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MTE 7: Advanced Calculus

Title Name	IGNOU MTE 7 SOLVED ASSIGNMENT HINDI
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	2025
Session	-
Language	English Medium
Assignment Code	MTE 7/Assignment-1/2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE-07 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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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 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 - 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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