BIT Basic Mathematics

i. What is even and odd function? Give example of each and write their symmetricity.

ii. Find the domain and range of the following functions.

a. \(f(x) = \frac{x^2 - 3x - 4}{x + 1}\)

b. \(f(x) = \sqrt{5x + 10}\)

Find the Tayllor’s series generated by \(f(x) = \frac{1}{x}\) at \(a = 2\) where it anywhere, does the series converges to \(\frac{1}{x}\).

i. Sketch the graph of the function \(f(x) = x^2\). Shifted vertically up to 1 and -2 units and horizontally up to 3 and -2 units.

(ii) Find the \(\delta\) algebraically of the following functions.

a) \(\lim_{x \to 5} \sqrt{x - 1}\), \(L = 2\) and \(\epsilon = 1\)

b) \(\lim_{x \to 2} 2x - 2\), \(L = 6\) and \(\epsilon = 0.02\).

Find the initial value problem in

\(\frac{dy}{dx} + 2y = 3,\ y(0) = 1\).

Determine the convergence or divergence of the series

\(\sum_{n=1}^{\infty} n^2 e^{-n}\)

Show the f(x) \(= \frac{x^2 + x - 6}{x^2 - 4}\), \(x \neq 2\) has a continuous extension to \(x = 2\).

Evaluate the

(i) \(\lim_{x \to \infty} (x - \sqrt{x^2 + 16})\)

(ii) \(\lim_{x \to 1} \left(\frac{\sqrt{6x + 10} - 5}{x^2}\right)\)

Find the derivatives of the function \(f(x, y) = x^3 - xy^2 + x^2y - y^3\) at \(p_0(5, 5)\) in the direction of \(\vec{u} = 4\vec{t} + 3\vec{j}\).

Integrate the following

\(\int_{0}^{\frac{x}{4}} \frac{dx}{1 - \sin x}\)

Find the area of the surface generated by revolving the curve \(y = 2\sqrt{x}\) \(1 \leq x \leq 2\), about x-axis.

Determine the concavity and find the inflection point of the function

\(f(x) = x^3 - 3x^2 + 2\)

Find \(\frac{dy}{dx}\)

(i) \(Y = 2u^3,\ u = 8x - 1\)

(ii) \(Y = \sin u,\ u = x - x\cos x\).