Using rectangles, estimate the area under the parabola $y = x^2$ from 0 to 1. A particle moves along a line so that its velocity v at time t is $v = t^2 + t + 6$. (i) find the displacement of the particle during the time period $1 \leq t \leq 4$. (ii) find the distance traveled during this time period. [5+5+0]

Find the area of the region bounded by $y = x^2$ and $y = 2x - x^2$. Using trapezoidal rule, approximate

$\int_1^2 \frac{1}{x} dx \text{ with n=5.}$

[5+5]

Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function $T = 0.03t + 8.50$, where T is temperature in degree centigrade and t represents years since 1900. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. [5]

Find the equation of tangent at (1,2) to the curve $y = 2x^3$. [5]

State Rolle's theorem and verify the Rolle's theorem for $f(x) = x^2 - 3x + 2$ in $[0, 3]$. [5]

Find the volume of the solid obtained by rotating about the y-axis the region between $y = x$ and $y = x^2$. [5]

Find the local maximum and minimum values, saddle points of $f(x, y) = x^4 + y^4 - 4xy + 1$. [5]

Find the derivative of $r(t) = (1 + t^2)\hat{i} - t e^{-t} \hat{j} + \sin 2t \hat{k}$ and find the unit tangent vector at $t=0$. [5]

If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{3-x}$, then find $fog$ and its domain and range. A rectangular storage container with an open top has a volume of $20m^3$. The length of its base is twice its width. Material for the base costs Rs 10 per square meter; material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base. [5+5]

Use Newton's method to find $\sqrt[6]{2}$, correct to five decimal places. [5]

What is a sequence? Is the sequence $a_n = \frac{n}{\sqrt{5+n}}$ convergent? [5]

Solve: $y' = \frac{x^2}{y^2}$, $y(0) = 2$. Solve the initial value problem: $y'' + y' - 6y = 0$, $y(0) = 0$, $y(0)' = 1$. [5+5]

Solve $y' + 2xy - 1 = 0$. [5]

Find the partial derivatives of $f(x, y) = x^2 + 2x^3y^2 - 3y^2 + x + y$ at (1,2). [5]

Find a vector perpendicular to the plane that passes through the points: $P(1, 4, 6)$, $Q(-2, 5, -1)$ and $R(1, -1, 1)$. [5]