Find the Maclaurin series for $\cos x$ and prove that it represents $\cos x$ for all x. Define initial value problem. Solve that initial value problem of $y' + 2y = 3$, $y(0) = 1$. Find the volume of a sphere of radius $r$. [4+4+2]
Applications of Derivatives
1.
Verify Mean value theorem of $f(x) = x^3 - 3x + 2$ for [-1,2]. [5]
2.
Find the volume of the resulting solid which is enclosed by the curve $y = x$ and $y = x^2$, is rotated about the x-axis. [5]
3.
Find the extreme values of the function $f(x, y) = x^2 + 2y^2$ on the circle $x^2 + y^2 = 1$. [5]
Function of One Variable
1.
A function is defined by
$f(x) = |x|$
Calculate f(-3), f(4), and sketch the graph. Prove that the limit does not exist.
$\lim_{x \to 2} \frac{|x-2|}{x-2}$
[5+0+5]
2.
Find the domain and sketch the graph of the function
$f(x) = x^2 - 6x$
Estimate the area between the curve $y = x^2$ and the line y = 1 and y = 2. [3+2]
3.
If $f(x) = \sqrt{2-x}$ and $g(x) = \sqrt{x}$, find $fof$ and $fog$. [5]
4.
Starting with $x_1 = 2$, find the third approximation $x_3$ to the root of the equation $x^3 - 2x - 5 = 0$. [5]
Infinite Sequence and Series
1.
If $f(x, y) = \frac{y}{x}$ $\lim_{(x,y)\to(0,0)} \frac{f(x,y)}{x}$ does not exist, justify. Calculate $\iint_R f(x, y) dA$, for $f(x, y) = 100 - 6x^2y$, and $R: 0 \leq x \leq 2, -1 \leq y \leq 1$. [5+5]
2.
Determine whether the series converges or diverges
$\sum_{n=1}^{\infty} \frac{n^2}{5n^2+4}$
[5]
Limits and Continuity
1.
Define continuity on an interval. Show that the function is continous on the interval [1,-1].
$f(x) = 1 - \sqrt{1 - x^2}$
[5]
Ordinary Differential Equations
1.
Find the solution of $y'' + 4y' + 4 = 0$. [5]
Partial Derivatives and Multiple Integrals
1.
Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ if z is defined as a function of x and y by the equation $x^3+y^3+z^3+6xyz=1$. [5]
Plane and Space Vectors
1.
If $a = (4,0,3)$ and $b = (-2,1,5)$, find $|a|$, the vector $a-b$ and $2a+b$. [5]