Define arc-length of the curve. Find the length of the curve $y = (\frac{x}{2})^{\frac{2}{3}}$ from $x = 0$ to $x = 2$. [1+4]
Application of Derivative
1.
Define the concavity of the function. The graph of the function is then $f(x) = x^4 - 4x^3 + 10$. Find the intervals on which $f$ is increasing and on which $f$ is decreasing. Find where the graph of $f$ is concave up and where it is concave down. Find the local maximum or local minimum value of function if exist. [2+3+3+2]
Differentiation
1.
Find $\frac{dy}{dx}$ of the following. $y^2 - x^2 = \cos(xy)$,$x^2 = \frac{9}{y^2}$. [5]
First Order Differential Equations
1.
What is a first order linear differential equation? Solve the initial value problem: $t \frac{dy}{dt} + 2y = t^3$, $t > 0$, $y(2) = 1$. [1+4]
Functions and their graphs
1.
If a function is defined by $$f(x) = \begin{cases} 1 + x & \text{if } x \leq -1 \\ x^2 & \text{if } x > -1 \end{cases}$$ Evaluate $f(-3)$, $f(-1)$, and $f(0)$ and sketch the graph. Define different types of discontinuity at a point. At what points the function becomes continuous of the function $f(x) = \frac{x-2}{x^2-7x+10}$ [5+5]
2.
Find the domain and range of the function $f(x) = \sqrt{5x + 10}$. Draw the graph of the function $y = x^2$ shifted up by 1 unit, down by 2 units, also shift 3 units to left, and 2 units right with new position of function. [2+3]
Infinite Sequence and Series
1.
Determine whether the following series are convergence or divergence $\sum_{n=1}^{\infty} \frac{5}{5n-1}$, $\sum_{n=0}^{\infty} \frac{1}{n!}$. [5]
Integration
1.
Find the area of the region between the x-axis and the graph of $f(x) = x^3 - x^2 - 2x$, $1 \leq x \leq 2$. [5]
2.
Evaluate the following integral. $\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \cos x}dx$,$\int \frac{3x^2-7x+1}{3x} , dx$ [5]
Limits and continuity
1.
Define horizontal and vertical asymptotes. Find the appropriate asymptotes to the function: $f(x) = x - \sqrt{x^2 + 16}$. [2+3]
Partial Derivatives
1.
Define Gradient vector and directional derivative.Find the direction in which $f(x, y) = \frac{x^2}{2} + \frac{y^2}{2}$ increases and decreases most rapidly at the point $(1,1)$. What is the direction of zero change in $f$ at $(1,1)$? Derivative of $f(x, y)$ at the point $(1,1)$ in the direction $v = 3i - 4j$. [5+2+3]
2.
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 $t = \frac{\pi}{2}$. [1+4]