Mathematics-1 (Calculus & Linear Algebra)

The curvature of a straight line is:

The surface area of a solid of revolution generated by rotating $ y=f(x) $ about the x-axis is given by:

Let f be twice differentiable and suppose that $ f^{\prime}(c)=0 $. Which condition indicates that f has a local minimum at $ x=c $?

Evaluate the limit: $ \lim_{x\rightarrow0}\frac{1-\cos x}{x^{2}} $

Parseval's theorem for a Fourier series relates the Fourier coefficients to the energy of the function. Which statement best describes Parseval's theorem?

Which of the following is the correct Taylor series expansion for $ \ln(1+x) $ valid for $ |x|<1 $?

Let $ f(x,y)=\frac{x+y}{x-y} $ if $ x \ne y $, and 0 if $ x=y $. For which points on the line $ x=y $ is $ f(x,y) $ continuous?

Let $ f(x,y)=e^{xy} $. Which of the following best describes the total derivative of f at a point (a, b)?

Let A be a $ 5 \times 7 $ matrix with $ rank(A)=4 $. According to the Rank-Nullity Theorem, what is the dimension (nullity) of the null space?

Consider the system $ \begin{cases}x+y=5\\ 2x+2y=10\end{cases} $. Which of the following best describes the solution set of the system?

Show that the Gamma function satisfies $ \Gamma(n+1)=n! $ For positive integers n.

Find the volume of the solid obtained by rotating the curve $ y=x^{2} $ about the x-axis from $ x=0 $ to $ x=a $.

Evaluate the improper integral $ \int_{0}^{\infty}e^{-ax}\sin bx~dx $ where $ a>0 $ and $ b>0 $.

Let $ f(x)=x(x-3)^{2}, x\in\mathbb{R} $.
i. Show that f is continuous on [0,3] and differentiable on (0,3).
ii. Verify that $ f(0)=f(3) $.
iii. Use Rolle's Theorem to show that there exists at least one c in (0,3) such that $ f^{\prime}(c)=0 $. Find the value of c.

Evaluate the limit $ \lim_{x\rightarrow0}\frac{\tan x-x}{x^{3}} $. Provide a detailed step-by-step application of L'Hôpital's rule.

Consider the function $ f(x)=x^{3}-6x^{2}+9x+5 $.
1. Find the critical points of f by solving $ f^{\prime}(x)=0 $.
2. Determine the nature (local maximum or minimum) of each critical point using the second derivative test.
3. Find the absolute maximum and minimum of f on the closed interval [0, 5].

Consider the series $ \Sigma_{n=1}^{\infty}\frac{(-1)^{n}\sqrt{n}}{n+1} $.
(i) Use the Alternating Series Test to show that the series converges.
(ii) Determine whether the series converges absolutely or only conditionally.

Write the Maclaurin series for $ \sin x $ up to (and including) the $ x^{5} $ term. Derive the Maclaurin series for $ \ln(1+x) $ and state its interval of convergence.

Consider the function $ f(x)=x $, $ 0

Let $ A=\begin{bmatrix}1&2&3\\ 0&1&4\\ 5&6&0\end{bmatrix} $.
i. Compute the determinant of A to verify whether A is invertible.
ii. Find the inverse of A.
iii. Determine the rank of A and use the Rank-Nullity theorem to find the nullity of the associated linear transformation $ T:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3} $ defined by $ T(x)=Ax $.

Let $ M=\begin{bmatrix}2&1\\ 1&2\end{bmatrix} $.
i. Show that M is symmetric.
ii. Find the eigenvalues and corresponding eigenvectors of M.
iii. Diagonalize M: that is, express it in the form $ M=PDP^{-1} $ where D is diagonal.

Find and classify the critical points of the function $ f(x,y)=x^{3}-3xy $.

Find the maximum and minimum values of $ f(x,y)=x+2y $ subject to the constraint $ g(x,y)=x^{2}-9=0 $.

Consider the series $ \sum_{n=2}^{\infty}\frac{1}{n(\ln n)^{p}}, p>0 $.
(i) Show that the function $ f(x)=\frac{1}{x(\ln x)^{p}}, x\ge2 $ is positive, continuous, and decreasing.
(ii) Use the Integral Test to determine for which values of p the series converges.

The curvature of a straight line is:

The surface area of a solid of revolution generated by rotating $y=f(x)$ about the x-axis is given by:

Let f be twice differentiable and suppose that $f^{\prime}(c)=0$. Which condition indicates that f has a local minimum at $x=c$?

Evaluate the limit: $\lim_{x\rightarrow0}\frac{1-\cos x}{x^{2}}$

Parseval's theorem for a Fourier series relates the Fourier coefficients to the energy of the function. Which statement best describes Parseval's theorem?

Which of the following is the correct Taylor series expansion for $\ln(1+x)$ valid for $ |x|<1 $?

Let $f(x,y)=\frac{x+y}{x-y}$ if $x \ne y$, and 0 if $x=y$. For which points on the line $x=y$ is $f(x,y)$ continuous?

Let $f(x,y)=e^{xy}$. Which of the following best describes the total derivative of f at a point (a, b)?

Let A be a $5 \times 7$ matrix with $rank(A)=4$. According to the Rank-Nullity Theorem, what is the dimension (nullity) of the null space?

Consider the system $\begin{cases}x+y=5\\ 2x+2y=10\end{cases}$. Which of the following best describes the solution set of the system?

Show that the Gamma function satisfies $\Gamma(n+1)=n!$ For positive integers n.

Find the volume of the solid obtained by rotating the curve $y=x^{2}$ about the x-axis from $x=0$ to $x=a$.

Evaluate the improper integral $\int_{0}^{\infty}e^{-ax}\sin bx~dx$ where a&gt;0 and b&gt;0.

Let $f(x)=x(x-3)^{2}, x\in\mathbb{R}$.
i. Show that f is continuous on [0,3] and differentiable on (0,3).
ii. Verify that $f(0)=f(3)$.
iii. Use Rolle's Theorem to show that there exists at least one c in (0,3) such that $ f^{\prime}(c)=0 $. Find the value of c.

Evaluate the limit $\lim_{x\rightarrow0}\frac{\tan x-x}{x^{3}}$. Provide a detailed step-by-step application of L'Hôpital's rule.

Consider the function $f(x)=x^{3}-6x^{2}+9x+5$.
1. Find the critical points of f by solving $f^{\prime}(x)=0$.
2. Determine the nature (local maximum or minimum) of each critical point using the second derivative test.
3. Find the absolute maximum and minimum of f on the closed interval [0, 5].

Consider the series $\Sigma_{n=1}^{\infty}\frac{(-1)^{n}\sqrt{n}}{n+1}$.
(i) Use the Alternating Series Test to show that the series converges.
(ii) Determine whether the series converges absolutely or only conditionally.

Write the Maclaurin series for $\sin x$ up to (and including) the $x^{5}$ term. Derive the Maclaurin series for $\ln(1+x)$ and state its interval of convergence.

Consider the function $f(x)=x$, $ 0<x< (that="" [-\pi,\pi]="" (="" $="" )).="" ).<="" \pi="" a="" are="" as="" b_{n}="\frac{2(-1)^{n+1}}{n}" be="" by="" coefficients="" expanded="" extension="" f(x)="" fourier="" given="" is="" is,="" odd="" of="" onto="" p="" series="" show="" sine="" that="" the="" to="" which=""> </x<>

Let $ A=\begin{bmatrix}1&2&3\ 0&1&4\ 5&6&0\end{bmatrix} $.
i. Compute the determinant of A to verify whether A is invertible.
ii. Find the inverse of A.
iii. Determine the rank of A and use the Rank-Nullity theorem to find the nullity of the associated linear transformation $T:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ defined by $T(x)=Ax$.

Let $M=\begin{bmatrix}2&amp;1\\ 1&amp;2\end{bmatrix}$.
i. Show that M is symmetric.
ii. Find the eigenvalues and corresponding eigenvectors of M.
iii. Diagonalize M: that is, express it in the form $M=PDP^{-1}$ where D is diagonal.

Find and classify the critical points of the function $f(x,y)=x^{3}-3xy$.

Find the maximum and minimum values of $f(x,y)=x+2y$ subject to the constraint $ g(x,y)=x^{2}-9=0 $.

Consider the series \sum_{n=2}^{\infty}\frac{1}{n(\ln n)^{p}}, p&gt;0.
(i) Show that the function $f(x)=\frac{1}{x(\ln x)^{p}}, x\ge2$ is positive, continuous, and decreasing.
(ii) Use the Integral Test to determine for which values of p the series converges.

1. Gradient, curl and divergence with applications.
2. Evaluation of definite and improper integrals with applications.
3. Applications of definite integrals to evaluate surface areas and volumes of revolutions.
4. System of linear equations with applications.

Lagrange's mean value theorem can be proved for a function f(x) by applying Roll's mean value theorem

The function $ f(x) = x(x + 3) e^{-\frac{x}{2}} $ satisfy all conditions of Roll's mean value theorem in the interval [-3, 0]. Then the value of c is:

if $ \begin{bmatrix} 4/3 \\ 1 \end{bmatrix} $ is an eigenvector of $ \begin{bmatrix} 3 & 4 \\ 3 & 1 \end{bmatrix} $. What is the associated eigenvalue?

if f is continuous on [-2.354, 2.354] then

let $ f(x) = |x|^{\frac{3}{2}}, x \in R $ then

if $ A(2) = 2i - j + 2k, A(3) = 4i - 2j + 3k $, then $ \int_{2}^{3} A \cdot \frac{dA}{dt} dt $ is

if $ \nabla \times \vec{F} $. Then it is called.

if $ 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 $ be a system of equations, then

If f(x) is continuous in the closed interval [a,b] and f(x) exists in (a,b) and f(a) = f(b), then there exist at least one value $ c(a < c < b) $ Such that $ f'(c)=0 $ is called

The rank of the matrix $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} $ is

show that : $ \int_{0}^{\infty} x^2 e^{-x^4} dx \times \int_{\infty}^{\infty} e^{-x^4} dx = \frac{\pi}{8\sqrt{2}} $

Use L' Hospital rule to find the following limits. $ \lim_{x \to 0} \frac{\cos - \ln (1+x) - 1 + xy^x}{\sin^2 x} $

Test the convergence of $ 1 + \frac{3}{7} x + \frac{3.6}{7.10} x^2 + \frac{3.6.9}{7.10.13} x^3 + \frac{3.6.9.12}{7.10.13.16} x^4 + \ldots $

Examine the convergence of the series of which the general term is $ 2^2 4^2 6^2 \cdots \frac{(2n-2)^2}{3.4.5 \cdots (2n-1)2n} x^{2n} $

Obtain the fourth-degree Taylor's polynomial approximation to $ f(x) = e^{2x} $ about $ x = 0 $. Find the maximum error when $ 0 \leq x \leq 0.5 $

It is given the Rolle's theorem holds the function $ f(x) = x^3 + bx^2 + cx, 1 \leq x \leq 2 $ at the Point $ x = 4/3 $. Find the value of b and c.

Evaluate $ \int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1+x^2} $

Discuss the convergence of the sequence whose n-th term is $ a_n = \frac{(-1)^n}{n} + 1 $.

if $ f(x) = \log (1 + x), x > 0 $ using Maclaurin's theorem, show that for $ 0 < \theta < 1 $, $ \log (1 + x)=x - \frac{x^2}{2} + \frac{x^3}{3(1+\theta x)^3} $. Deduce that $ \log (1+x)=< x - \frac{x^2}{2} + \frac{x^3}{3} $ for $ x> 0 $

Using Taylor's theorem, express the polynomial $ 2x^3 + 7x^2 + x - 6 $ in powers of (x - 1).

Find the values of a,b,c if A = $ \begin{bmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{bmatrix} $ is orthogonal?

Verify Cayley-Hamilton theorem for the matrix A = $ \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} $ and find its inverse. Also express $ A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I $ as a linear polynomial in A.

Expand $ \ln x $ in power of $ (x - 1) $ by Taylor's theorem.

By using Beta function evaluate $ \int_{0}^{1} x^5 (1 + x^3)^{10} dx $.

Find the Fourier series to represent the function defined as $ f(x) = \begin{cases} x + \frac{\pi}{2}, & -\pi < x < 0 \\ \frac{\pi}{2} - x, & 0 < x < \pi \end{cases} $

Evaluate: (i) div F and Curl F, where F= grad $ (x^3 + y^3 + z^3 - 3xyz) $. (ii) If $ F = (x + y + z) i + j - (x + y)k $ show that $ F \cdot \text{Curl} F = 0 $.

Lagrange's mean value theorem can be proved for a function f(x) by applying Roll's mean value theorem

The function $ f(x) = x(x + 3) e^{-\frac{x}{2}} $ satisfy all conditions of Roll's mean value theorem in the interval [-3, 0]. Then the value of c is:

if $ \begin{bmatrix} 4/3 \\ 1 \end{bmatrix} $ is an eigenvector of $ \begin{bmatrix} 3 & 4 \\ 3 & 1 \end{bmatrix} $. What is the associated eigenvalue?

if f is continuous on [-2.354, 2.354] then

let $ f(x) = |x|^{\frac{3}{2}}, x \in R $ then

if $ A(2) = 2i - j + 2k, A(3) = 4i - 2j + 3k $, then $ \int_{2}^{3} A \cdot \frac{dA}{dt} dt $ is

if $ \nabla \times \vec{F} $. Then it is called.

if $ 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 $ be a system of equations, then

If f(x) is continuous in the closed interval [a,b] and f(x) exists in (a,b) and f(a) = f(b), then there exist at least one value $ c(a < c < b) $ Such that $ f'(c)=0 $ is called

The rank of the matrix $ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} $ is

show that : $ \int_{0}^{\infty} x^2 e^{-x^4} dx \times \int_{\infty}^{\infty} e^{-x^4} dx = \frac{\pi}{8\sqrt{2}} $

Use L' Hospital rule to find the following limits. $ \lim_{x \to 0} \frac{\cos - \ln (1+x) - 1 + xy^x}{\sin^2 x} $

Test the convergence of $ 1 + \frac{3}{7} x + \frac{3.6}{7.10} x^2 + \frac{3.6.9}{7.10.13} x^3 + \frac{3.6.9.12}{7.10.13.16} x^4 + \ldots $

Examine the convergence of the series of which the general term is $ 2^2 4^2 6^2 \cdots \frac{(2n-2)^2}{3.4.5 \cdots (2n-1)2n} x^{2n} $

Obtain the fourth-degree Taylor's polynomial approximation to $ f(x) = e^{2x} $ about $ x = 0 $. Find the maximum error when $ 0 \leq x \leq 0.5 $

It is given the Rolle's theorem holds the function $ f(x) = x^3 + bx^2 + cx, 1 \leq x \leq 2 $ at the Point $ x = 4/3 $. Find the value of b and c.

Evaluate $ \int_{0}^{\infty} \log \left( x + \frac{1}{x} \right) \frac{dx}{1+x^2} $

Discuss the convergence of the sequence whose n-th term is $ a_n = \frac{(-1)^n}{n} + 1 $.

if $ f(x) = \log (1 + x), x > 0 $ using Maclaurin's theorem, show that for $ 0 < \theta < 1 $, $ \log (1 + x)=x - \frac{x^2}{2} + \frac{x^3}{3(1+\theta x)^3} $. Deduce that $ \log (1+x)=< x - \frac{x^2}{2} + \frac{x^3}{3} $ for $ x> 0 $

Using Taylor's theorem, express the polynomial $ 2x^3 + 7x^2 + x - 6 $ in powers of (x - 1).

Find the values of a,b,c if A = $ \begin{bmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{bmatrix} $ is orthogonal?

Verify Cayley-Hamilton theorem for the matrix A = $ \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} $ and find its inverse. Also express $ A^5 - 4A^4 - 7A^3 + 11A^2 - A - 10I $ as a linear polynomial in A.

Expand $ \ln x $ in power of $ (x - 1) $ by Taylor's theorem.

By using Beta function evaluate $ \int_{0}^{1} x^5 (1 + x^3)^{10} dx $.

Find the Fourier series to represent the function defined as $ f(x) = \begin{cases} x + \frac{\pi}{2}, & -\pi < x < 0 \\ \frac{\pi}{2} - x, & 0 < x < \pi \end{cases} $

Evaluate: (i) div F and Curl F, where F= grad $ (x^3 + y^3 + z^3 - 3xyz) $. (ii) If $ F = (x + y + z) i + j - (x + y)k $ show that $ F \cdot \text{Curl} F = 0 $.