Show the integral coverages $\int_0^3 \frac{dx}{x-1}$. [5]
2.
Use Trapezoidal rule to approximate the integral $\int_1^2 \frac{dx}{x}$, with n=5. [5]
3.
Solve $\int_0^3 \int_1^2 x^2y \, dx \, dy$ [5]
Applications of Derivatives
1.
Estimate the area between the curve $y = x^2$ and the lines $x = 0$ and $x = 1$, using rectangle method, with four sub intervals. A particle moves a line so that its velocity $v$ at time $t$ is
(1) Find the displacement of the particle during the time period $1 \leq t \leq 4$
(2) Find the distance travelled during this time period. [10+0]
2.
Dry air is moving upward. If the ground temperature is $20^\circ$ and the temperature at a height of 2km is $10^\circ$, express the temperature $T$ in $^\circ$C as a function of the height $h$(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km? [5]
3.
Find the equation of the tangent at (1,3) to the curve $y = x^2 + 1$. [5]
4.
State Rolle's theorem and verify the theorem for $f(x) = x^2 - 9$, $x \in [-3,3]$. [5]
Derivatives
1.
Find the derivative of $\mathbf{r}(t) = t^2\mathbf{i} - te^{-t}\mathbf{j} + \sin(2t)\mathbf{k}$ and find the unit tangent vector at $t = 0$. [5]
Function of One Variable
1.
If a function is defined by $f(x) = \begin{cases} 1 + x, & x \leq -1 \\ x^2, & x > -1 \end{cases}$, evaluate $f(-3)$, $f(-1)$ and $f(0)$ and sketch the graph. Prove that $\lim_{x \to 0} \frac{|x|}{x}$ does not exist. [10+0]
2.
Sketch the curve $y = x^2 + 1$ with the guidelines of sketching. If $z = xy^2 + y^3$, $x = \sin t$, $y = \cos t$, find $\frac{dz}{dt}$ at $t = 0$. [10+0]
3.
Starting with $x_1 = 1$, find the third approximate $x_3$ to the root of the equation $x^3 - x - 5 = 0$. [5]
Infinite Sequence and Series
1.
What is sequence? Is the sequence $a_n = \frac{n}{\sqrt{5+n}}$ convergent? [5]
Ordinary Differential Equations
1.
Define initial value problem. Solve: $y'' + 4y' - 6y = 0$, $y(0) = 1$, $y'(0) = 0$ Find the Taylor's series expansion for $\cos x$ at $x = 0$. [10+0]
Partial Derivatives and Multiple Integrals
1.
Find the partial derivative $f_{xx}$ and $f_{yy}$ of $f(x,y) = x^2 + x^3y^2 - y^2 + xy$ at (1,2). [5]
Plane and Space Vectors
1.
Find the angle between the vectors $a = (2, 2, -1)$ and $b = (1, 3, 2)$. [5]