Semester 1 Assignment 3, 2021

School of Mathematics and Statistics

MAST10006 Calculus 2

Submission deadline: 6pm, Monday 12 April 2021

This assignment consists of 6 pages (including this page)

Instructions to Students

• If you have a printer, print the assignment one-sided.

Writing

• This assignment is worth 2.22% of your final MAST10006 mark.

• You should answer all questions in the answer boxes below. However only one question will be chosen for marking.

• Marks may be awarded for:

◦ Correct use of appropriate mathematical techniques

◦ Accuracy and validity of any calculations or algebraic manipulations

◦ Clear justification or explanation of techniques and rules used

• You must explicitly state if you use the Sandwich Theorem, l’Hˆopital’s Rule, limit laws, continuity or standard limits in your answers when evaluating limits.

• You must specify the standard limits that you use.

• If you claim that a function is continuous you must reference the type of function as listed in Continuity Theorem 3.

• In your assignments, you should focus on communicating how you obtained your answers.

• You must use methods taught in MAST10006 Calculus 2 to solve the assignment questions.

• Write your answers in the boxes provided on the assignment that you have printed. If you need more space, you can use blank paper. Note this in the answer box, so the marker knows. The extra pages can be added to the end of the exam to scan.

• If you have been unable to print the assignment write your answers on A4 paper. The first page should contain only your student number, the subject code and the subject name. Write on one side of each sheet only. Start each question on a new page and include the question number at the top of each page.

Scanning

• Put the pages in number order and the correct way up. Add any extra pages to the end. Use a scanning app to scan all pages to PDF. Scan directly from above. Crop pages to A4. Check PDF is readable.

Submitting

• Go to the Gradescope window. Choose the Canvas assignment for this assignment. Submit your file. Get Gradescope confirmation on email.

Question 1

Consider the series

(a) Calculate the first 11 partial sums, SN for N ∈ {0, 1 . . . , 10}. You can write the values correct to 3 decimal places.

Sketch the partials sums on a graph with N on the horizontal axis and SN on the vertical axis, labelling each point. You can hand draw or use an app of your choice such as Desmos or Mathematica, though the graph must be of a good enough quality to distinguish between the points. Make an educated guess about its convergence (or divergence) behaviour. You don’t need to justify your guess.

(b) Can you apply any of the tests, as discussed in class, to this series to determine its convergence properties? For each test, explain briefly why they can or cannot be used to determine the convergence of the series.

(c) Look up the “Alternating series test” on Wikipedia. State a version of the test from Wikipedia, giving the conditions needed to be checked and the conclusion that can be drawn. Then use it to test convergence of the series.

Question 2

Consider the function

f(x) = sinh(2x) − cosh(x).

In this question give numerical answers as exact values, in terms of inverse hyperbolic functions if necessary. In your graphs, label all curves, axis intercepts, stationary points and asymptotes.

(a) Find the axis intercepts of the graph of y = f(x).

(b) For which value(s) of x ∈ R is the function f continuous? Justify your answer with reference to continuity theorems from lectures.

(c) Find the stationary points of y = f(x), or show that f does not have any.

(d) Determine if f is odd, even or neither.

(e) Hence sketch the graph of y = f(x).

Question 3

Use the complex exponential to evaluate the indefinite integral

End of Assignment