Mathematics I TU Board 2080 question paper

• a) If $\vec{a}=(4,0,3)$ and $\vec{b}=(-2,1,5)$, find $|\vec{a}|$, $3\vec{b}$, $\vec{a}+\vec{b}$ and $\overrightarrow{2a}+5\vec{b}$

(a)The area of the parabola y = x2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulitng surface.

As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at height of 1 km is 10C, express the temeperature T(in °C) as a function of height h(in kilometer), assuming that the linear model is appropriate.

Estimate the value of lim_x→0 ( (√(x^2 + 9) ) – 3 )/ x^2

• b) Find the solution of the equation y^2dy = x^2dx that satisfies the initial condition y(0) = 2.

• b) Draw a graph of the function in part(a). What does the slope represent?

• c) What is the temperature at a height of 2.5 km?

Integrate _0∫^1 x^2√(x^3 + 1) dx.

Find the Maclaurin series expansion of f(x) = e^x at x =0.

Find where the function f(x) = 3×4 – 4×3 – 12 x2 + 5 is increasing and where it is decreasing.

Find y' if x^3 + y^3 = 6xy.

Show that y = x – 1/x is a solution of the differential equation xy' + y = 2x.

Sketch the graph and find the domain and range of the function f(x) =2x-1.

Determine whether the series _n=1∑^∞ n^2 / (5n^2 + 4) converges or diverges.

If f(x,y) = x^3 + x^2y^3 – 2y^2, find f_x(2,1) and f_y(2,1).

Show that the function f(x) = x^2 + √(7-x) is continuous at x = 4.