IGNOU BSC MTE 7 Advanced Calculus Solved Assignment English

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MTE 7: Advanced Calculus

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Title Name	IGNOU BSC MTE 7 Advanced Calculus Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 7
Subject Name	Advanced Calculus
Year	2026
Session	-
Language	English
Assignment Code	MTE 7/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 07 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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MTE 7 2026 - English

ASSIGNMENT

Course Code: MTE-07

Assignment Code: MTE-07/TMA/2026

Maximum Marks: 100

1. State whether the following statements are true or false. Give reasons for your answers.

(i) $\lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F : \mathbb{R}^2 \to \mathbb{R}^2$ , defined by $F(x, y) = (y + 2, x + y)$ , is locally invertible at any $(x, y) \in \mathbb{R}^2$ .

(iv) $f : [-1, 1] \times [-2, 2] \to \mathbb{R}$ , defined by
$$f(x, y) = \begin{cases} x, & \text{if } y \text{ is rational} \\ 0, & \text{if } y \text{ is not rational} \end{cases}$
is integrable.

(v) The function $f : \mathbb{R}^2 \to \mathbb{R}$ , defined by $f(x, y) = 1 - y^2 + x^2$ , has an extremum at (0, 0).

2) (a) Find the following limits:

$\quad$ (i) $\lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$

$\quad$ (ii) $\lim_{x \to 0^+} (\sin x)^{\sin x}$

$\quad$ (b) Find the third Taylor polynomial of the function $f(x, y) = 1 + 5xy + 3^2y$ at (1, 2).

$\quad$ (c) Using only the definitions, find f_xy(0, 0) and f_yx(0, 0), if they exists, for the function

$$f(x, y) = \begin{cases} \frac{x^2 y}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

3) (a) Let the function f be defined by
$$f(x, y) = \begin{cases} \frac{3x^2 y^4}{x^4 + y^8}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}$

$\quad$ Show that f has directional derivatives in all directions at (0, 0).

(b) Let $x = e^r \cos \theta$ , $y = e^r \sin \theta$ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$$\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$

(c) Let $a = (1, 2, 3)$ , $b = (-5, 3, -2)$ , $c = (2, -4, 1)$ be three points in $\mathbb{R}^3$ .

Find |2b - a + 3c|.

4. (a) Find the centre of gravity of a thin sheet with density $\delta(x, y) = y$ , bounded by the curves $y = 4x^2$ and $x = 4$ .

(b) Find the mass of the solid bounded by $z = 1$ and $z = x^2 + y^2$ , the density function being $\delta(x, y, z) = |x|$ .

5. (a) State Green's theorem, and apply it to evaluate

$$\int_C (3x^2 - 4y) dx - (2x + y^3) dy,$

Where C is the ellipse $4x^2 + 9y^2 = 36$ .

(b) Find the extreme values of the function

$f(x, y) = x^2 + y$ on the surface $x^2 + 2y^2 = 1$ .

6. (a) State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of $\mathbb{R}^2$ . Verify this theorem for the functions f and g, defined by

$$f(x, y) = \frac{y - x}{y + x}, \quad g(x, y) = \frac{x}{y}.$

(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that $g(1) = 2$ and $F(g(y), y) = 0$ in a neighbourhood of (2, 1), where

$$F(x, y) = x^5 + y^5 - 16xy^3 - 1 = 0$

defines the function F. Also find g'(y).

(c) Check the local invertibility of the function f defined by $f(x, y) = (x^2 - y^2, 2xy)$ at (1, -1). Find a domain for the function f in which f is invertible.

7. (a) Check the continuity and differentiability of the function at (0, 0) where

$$f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

(b) Find the domain and range of the function f, defined by $f(x, y) = \frac{2xy}{x^2 + y^2}$ . Also find two level curves of this function. Give a rough sketch of them.

8. (a) Evaluate $\int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$x(t) = at^2, y(t) = 2at, 0 \le t \le 1$ .

(b) Use double integration of find the volume of the ellipsoid

$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$ .

9. (a) Find the values of a and b, if

$$\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$

(b) Suppose S and C are subsets of $\mathbb{R}^3$ . S is the unit open sphere with centre at the origin and C is the open cube

$= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ .

Which of the following is true. Justify your answer.

(i) $S \subset C$

(ii) $C \subset S$

(c) Identify the level curves of the following functions:

$\quad$ (i) $\sqrt{x^2 + y^2}$

$\quad$ (ii) $\sqrt{4 - x^2 - y^2}$

$\quad$ (iii) x - y

$\quad$ (iv) y / x

10. (a) Does the function

$$f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}, x \neq 0, y \neq 0$

satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer.

$\quad$ (b) Locate and classify the stationary points of the following:

$\quad$ (i) $f(x, y) = 4xy + x^4 - y^4$

$\quad$ (ii) $f(x, y) = xy + \frac{2}{x} + \frac{4}{y}, x > 0, y > 0$

MTE 7 2025 - English

Course Code: MTE-07

Assignment Code: MTE-07/TMA/2025

Maximum Marks: 100

1. State whether the following statements are true or false. Give reasons for your answers.

(i) $lim_{x ightarrow 0}frac{x^2sinfrac{1}{x}}{sin x}=1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F:mathbf{R}^2 ightarrowmathbf{R}^2$ , defined by F(x, y) = ( y + ,2 x + y) at any (x, y) ∈ $mathbb{R}^{2}$

(iv) $f:[-1,1] imes[-2,2] ightarrowmathbf{R}$ ,defined by

$f(x,y)=egin{cases}x,& ext{if,,,,}y,,,, ext{is rational}\0,& ext{if,,,,}y,,,, ext{is not rational}end{cases}$

is integrable.

(v) The function $f:mathbf{R}^2 ightarrowmathbf{R}$ defined by $f(x,y)=1-y^2+x^2$ has an extremum at (0,0).

2) (a) Find the following limits:

(i) $lim_{x ightarrow+infty}left(frac{x^2}{8x^2-3} ight)^{1/3}$

(ii) $lim_{x ightarrow0^+}(sin x)^{sin x}quad ext{}$

(b) Find the third Taylor polynomial of the function f (x,y) = 1 + 5xy + 3²y at (1,2).

$f(x,y)=egin{cases}frac{x^2 y}{sqrt{x^2+y^2}},&(x,y) e(0,0)\0,& ext{}end{cases}$

3) (a) Let the function f be defined by

$f(x,y)=egin{cases}frac{3x^{2}y^{4}}{x^{4}+y^{8}},&(x,y) e(0,0)\0,&(x,y)=(0,0)end{cases}$

Show that f has directional derivatives in all directions at (0,0).

(b) Let x e^r = cosθ, y = e^rsin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$frac{partial^2 f}{partial r^2}+frac{partial^2 f}{partial heta^2}=(x^2+y^2)left(frac{partial^2 f}{partial x^2}+frac{partial^2 f}{partial y^2} ight)$

4. (a) Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the curves y = 4x² and x = 4.

(b) Find the mass of the solid bounded by z =1 and , z = x² + y² the density function being δ (z,y,x) = | x | .

5. (a) State Green’s theorem, and apply it to evaluate

$int_C(3x^2-4y)dx-(2x+y^3)dy$

Where C is the ellipse $4x^2+9y^2=36$

(b) Find the extreme values of the function

$f(x,y)=x^2+y$ on the surface $x^2+2y^2=1$

6. (a) State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . $mathbb{R}^{2}$ Verify this theorem for the functions f and g, defined by

$f(x,y)=frac{y-x}{y+x},quad g(x,y)=frac{x}{y}$

(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that g (1) = 2 and F(g( y), y) = 0 in a neighbourhood of (1,2), where

$F(x,y)=x^5+y^5-16xy^3-1=0$

defines the function F. Also find g′( y).

(c) Check the local inevitability of the function f defined by f(x,y) =(x²-y²,2xy) at (1,1) Find a domain for the function f in which f is invertible.

7. (a) Check the continuity and differentiability of the function at (0,0) where

$f(x,y)=egin{cases}frac{2x^3y}{x^2+y^2},&(x,y) eq(0,0)\0,& ext{otherwise}end{cases}$

(b) Find the domain and range of the function f , defined by $f(x,y)=frac{2xy}{x^2+y^2}.$ Also find two level curves of this function. Give a rough sketch of them.

8. (a) Evaluate $int_C(2x^2+3y^2)dx$ where C is the curve given by

$x(t)=at^2,y(t)=2at,0le tle 1.$

(b) Use double integration of find the volume of the ellipsoid

$frac{x^2}{4}+frac{y^2}{9}+frac{z^2}{16}=1.$

9. (a) Find the values of a and b, if

$lim_{x ightarrowinfty}frac{x(1+acos x)-bsin x}{x^3}=1$

(b) Suppose S and C are subsets of $mathbb{R}^{3}$ . S is the unit open sphere with centre at the origin and C is the open cube = ${P(x,y,z)mid-1<x<1,-1<y<1,-1<z<1}.$

Which of the following is true. Justify your answer.

(i) S ⊂ C

(ii) C ⊂ S

(i) $sqrt{x^2+y^2}$

(ii) $sqrt{4-x^2-y^2}$

(iii) x − y

(iv) y / x

10. (a) Does the function

$f(x,y)=frac{x^2-y^2}{x^2+y^2},quad x e 0,y e 0$ satisfy the requirement of Schwarz’s theorem at (1,1) ? Justify your answer.

(b) Locate and classify the stationary points of the following:

$ext{(i)}quad f(x,y)=4xy+x^4-y^4$

$ext{(ii)}quad f(x,y)=xy+frac{2}{x}+frac{4}{y},quad x>0,y>0$

MTE 7 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: एम टी इ-07

सत्रीय कार्य कोड: एम टी इ-07/ टी एम ए/2025

अधिकतम अंकः 100

1. बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए।

(i) $lim_{x ightarrow0}frac{x^{2}sinfrac{1}{x}}{sin x}=1quad ext{(i)}$

(ii) तीन घरों वाला एक वास्तविक मान फलन, जो सर्वत्र संतत है, अवकलनीय होता है।

(iii)F(x, y) = ( y + ,2 x + y) से परिभाषित फलन $F:mathbf{R}^2 ightarrowmathbf{R}^2$ किसी भी बिन्दु (x, y) ∈R²पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv) $f(x,y)=egin{cases}x,& ext{if,,,,}y,,,, ext{is rational}\0,& ext{if,,,,}y,,,, ext{is not rational}end{cases}$ है फ(एक्स,य)= 10, यदि वाइ परिमेय नहीं है. से परिभाषित फलन $f:[-1,1] imes[-2,2] ightarrowmathbf{R}$ समाकलनीय होता है।

(v) $f(x,y)=1-y^2+x^2$ से परिभाषित फलन $f:mathbf{R}^2 ightarrowmathbf{R}$ का (0,0)पर एक चरम मान होता है।

2. (क) निम्नलिखित सीमा ज्ञात कीजिए:

(i) $lim_{x ightarrow+infty}left(frac{x^2}{8x^2-3} ight)^{1/3}$

(ii) $lim_{x ightarrow0^+}(sin x)^{sin x}quad ext{}$

(ख) बिन्दु (1,2) पर फलन $f(x,y)=1+5xy+3^2 y$ का तृतीय टेलर बहुपद ज्ञात कीजिए।

(ग) केवल परिभाषाओं को लागू करके f_xy (0,0) और f_yx, (0,0) ज्ञात कीजिए, जबकि फलन $f(x,y)=egin{cases}frac{x^2 y}{sqrt{x^2+y^2}},&(x,y) e(0,0)\0,& ext{}end{cases}$ अन्यथा

के लिए इनका अस्तित्य होता हो।

3) (क) मान लीजिए

$f(x,y)=egin{cases}frac{3x^{2}y^{4}}{x^{4}+y^{8}},&(x,y) e(0,0)\0,&(x,y)=(0,0)end{cases}$ दिखाइए कि (0,0) पर सभी दिशाओं में ई दिक् अवकलज होते हैं।

(ख) मान लीजिए

$x=e^tcos heta,quad y=e^tsin heta$ और f xऔर yका एक संततः अवकलनीय फलन है जिसके आंशिक अवकलज भी संततः अवकलनीय हैं। दिखाइए कि $frac{partial^2 f}{partial r^2}+frac{partial^2 f}{partial heta^2}=(x^2+y^2)left(frac{partial^2 f}{partial x^2}+frac{partial^2 f}{partial y^2} ight)$

(ग) माग जीजिए $a=(1,2,3),quad b=(-5,3,-2),quad c=(2,-4,1)inmathbf{R}^3$ के तीन बिन्दु हैं। 126-a+31 ज्ञात कीजिए।

4 (क) वक्रो y = 4x² और 4 से पसिद्ध और $delta$ (x, y) के गात वाजे एक पतली सीट का गुरुत्व केन्द्र ज्ञात कीजिए। (5)

(ख) z=1 और z=x+y² से पषिद्ध ठोस घनाकृति का इमान ज्ञात कीजिए जबकि धनन्त्तव $delta(x,y,z)=|x|$ हौ

5. (क) ग्रीन प्रमेय का कथनदीजिए और इसकी सहायता से

$int_C(3x^2-4y)dx-(2x+y^3)dy$ मान निकालिए जहाँ सी. दीर्घवृत $4x^2+9y^2=36$ है।

(ख) पृष्ठ $x^2+2y^2=1$ पर फलन $f(x,y)=x^2+y$ के चरम माग ज्ञात कीजिए।

6 (क) R²के एक वितृत उपसमुचय D पर दो अवकलनीय फलनों F और g है की फजनक आश्रितता का आरक प्रतिबंध बताने वाले प्रमेय का कब दीजिए। निम्नलिखित फलनों f तथा g परिभाषित इस प्रमेय को सत्यापित कीजिए।

$f(x,y)=frac{y-x}{y+x},quad g(x,y)=frac{x}{y}$

(ख) अस्पष्ट फल प्रमेय की सहायता सेवा दिखाइए कि 1 के प्रति में एक ऐसा वीयफजन होता है, जिससे कि (2,1) के प्रतिवेश में g(1)= 2 और $F(g(y),y)=0$ जह $F(x,y)=x^5+y^5-16xy^3-1=0$ परिभाषित है।g' (y) भी ज्ञात कीजिए।

(ग) $f(x,y)=(x^2-y^2,2xy)$ द्वारा परिभाषित फलन f की। (1-1) के पर स्थानीय की लिए जाँच कीजिए फलन f के लिए एक प्रांत ज्ञात कीजिए जिसमें f व्युक्रमणीयता है।

7. (क) (0.0) पर निम्नलिखित फलन f के सांतत्य और अवकलनीयता की जाँच कीजिए, जहाँ

$f(x,y)=egin{cases}frac{2x^3 y}{x^2+y^2},&(x,y) e(0,0)\0,& ext{}end{cases}$

8.(क) $int_C(2x^2+3y^2)dx$ के मान निकालिए, जहाँ $C:quad x(t)=at^2,quad y(t)=2at,quad 0le tle 1$ से प्राप्त वक्र है।

(ख) दितः समाकलन का प्रयोग करके दीर्घवृज

$frac{x^2}{4}+frac{y^2}{9}+frac{z^2}{16}=1$ का आयतन ज्ञात कीजिए।

(क) यदि $lim_{x ightarrowinfty}frac{x(1+acos x)-bsin x}{x^3}=1$ तो a और b मान ज्ञात कीजिए। (5)

(ख) मान लीजिए कि S और C R³के उपसमुच्चय हैं। ऍस मूल-बिन्दु पर केन्द्र वाला एकक विवृत तथा क विवृत घन = ${P(x,y,z)mid-1<x<1,-1<y<1,-1<z<1}$

निम्नलिखित में से कौनसा कथन सत्य है? अपने उत्तर की पुष्टि कीजिए।

(i) S ⊂ C

(ii)C ⊂ S

(i) $sqrt{x^2+y^2}$

(ii) $sqrt{4-x^2-y^2}$

(iii) x − y

(iv) y / x

10. (क) क्या निम्नलिखित फलन $f(x,y)=frac{x^2-y^2}{x^2+y^2},quad x e 0,y e 0$ श्वार्ज-प्रमेय आवश्यकताओं को पर संतुष्ट करता है? अपने उत्तर की पुष्टि कीजिए।

((ख) निम्नलिखित के स्तब्ध बिन्दु निर्धारित करके उनका वर्गीकरण कीजिए :

$ext{(i)}quad f(x,y)=4xy+x^4-y^4$

$ext{(ii)}quad f(x,y)=xy+frac{2}{x}+frac{4}{y},quad x>0,y>0$

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IGNOU BSC Assignments 2026

The BSC (Bachelor in Science) programme includes assignment work as part of the academic requirements applicable to students. Students should always verify the latest assignment questions, instructions and submission requirements from the official IGNOU website.

Our MTE 7 Solved Assignment is independently prepared reference material intended to assist students with understanding and preparing their answers. It is not an official IGNOU publication.

Why Keep a Solved Assignment Reference?

A useful reference is more than simply a collection of answers. It can help you see how a question can be approached, what important points need attention and how the subject matter can be organised into a clear answer.

Save preparation time with organised reference material
Understand difficult questions more easily
Review important concepts before writing your answers
Improve answer organisation with a structured reference
Study conveniently whenever and wherever you need it

Product Information

Course Name	Bachelor in Science
Course Code	BSC
Subject Code	MTE 7
Subject	Advanced Calculus
Programme	BACHELOR DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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Get the 2026 MTE 7 Advanced Calculus Solved Assignment as a convenient reference for your assignment preparation.

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IGNOU MTE 7 2026 Solved Assignment
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Important Note for IGNOU Students

Always check the latest assignment questions and submission instructions on the official IGNOU website before preparing and submitting your assignment. Use this solved assignment as a study and reference resource and prepare your own submission according to the applicable IGNOU requirements.

Why Choose IGNOU Academy for Your Assignments?

Getting your assignments right is an important part of your IGNOU academic journey.IGNOU Academy provides carefully prepared reference materials designed to makeassignment preparation simpler, clearer and more convenient for students.

Latest Curriculum: Materials are prepared with the current IGNOU syllabus and assignment requirements in mind.
Proper Assignment Format: Understand the structure, presentation and answer style generally expected in an IGNOU assignment.
Simple Explanations: Difficult topics are explained in clear and easy-to-understand language.
Exam Preparation: Assignment material can also be useful for revising important concepts before the Term-End Examination.
Student-Friendly Language: Clear English/Hindi content helps students understand topics without unnecessary complexity.
Trusted by IGNOU Learners: Students across India use IGNOU Academy as a convenient source of assignment and study-reference material.

These materials are provided as reference study guides. Students should understandthe concepts, follow IGNOU guidelines and prepare their own original assignmentswhere required.

How to Get Your Solved Assignment PDF

Visit IGNOU Academy:Go to www.ignouacademy.com.
Find Your Course:Search for your programme, course code and subject.
Select the Session:Choose the latest available reference material for your academic session.
Complete Your Order:Add the required material to your cart, log in or register, and complete the payment.
Access Your Material:After successful payment, access your purchased study material from your account.

Step-by-Step: Finding IGNOU Assignment Question Papers

Open the Website:Visit www.ignouacademy.com.
Open Assignment Question Papers:Select the relevant IGNOU assignment question paper section.
Filter Your Requirement:Select your course, session and medium such as English or Hindi.
Open the Paper:View or download the required question paper for your preparation.

How to Submit Your IGNOU Assignments

Prepare Neatly:Use the format and submission instructions applicable to your programme and study centre.
Complete the Front Page:Enter your Name, Enrollment Number, Programme/Course Code, Subject and Study Centre details correctly.
Offline Submission:If your study centre requires physical submission, submit the assignment as instructed and retain the acknowledgement.
Online Submission:Where online submission is permitted, follow the instructions issued by your Regional Centre or Study Centre. Keep proof of submission for your records.

Tracking Your Assignment Submission Status

After submitting your assignments, students can check whether their assignmentmarks and submission details have been updated through the applicable official IGNOU system.

Open the Official IGNOU Student Area:Visit the official IGNOU website and access the relevant student services.
Find Assignment Status:Open the assignment-status service applicable to your programme.
Enter Your Details:Provide your Enrollment Number and other requested programme details.
Check the Status:Review the available submission, evaluation and marks information.

A Quick Tip for IGNOU Students

Assignments are an important part of the IGNOU evaluation process. Preparing themcarefully, understanding the questions, following the prescribed format and submittingthem within the applicable deadline can help you stay on track academically.

Smart Study + Proper Preparation + Timely Submission = Better Academic Planning

Need Help? Contact IGNOU Academy WhatsApp: +91 9199852182 | Website: www.ignouacademy.com

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