(a) Determine whether each of the following functions is even, odd, or neither even nor odd. Also, sketch the graph of each of the functions:
(i) \(f(x)=x^5+x\)
(ii) \(g(x)=2x-x^2\).
(b) Find the value of \(\displaystyle \lim_{x\to4}\frac{\sqrt{x}-2}{x^2-16}\).
(a) Use Newton’s method to find \(\sqrt[6]{2}\) correct to five decimal places.
(b) What do you mean by ordinary differential equation of first order? Differentiate linear and non linear differential equation. Find the order of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^3=\left(\frac{dy}{dx}\right)^4\).
(a) Find the Maclaurin series of the function \(f(x)=e^x\) and its radius of convergence.
(b) Find a vector equation and parametric equations for the line that passes through the point \((5,1,3)\) and is parallel to the vector \(\vec{i}+4\vec{j}-2\vec{k}\). Also, find two other points on the line.
Attempt any Eight questions
[8×5=40]Integrate \(\displaystyle \int_1^9 \frac{x-1}{\sqrt{x}}\,dx\).
State Mean value theorem and verify this for \(f(x)=x^2-4x-3\) in the interval \([1,4]\).
Find two positive numbers whose product is \(100\) and whose sum is a minimum.
Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) if \(z\) is defined implicitly as a function of \(x\) and \(y\) by the equation \(x^3+y^3+z^3+6xyz=1\).
Compare the value of \(\Delta y\) and \(dy\) if \(y=x^3+x^2-2x+1\) and \(x\) changes from \(1\) to \(1.05\).
Find the area of the region enclosed by the parabola \(y=x^2\) and the line \(y=2\).
Determine whether the series \(\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2+1}\) converges or diverges.
If \(f(x)=x^2+2\), then find the range and domain of \(f(x)\).
Identify the separable differential equations
(i) \(y'=x+2\)
(ii) \(y'=xy+4x-7y\)
(iii) \(y'=\ln y\)
(iv) \(y'=\sin(xy)\).