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]

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]

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]