A pyramid $3\text{ m}$ high has a square base that is $3\text{ m}$ on a side. The cross section of the pyramid perpendicular to the altitude $x\text{ m}$ down from the vertex is a square $x\text{ m}$ on a side. Find the volume of the pyramid. [5]
Application of Derivative
1.
Find the positive root of the equation $f(x) = x^2 - 2 = 0$. Find the Taylor series and the Taylor polynomials generated by $f(x) = e^x$ at $x = 0$. Use the Trapezoidal Rule with $n = 4$ to estimate $\int_{1}^{2} x^2 dx$. Compare the estimate with the exact value. [3+3+4]
2.
Water runs into a conical tank at the rate $9\text{ ft}^3/\text{minutes}$. The tank stands point down and has a height of $10\text{ ft}$ and a base radius of $5\text{ ft}$. How fast is the water level rising when the water is $6\text{ ft}$ deep? [5]
3.
Find the absolute maximum and minimum values of $f(x) = x^{2/3}$ on the interval $[-2, 3]$. [5]
Differentiation
1.
A rock breaks loose from the top of a tall cliff. Find average speed during the first 2 sec of fall. What is its average speed during the 1sec interval between second 1 and second 2? Find the speed of the falling rock at $t = 1$ and $t = 2$. [3+3+4]
2.
Find the slope of the curve $y = 1/x$ at any point $x = a$, $a \neq 0$. What is the slope at the point $x = -1$? Where does the slope equal $-1/4$? What happens to the tangent to the curve at the point $(a, 1/a)$ as $a$ changes? [2+1.5+1.5]
First Order Differential Equations
1.
Draw a phase line for the equation $\frac{dy}{dx} = (y + 1)(y - 2)$ and use it to sketch solutions to the equation. [5]
Functions and their graphs
1.
In 2000, 100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5.5$\%$. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount $A$ in the account after $x$ years have elapsed. Define when the function $f(x)$ is odd and even. Also, define when a function $f(x)$ is increasing and decreasing? If $y = x^2$ is a given function then determine the interval in which the function is increasing and decreasing and draw the graph of the given function. [5+5]
Integration
1.
Find the area between the curves $y = x^2 - 2$ and $y = 2$. [5]
Limits and continuity
1.
Define horizontal asymptote to a curve $y = f(x)$. Find the horizontal asymptote to the curve $f(x) = \frac{5x^2 + 8x - 3}{3x^2 + 2}$ and draw the curve. [2+3]
Partial Derivatives
1.
Find the second order derivative $\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial y^2}, \frac{\partial^2 f}{\partial x\partial y}, \frac{\partial^2 f}{\partial y\partial x}$ of $f(x,y) = x \cos y + ye^x$ [5]
2.
Find the local extreme values of the function $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$. [5]