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February Standard 2023 Mathematics 1

COVER PAGE

Assignment 2 (5%)

Due Date and Time: 5:30pm, 24 July 2023

Assignment Overview

This assignment contains three questions from the topic of Applications of Differentiation.  You are re- quired to answer each question in the assignment. You will be marked on the correctness of your responses, your working and the quality of your mathematical communication. You are also required to discuss the assignment in an interview (as described in the Mathematics 1 Assignment Guide for Students available on the TCOLE page).

Assignment Instructions

Submission:

• Include this cover page in your submission.

• You must submit your assignment via GradeScope and accurately assign responses to questions.

• You must only submit responses to the assignment version you have been allocated on TCOLE; sub- mitting the wrong version will incur a 1 mark penalty.

• Your entire assignment must be submitted as a single submission on gradescope.  Pictures, graphs, etc. must be included in the same submission as the rest of your assignment.

• Your responses maybe typed or handwritten in either pen or pencil.

• Check that your submission displays properly on gradescope; ensuring that your submission is cor- rect and legible is your responsibility.

Late Submission:  Late submissions will receive a marks penalty of 1 mark per day past the due date. Assignments will not be accepted more than four days past the due date.

Academic integrity:  This assignment must be completed in accordance with the TCFS Academic Integrity Policy.

In this assignment we will be using Excel and Desmosto investigate linear and quadratic approximating functions. For a given function f, an approximating function is a function g which satisfies f(x) ≃ g(x) in an appropriate region.  Usually, the function g is chosen to be easier to calculate than f.  The simplest, and most widely used, approximating functions are linear approximations, which are calculated in exactly the same way as a tangent line:

The linear approximation of a function f when x = a is

L(x) = f(a) + f′ (a)(x − a).

One way to improve on this approximation is to use a quadratic approximation

Q(x) = f(a) + f′ (a)(x − a) +  (x − a)2

Where f′′ is the is the second derivative off. i.e. f′′ (x) = (f′ (x))′

In both the these cases, the approximating function should remain closetof as long the value of x does not stray too far from a.

1. Let fbe defined as

f(x) = xsin(x) + 10  ai.

Where ai is the ith digit of your student number.

(a) Find the linear approximation, L(x), off(x) atx = π . Show full working.

(b) Find the quadratic approximation, Q(x), off(x) atx = π . Show full working.   [3 marks]

2.    (a) Using the functions f(x), L(x) and Q(x) from Question 1, create an Excel spreadsheet and name it Values. Fill in the headings below and enter your name, student ID and question number.

Then using Excel formulas fill in the columns as specified below. Answers should be displayed correct to 7 decimal places.

Column A: 21 evenly-spaced x-values from the interval [2.6, 3.6]

Column B: the values off(x) at those 21 x-values

Column C: the values of L(x) at those 21 x-values

Column D: the values of Q(x) at those 21 x-values

Column E: the error between f(x) and L(x) at those 21 x-values

Column F: the error between f(x) and Q(x) at those 21 x-values

(b) Copy and paste the cells from the Values sheet into a new spreadsheet and name it Formulas. Convert this sheet so that it displays as formulas (ensuring that the columns are wide enough to display the formulas fully).  Submit screenshots of both the Values and Formulas spreadsheets as part of your assignment. Please ensure that each sheet fits onto a single page.

(c) By referring to your sheet of Values created in response to Question 2(a),briefly describe (in one or two sentences) which x-values seem to correspond to the smallest error when L(x) is used to approximate f(x).

(d) By referring to your sheet of Values created in response to Question 2(a), briefly comment (in one or two sentences) on whether L or Q has less associated error when approximating f on the interval [2.6, 3.6]. [3 marks]

3. Edgar the engineer uses many complicated equations in his work.  Before doing lengthy and time consuming calculations he often uses a linear approximation to investigate the problem.  Edgar’s current project requires the error in making this approximation to be less than 1 to provide valid

results. In this instance, Edgar is working with the function

h(x) = 100 + 3xsin x2 ）.

over the interval [−0.5, 7]. Let P(x) be the linear approximation of h(x) atx = 3.

(a) Find P(x). Show full working.

(b) To check the accuracy of Edgar’s approximation, divide the interval [−0.2, 6.6] into halves to obtain 3 x-values (the middle and the two endpoints). Create an Excel spreadsheet called Values and enter your name, Student ID and question number, then repeat the process from question 2(a) to calculate the error for P (and only P) at those 3 positions.

(c) Copy and paste the cells from the Values sheet into a new spreadsheet and name it Formulas. Convert this sheet so that it displays as formulas (as in question 2) ensuring that the columns are wide enough to display the formulas fully. Submit screenshots of both the Values and Formulas spreadsheets as part of your assignment. Please ensure that each sheet fits onto a single page.

(d) Now use the Desmos graphics calculator to display the graphs of y = h(x) andy = P(x). Use a suitable viewing window and include DESMOS labels showing the points (x,h(x)) for the 3 x-values considered in the spreadsheet in Question 3(b).  Submit a screenshot of your Desmos graph (including the sidebar containing the formulas) as part of your assignment.

(e) Briefly comment on whether or not the approximation satisfies Edgar’s error requirement over the interval  [−0.5, 7].   Use the information from both the spreadsheet in Question 3(b) and Desmos graph in Question 3(d) to as part of your answer. [3 marks]