MTE-07 Solved Assignment January 2026 | ADVANCED CALCULUS | IGNOU BDP

Q: Answer the following:

Answer available — download the full PDF for complete details.

Q: Answer the following:

Answer available — download the full PDF for complete details.

Q1. State whether the following statements are true or false. Give reasons for your answers.

(i)) lim x->0 (x^2 sin(1/x)) / (sin x) = 1 (80 words)
(ii)) A real-valued function of three variables which is continuous everywhere is differentiable. (80 words)
(iii)) The function F : R^2 -> R^2, defined by F(x, y) = (y + 2, x + y), is locally invertible at any (x, y) ∈ R^2. (80 words)
(iv)) f: [-1,1]×[-2,2] → R, defined by f (x, y) = { x, if y is rational; 0, if y is not rational } is integrable. (80 words)
(v)) The function f : R^2 → R, defined by f (x, y) = 1 − y^2 + x^2, has an extremum at (0,0). (80 words)

Key Points:

lim (x/sin x) = 1, but lim (x sin(1/x)) = 0 by Squeeze Theorem.
Continuity does not imply differentiability in multivariable calculus.
Inverse Function Theorem: Local invertibility requires non-zero Jacobian determinant.
Riemann Integrability: Requires set of discontinuities to have measure zero.

Answer: As an expert IGNOU tutor, I will analyze each statement in the context of advanced calculus concepts covered in MTE-07, providing a 'True' or 'False' determination along with a comprehensive justification. The evaluations will draw upon fundamental definitions and theorems such as limits, continuity, differentiability, integrability, and multivariable extremum conditions, adhering to the specified word limits for each sub-question.

Q2. Answer the following:

(a)) Find the following limits: (160 words)
(a)(i)) lim x->+∞ (x^2 / (8x^2 - 3))^(1/3) (80 words)
(a)(ii)) lim x->0+ (sin x)^(sin x) (80 words)
(b)) Find the third Taylor polynomial of the function f (x, y) = 1+5xy +3y^2 at (1,2). (120 words)
(c)) Using only the definitions, find fxy(0,0) and fyx(0,0), if they exists, for the function f(x, y) = { (x^2 y) / (sqrt(x^2 + y^2)), (x, y) ≠ (0,0); 0, otherwise } (120 words)

Key Points:

Limit of rational function at infinity is found by dividing by the highest power of x.
Indeterminate form 0^0 requires transformation to e^(g ln f) and L'Hopital's Rule.
Taylor polynomial of degree n for a polynomial function (degree ≤ n) is the function itself.
Third Taylor polynomial involves partial derivatives up to the third order.

Answer: This response comprehensively addresses the given problems from Advanced Calculus, MTE-07, covering limits, Taylor polynomials for multivariable functions, and calculation of mixed partial derivatives using definitions. The solutions adhere to the specified word limits for each sub-question where possible, showing step-by-step calculations as required by the 'Apply' Bloom's Level.

Q3. Answer the following:

(a)) Let the function f be defined by f(x, y) = { (3x^2 y^4) / (x^4 + y^8), (x, y) ≠ (0,0); 0, (x, y) = (0,0) }. Show that f has directional derivatives in all directions at (0,0). (120 words)
(b)) Let x = e^t cos θ, y = e^t sin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that ∂^2 f / ∂t^2 + ∂^2 f / ∂θ^2 = (x^2 + y^2) (∂^2 f / ∂x^2 + ∂^2 f / ∂y^2). (200 words)
(c)) Let a = (1,2,3), b = (−5, 3, − 2), c = (2,– 4,1) be three points in R^3. Find | 2b-a + 3c |. (80 words)

Key Points:

Directional derivative at (0,0) is found using the limit definition D_u f(0,0) = lim (t→0) [f(th, tk) - f(0,0)] / t.
The chain rule for multivariable functions is crucial for transforming partial derivatives between coordinate systems.
Second-order partial derivatives transformations require careful application of the product and chain rules multiple times.
Continuously differentiable functions guarantee equality of mixed partial derivatives (e.g., ∂²f/∂x∂y = ∂²f/∂y∂x).

Answer: This response addresses a multi-part question from the MTE-07 Advanced Calculus course, covering directional derivatives, multivariable chain rule applications for second-order partial derivatives, and basic vector algebra. Each sub-question is answered comprehensively with step-by-step calculations and adherence to specified word limits, reflecting concepts typically found in advanced calculus curricula. Sub-question (a) demonstrates the existence of directional derivatives at a singular point...

Q4. Answer the following:

(a)) Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the curves y = 4x^2 and x=4. (200 words)
(b)) Find the mass of the solid bounded by z=1 and z=x^2+y^2, the density function being δ(x,y,z)=|x|. (200 words)

Key Points:

Centre of gravity (x̄, ȳ) of a 2D sheet involves calculating mass (M) and moments (M_x, M_y) using double integrals.
Formulas for centre of gravity are: x̄ = M_y / M and ȳ = M_x / M.
Mass of a 3D solid is found by integrating the density function over the volume using a triple integral: M = ∫∫∫δ dV.
Cylindrical coordinates (r, θ, z) are ideal for solids with circular symmetry, with dV = r dz dr dθ.

Answer:

Q5. Answer the following:

(a)) State Green's theorem, and apply it to evaluate ∫_C [(3x^2-4y) dx - (2x+y^3) dy, Where C is the ellipse 4x^2+9y^2=36. (160 words)
(b)) Find the extreme values of the function f(x, y) = x^2 + y on the surface x^2 + 2y^2 = 1. (240 words)

Key Points:

Green's Theorem relates a line integral over a closed curve to a double integral over its enclosed region.
The integrand for Green's Theorem's double integral is (∂Q/∂x - ∂P/∂y).
Area of ellipse x²/a² + y²/b² = 1 is πab.
Lagrange Multipliers find extreme values of f(x,y) subject to g(x,y)=0.

Answer: This response addresses two fundamental concepts in Advanced Calculus: Green's Theorem for evaluating line integrals and the method of Lagrange Multipliers for finding extreme values of functions subject to constraints. Green's Theorem transforms a line integral over a closed curve into a double integral over the region it encloses, simplifying evaluation. Lagrange Multipliers provide a systematic approach to optimize a function when its variables are linked by one or more constraint equations, ...

Q6. Answer the following:

(a)) State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of R^2. Verify this theorem for the functions f and g, defined by f(x, y) = (x/y) + x and g(x, y) = x/y. (160 words)
(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). (120 words)
(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. (120 words)

Key Points:

Functional dependence requires the Jacobian determinant J(f,g) to be identically zero.
Jacobian J(f,g) = (∂f/∂x)(∂g/∂y) - (∂f/∂y)(∂g/∂x).
IFT conditions: F is C¹, F(x₀,y₀)=0, and ∂F/∂x(x₀,y₀)≠0 for implicit function existence.
Implicit function derivative g'(y) = - (∂F/∂y) / (∂F/∂x).

Answer:

Q7. Answer the following:

(a)) Check the continuity and differentiability of the function at (0,0) where f(x, y) = { (2x^3 y) / (x^2 + y^2), (x, y) ≠ (0,0); 0, otherwise } (240 words)
(b)) Find the domain and range of the function f, defined by f (x, y) = (2xy) / (x^2 + y^2). Also find two level curves of this function. Give a rough sketch of them. (160 words)

Key Points:

Continuity of `f(x,y)` at `(a,b)` requires `lim_((x,y)→(a,b)) f(x,y) = f(a,b)`.
Polar coordinates `(x=rcosθ, y=rsinθ)` are effective for evaluating limits at `(0,0)`.
Differentiability requires partial derivatives `f_x`, `f_y` to exist and a specific limit to be zero.
Domain of `f(x,y) = (2xy)/(x^2+y^2)` is `R²` excluding the origin `(0,0)`.

Answer: This response addresses the continuity and differentiability of a multivariable function at the origin, and then determines the domain, range, and sketches level curves for another function, as per the MTE-07 Advanced Calculus course requirements. The solutions employ limit definitions, polar coordinates, and algebraic manipulation to thoroughly analyze the functions at specific points and across their domains. Each sub-question adheres to the specified word limits and formatting guidelines, pro...

Q8. Answer the following:

(a)) Evaluate ∫_C (2x^2+3y^2) dx, where C is the curve given by x(t) = at^2, y(t) = 2at, 0 ≤ t ≤ 1. (200 words)
(b)) Use double integration to find the volume of the ellipsoid x^2/4 + y^2/9 + z^2/16 = 1. (200 words)

Key Points:

Line integrals transform into definite integrals using curve parametrization.
For ∫_C P dx, substitute x(t), y(t), and dx = x'(t)dt.
Volume of a solid can be found using double integration: V = ∫∫_R 2z dA.
Ellipsoid volume integral requires expressing z in terms of x and y.

Answer: This response addresses two fundamental concepts in Advanced Calculus: evaluating a line integral along a parametric curve and calculating the volume of an ellipsoid using double integration. These techniques are crucial for understanding vector calculus and its applications in various scientific and engineering fields, as covered in courses like MTE-07. The first part involves transforming a line integral into a definite integral using parametrization, while the second part demonstrates how to ...

Q9. Answer the following:

(a)) Find the values of a and b, if lim x->∞ (x(1+a cos x) -b sin x) / x^3 = 1. (200 words)
(b)) Suppose S and C are subsets of R^3. S is the unit open sphere with centre at the origin and C is the open cube= {P(x, y, z) | − 1 < x < 1,−1 < y < 1,−1 < z <1}. Which of the following is true. Justify your answer. (120 words)
(c)) Identify the level curves of the following functions: (80 words)
(c)(i)) sqrt(x^2 + y^2) (20 words)
(c)(ii)) sqrt(4 - x^2 - y^2) (20 words)
(c)(iii)) x - y (20 words)
(c)(iv)) y/x (20 words)

Key Points:

Taylor series expansion simplifies limits of indeterminate forms at x=0.
Limit `x->∞` of bounded oscillation divided by `x^n` tends to 0.
Open sphere `x²+y²+z²<1` is a subset of open cube `-1<x,y,z<1`.
A counterexample disproves a subset relationship.

Answer: This response addresses all sub-questions from MTE-07 Advanced Calculus, focusing on limits, set theory in R³, and level curves. For sub-question (a), a crucial assumption about the limit variable's behavior is made to ensure a solvable problem, as the stated limit to infinity would lead to a trivial result. The solution utilizes Taylor series expansion for limit calculation. Sub-question (b) distinguishes between an open unit sphere and an open unit cube, demonstrating subset relationships with...

Q10. Answer the following:

(a)) Does the function f(x, y) = { (x^2-y^2)/(x^2+y^2), x ≠ 0, y ≠ 0; satisfy the requirement of Schwarz's theorem at (1,1)? Justify your answer. (160 words)
(b)) Locate and classify the stationary points of the following: (240 words)
(b)(i)) f(x, y) = 4xy + x^4 – y^4 (120 words)
(b)(ii)) f (x, y) = xy + (2/x) + (4/y), x > 0, y > 0 (120 words)

Key Points:

Schwarz's theorem: `f_xy = f_yx` if mixed partial derivatives are continuous.
Stationary points: Where first partial derivatives `f_x` and `f_y` are simultaneously zero.
Second Derivative Test uses Hessian determinant `D = f_xx * f_yy - (f_xy)^2`.
Local minimum: `D > 0` and `f_xx > 0`.

Answer: Schwarz's theorem provides conditions under which the order of differentiation for mixed partial derivatives does not matter, meaning `f_xy = f_yx`. This typically requires the continuity of these mixed partial derivatives in an open region around the point of interest. In multivariable calculus, identifying stationary points is crucial for optimization, determining local maxima, local minima, or saddle points of a function. Stationary points are found by setting the first partial derivatives o...

MTE-07 Solved Assignment — January 2026 / July 2026

ADVANCED CALCULUS | IGNOU BA CHELORS DEGREE PROGRAMME (BDP)

MTE-07—January 2026 / July 2026

ADVANCED CALCULUS

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MTE-07 Assignment Questions — January 2026

Q1. State whether the following statements are true or false. Give reasons for your answers.

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