Mathematics I TU Board 2077 question paper

If f(x) = x^2 then find $\frac{f(2+h)–f(2)}{h}$

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?

Show that the converges ${\int }_{\infty }^1\frac{1}{x^2}dx$ and diverges ${\int }_{\infty }^1\frac{1}{x}dx$

Evaluate ${\int }_3^2{\int }_0^{\frac{\pi }{2}}(y+y^{2}cosx)dxdy$

Dry air is moving upward. If the ground temperature is 20^0 and the temperature at a height of 1km is 10^0 C, express the temperature T in ^0C 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)

Sketch the curve $y=\frac{1}{x–3}$

If $f(x,y)=\frac{xy}{x^2+y^2}$, does f(x, y) exist, as (x, y) → (0, 0)?

A particle moves in a straight line and has acceleration given by a(t) = 6t^2 + 1. Its initial velocity is 4m/sec and its initial displacement is s(0) = 5cm. Find its position function s(t).

Find the Maclaurin series for cos x and prove that it represents cos x for all x.

If f(x) = x^2 – 1, g(x) = 2x + 1, find fog and gofand domain of fog.

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].

State Rolle's theorem and verify the Rolle's theorem forf(x) = x^3 – x^2 – 6x + 2 in [0, 3]

Find the third approximation x_3 to the root of the equation f(x) = x^3 – 2x – 7, setting x_1 = 2.

Find the derivatives of r(t) = (1 + t^2)i – te^-tj + sin 2tk and find the unit tangent vector at t=0.

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

Solve: y" + y' = 0, y(0) = 5, y(π/4) = 3

Show that the series ${\sum }_{n=0}^{\infty }\frac{1}{1+n^2}$ converges.

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)

Find the partial derivative of f(x, y) = x^3 + 2x^3y^3 – 3y^2 + x + y, at (2,1)

Find the local maximum and minimum values, saddle points of f(x,y) = x^4 + y^4 – 4xy + 1