## AMA1100 2023/24 Semester 1

## Assignment 2

**AMA1100 **

**Basic Mathematics – **

**An Introduction To Algebra And Differential Calculus **

**Assignment 2**

代写Mathematics作业 You may re-submit the assignment again, to a maximum of 10 times, before the due date. Only the last attempt will be counted.

**A Guideline **

**A.1 Due Date: ****Monday, November 27, 2023 before 17:00 HKT (GMT+8) **

**A.2 Due Date: ****Friday, December 1, 2023 before 17:00 HKT ****(GMT+8) **

**A.3 Instructions for uploading the Assignment through Black-****board **

1.Please download the “Assignments” from your Blackboard website in the link.

https://learn.polyu.edu.hk/webapps/blackboard/content/listContentEditable.jsp?content_id=_6350357_1&course_id=_109017_1

2.Please complete all questions. 代写Mathematics作业

3.Sign the covering declaration statement and write your answers with detailed steps.

4.Please scan your solutions onto a PDF using the app “CamScanner” on Google

Play or Apple Store:

https://play.google.com/store/apps/details?id=com.intsig.camscanner&hl=en

OR

https://apps.apple.com/us/app/camscanner-scanner-to-scan-pdf/id388627783 You may use other app scanners, like “Tiny Scanner”, “Genius Scanner” etc.

5.Please upload/attach your assignment solutions (a single) PDF file with the first page “covering declaration statement” at the same place you’ve downloaded this assignment by pressing the “Browse My Computer”, choosing your PDF file you want to upload, and then press Submit. You may re-submit the assignment again, to a maximum of 10 times, before the due date. Only the last attempt will be counted.

1AMA1100 2023/24 Semester 1

Assignment 2

**A.4 Covering Declaration 代写Mathematics作业**

By submitting this work through the online system, I affirm on my honour that I am aware of the Regulations on Academic Integrity in Student Handbook1 and

1.have not given nor received any unauthorized aid to/from any person or persons,and

2.have not used any unauthorized materials in completing my answers to this submission

**Subject Lecturer : **Dr. Charles K. F. LEE

**Name : **

**Signature : **

**Student I.D. :**

1https://www.polyu.edu.hk/ar/web/en/for-polyu-students/student-handbook

**B Questions 代写Mathematics作业**

Please complete the following questions and upload the solutions through Blackboard.Remember, combine all your solutions into one single PDF file.

1.Differentiate the function *y *with respect to *x*.

### 2.Find the derivative in terms of *y *for the following functions by applying the differentiation of inverse function. 代写Mathematics作业

(a) *y *= *x*^{3} *− *9*x*^{2} + 27*x **− *20, *x **≥ *3

(b) *y *= tan^{−}^{1} *x *

[10 marks]

3.Let *y *=

[10 marks]

### 4.Show that the point *P*(1*, *1) lies on the curve defined by the equation

**K **: *x ^{2}*

*y*+

*y*=

^{3}*x*+

^{3}*y*

^{2}and find the tangent to the curve **K **at *P*(1*, *1). 代写Mathematics作业

[8 marks]

5.Use the implicit differentiation to find the derivativefor the following func-tions.

(a) sin(*xy*) = *x ^{2}*

*−*

*y*;

(b) *y *= tan* ^{2}* (

*x*+

*y*).

[8 marks]

### 6.Evaluate the following limits by l’Hôpital’s rule. 代写Mathematics作业

[14 marks]

7.Consider the function *y *= *f*(*x*) implicitly defined by the curve

**C **: 1 + 16*x ^{2}*

*y*= tan(

*x*

*−*2

*y*)

*.*

Show that the point *Q*(*, *0) lies on the curve **C**. Find *y **′ *by implicit differentia-tion. Hence, find the equation of the tangent and the equation of the normal to the curve at *Q*(*, *0) in terms of *π*.

[12 marks]

### 8.A function *f*(*x*) and its first and second order derivatives *f **′ *(*x*) and *f **′′*(*x*) are shown in the following table.

If *g*(*x*) = *e **x **f*(*x ^{2}* ), find the values of

*g*(

*x*) and its first and second order derivatives

*g*

*′*(

*x*) and

*g*

*′′*(

*x*) for

*x*= 0 and

*x*= 1. 代写Mathematics作业

[6 marks]

9.Consider the function

(a) Find *f **′ *(*x*) and *f **′′*(*x*) for *x ≠* 0*, **−*1.

(b) Discuss the existence of *f **′ *(*x*) and *f **′′*(*x*) when *x *= 0*, **−*1.

(c) Find the local maxima and minima (if any) of *f*(*x*) by **the first derivative ****test**.

(d) Find the local maxima and minima (if any) of *f*(*x*) by **the second deriva-****tive test**.

[18 marks]

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