BIT Basic Mathematics

i. If a function \(f(x)\) is defined by

\(f(x) = \begin{cases} 1 + x \ \text{ if } x \le -1 \\ x^2 \ \text{ if } x \gt -1 \end{cases}\)

Evaluate \(f(-3)\), \(f(-1)\) and \(f(0)\) and sketch the graph.

ii. Define different types of discontinuity at a point. At what points the function becomes continuous of the function \(f(x) = \frac{x-2}{x^2-7x+10}\)

Define Gradient vector and directional derivative.

Find the direction in which \(f(x, y) = \frac{x^2}{2} + \frac{y^2}{2}\)

i. Increases and decreases most rapidly at the point (1,1)

ii. What is the direction of zero change in \(f\) at (1,1)?

iii. Derivative of \(f(x, y)\) at the point (1,1) in the direction \(v = 3i - 4j\).

Define the concavity of the function.

The graph of the function is \(f(x) = x^4 - 4x^3 + 10\) then

i. Find the intervals on which \(f\) is increasing and on which \(f\) is decreasing.

ii. Find where the graph of \(f\) is concave up and where it is concave down.

iii. Find the local maximum or local minimum value of function if exist.

i. Find the domain and range of the function.

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

ii. Draw the graph of the function \(y = x^2\) sift up by 1 unit, down by 2 units, also shift 3 units to left, and 2 units right with new position of function.

Define horizontal and vertical asymptotes.

Find the appropriate asymptotes to the function:

\(f(x) = x - \sqrt{x^2 + 16}\)

Find the area of the region between the x-axis and the graph of

\(f(x) = x^3 - x^2 - 2x\). \(1 \le x \le 2\).

Define arc-length of the curve. Find the length of the curve \(y = \left(\frac{x}{2}\right)^{\frac{2}{3}}\) from \(x = 0\) to \(x = 2\).

Evaluate the following integral.

(i) \(\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \cos x}\, dx\).

(ii) \(\int \frac{3x^2 - 7x + 1}{3x}\, dx\)

What is a first order linear differential equation?

Solve the initial value problem.

\(t\frac{dy}{dt} + 2y = t^3,\ t \gt 0,\ y(2) = 1\)

Determine whether the following series are convergence or divergence

(i) \(\sum_{n=1}^{\infty} \frac{5}{5n-1}\)

(ii) \(\sum_{n=0}^{\infty} \frac{1}{n!}\).

What is chain rule for function \(w = f(x, y)\)?

To use this rule, find the derivative of \(w = xy\) w.r.t. \(t\) along the path \(x = \cos t,\ y = \sin t\).

Also, find derivative of \(w\) at \(t = \frac{\pi}{2}\).

Find \(\frac{dy}{dx}\) of the following.

(i) \(y^2 - x^2 = \cos(xy)\)

(ii) \(x^2 = \frac{9}{y^2}\)