully...
-
The following additional information is available for the Dr. Ivan and Irene Incisor family. Ivan and Irene have the following investment income, in addition to that reported in Chapter 1: Dividends...
-
In the 2012 tax year, Michelle paid the following amounts relating to her 2010 tax return: Tax deficiency..........................................$5,000 Negligence...
-
A company expects to receive which of the following benefits when it uses its budgeting process? a. The planning required to develop the budget helps managers foresee and avoid potential problems...
-
The following budgets are all financial budgets except for the a. combined cash budget. b. budgeted balance sheet. c. budgeted income statement. d. capital expenditures budget.
-
The income statement is part of which element of a companys comprehensive budget? a. The operating budgets b. The capital expenditures budget c. The financial budget d. The cash budget
Study smarter with the SolutionInn App