A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Sketch the curve
$y = \frac{1}{x-3}$
[5+5]
2.
State Rolle's theorem and verify the Rolle's theorem for $f(x) = x^3 - x^2 - 6x + 2$ in [0, 3]. [5]
3.
Find the volume of the solid obtained by rotating about the y-axis the region between $y = x$ and $y = x^2$. [5]
4.
Find the local maximum and minimum values, saddle points of $f(x, y) = x^4 + y^4 - 4xy + 1$. [5]
Derivatives
1.
Find the derivatives of $r(t) = (1 + t^2)\hat{i} - te^t\hat{j} + \sin 2t\hat{k}$ and find the unit tangent vector at $t=0$. [5]
Function of One Variable
1.
If $f(x) = x^2$ then find
$\frac{f(2+h)-f(2)}{h}$
Dry air is moving upward. If the ground temperature is $20^{\circ}C$ and the temperature at a height of 1km is $10^{\circ}C$, express the temperature T in $^{\circ}C$ as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part(a). What does the slope represent? (c) What is the temperature at a height of 2km? Find the equation of the tangent to the parabola $y = x^2 + x + 1$ at (0, 1). [2.5+5+5]
2.
If $f(x) = x^2 - 1$, $g(x) = 2x + 1$, find $fog$ and $gof$ and domain of $fog$. [5]
3.
Find the third approximation $x_3$ to the root of the equation $f(x) = x^3 - 2x - 7$, setting $x_1 = 2$. [5]
Infinite Sequence and Series
1.
Show that the following integrals converge and diverge respectively.
$\int_1^{\infty} \frac{1}{x^2} dx \text{ and } \int_1^{\infty} \frac{1}{x} dx$
If $f(x, y) = xy/(x^2 + y^2)$, does $f(x, y)$ exist as $(x, y) \to (0, 0)$? A particle moves in a straight line and has acceleration given by $a(t) = 6t^2 + t$. Its initial velocity is 4m/sec and its initial displacement is $s(0) = 5$cm. Find its position function $s(t)$. [2+3+5]
2.
Show that the series converges.
$\sum_{n=0}^{\infty} \frac{1}{1+n^2}$
[5]
Limits and Continuity
1.
Define continuity of a function at a point $x = a$. Show that the function $f(x) = \sqrt{1-x^2}$ is continuous on the interval $[1, -1]$. [5]