Find the Maclaurin series expansion of $f(x) = \sin x$ for all x. [5]
Applications of Derivatives
1.
Show that every member of the family of function $y = \frac{1 + ce^t}{1 - ce^t}$ is a solution of the differential equation $y' = \frac{1}{2}(y^2 - 1)$. [5]
2.
Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve $y = \sqrt{x}$ for $0$ to $1$. [5]
Derivatives
1.
Find the unit normal and binormal vectors for the circular helix $r(t) = \cos t \hat{i} + \sin t \hat{j} + t \hat{k}$. [5]
2.
The position vector of an object moving in a plane is given by $r(t) = t^2 \hat{i} + t^2 \hat{j}$. Find its velocity, speed, and acceleration when $t = 1$ and illustrate geometrically. [5]
Function of One Variable
1.
Sketch the graph of $f(x) = x^2$. Find its domain and range. Evaluate
Where the function $f(x) = |x|$ is differentiable? Discuss. A farmer has 1200 m. of fencing and wants to fence off a rectangular field that borders a straight river. He needs to fence along the river. What are the dimensions of the field that has the largest area? [5+5]
Infinite Sequence and Series
1.
Determine whether the sequence $a_n = (-1)^n$ is convergent or divergent. [5]
Ordinary Differential Equations
1.
Find the solution of the initial value problem $x^2 y' + x y = 1$, $y(1) = 2$, $x > 0$. Find the area enclosed by the line $y = x - 1$ and the parabola $y^2 = 2x + 6$. [5+5]
Partial Derivatives and Multiple Integrals
1.
If $f(x, y) = \frac{xy}{x^2 + y^2}$, does $\lim_{(x, y) \to (0, 0)} f(x, y)$ exist? Justify. [5]
2.
If $f(x, y) = 2x^3 + x^2y^2 - y^4$, find $f_x(1, -2)$, $f_y(1, -1)$ and $f_{yx}(1, -1)$. [5]