代做Math 1152 - Written Homework 4 - Spring 2024代做Python编程

日期：2024-04-05 11:21

Math 1152 - Written Homework 4 - Spring 2024

Due: Upload to Gradescope by 11:59pm of April 2, 2024 (OSU time)

Instructions:

❼ Note that the point values of problems may change slightly when the final rubrics are determined in Gradescope.

❼ You may work in groups of up to three students. All group members must be in the same lecture section. When uploading your submission to Gradescope, upload one file per group, selecting all the group members. See this Gradescope help page for details on how to select group members when uploading your assignment. If a group member is not selected, they will be counted as having no submission.

❼ Show all relevant supporting work. You must show appropriate justification for your solutions. Incorrect answers with substantially correct work may receive partial credit. Correct answers with no supporting work may receive no credit. You are being graded both on your ability to solve math problems and your ability to communicate your mathematical reasoning to others. Your solutions should only use concepts and techniques covered in this course (up to this point in the course) or prerequisite courses.

❼ Give exact answers unless instructed to do otherwise. Calculators are permitted except those that have symbolic algebra or calculus capabilities.

❼ All pages of this assignment must be uploaded in the correct order to Gradescope as a single pdf file. Your solutions should be written on this document in the spaces provided. Note that failure to follow the formatting guidelines outlined in the Gradescope Formating Policies and Guidelines page on Carmen will result in a 5 point penalty.

❼ Note that the point values of problems may change slightly when the final rubrics are determined in Gradescope.

Some Important Notes about Expections for this Assignment:

❼ You can discuss these problems with other Math 1152 students and you can ask your lecturer and recitation instructor for help during office hours, but ultimately the solutions you submit must be created by the members of your group.

❼ While you can talk about the problems with people in other groups, you should not share your written work with people in other groups.

❼ All members of the group should attempt to solve all problems. You are NOT to just split the problems up among the group members. You can all work together in real time to solve the problems or you can each solve them and then compare your solutions. You are each responsible for making sure you understand all the solutions your group submits; everyone will need to understand these ideas for the exams.

❼ MSLC tutors may help you with the mathematics behind this assignment, but should not solve these particular problems for you.

❼ You are not allowed to post these questions online or ask for help from other online sources.

Improper Markings: Incomplete bubbles, x’s, checkmarks, and other markings will not be recognized. Do not use these marks to indicate correct or incorrect answers.

❼ Acceptable Marking for selected answers: A

❼ Acceptable Marking for unselected answers:

❼ Unacceptable Markings: X •

1. [20 pts] Directions: Fill in the circle next to the correct response(s).

Q1: Multiselect [3 pts] Suppose f(x) is an infinitely differentiable function for which f(3)(0) = −24. Of the polynomials listed below, select ALL which could be a Taylor polynomial centered at x = 0 for f(x)?

Q2: Multiple Choice [3 pts] Fill in the circle next to the best response.

Q3: Multiple Choice [3 pts] Fill in the circle next to the best response.

Q4: Multiple Choice [3 pts] Fill in the circle next to the best response.

The third degree Taylor polynomial centered at x = 0 for f(x) = xe2x is:

Q5: True or False [2 pts each] Determine whether each statement below is true or false.

2. Suppose that and suppose it is known that f(9) diverges but that the series represented by converges.

A. [6 pts] Indicate whether the following series must converge, must diverge, or could converge or diverge by filling in the appropriate circle. No justification is necessary.

(i) The series represented by f(1): must converge   must diverge   could converge or diverge.

(ii) The series f(5):   must converge   must diverge   could converge or diverge.

(iii) The series represented by :   must converge   must diverge   could converge or diverge.

B. [2 pts] State the largest possible interval of convergence for the series consistent with the given information. Write your answer in the box provided.

3. [6 pts] Find the third degree Taylor polynomial of f centered at x = 2 for

4. [26 pts] Let

a. Use the Ratio Test to determine the radius of convergence of the power series.

b. Determine the interval of convergence of the given power series. If applicable, determine if the power series converges at the endpoints of the interval. Justify any convergence tests you use.

c. Without doing any additional calculations, determine if f′(0) converges or diverges. You must justify your answer.

d. Determine f(37)(−2). You must justify your answer.

5. [24 pts] Let f(x) = ln(1 + 4x 2 )

a. Determine the Taylor Series for centered at x = 0 and then determine the radius of convergence for this Taylor Series.

In order to receive full credit, you must explicitly show all the relevant steps and work entirely in summation notation. Make sure you write your Taylor Series in standard form. Do not use the Ratio Test to determine the radius of convergence.

b. Now use your answer from part a to determine the Taylor Series of f(x) = ln(1 + 4x 2 ) centered at x = 0 and determine the radius of convergence for this Taylor Series.

In order to receive full credit, you must explicitly show all the relevant steps and work entirely in summation notation. You must determine the Taylor series for f by using your work in part a. Make sure you write your Taylor Series in standard form. Do not use the Ratio Test to determine the radius of convergence.

c. Use the result from part a to calculate or explain why the limit does not exist.

In order to receive credit, you must use the Taylor Series of f ′ (x) to determine the limit.