Mathematics I TU Board 2078 question paper

If $f(x)=\sqrt{x}$ and $g(x)=\sqrt{3–x}$ then find f0g and its domain and range.

Using rectangular, estimate the area under the parabola y = x^2 from 0 to 1.

Find the area of the region bounded by y = x^2 and y = 2x – x^2

Solve y' = x^2/y^2, y(0) = 2

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.

A particle moves along a line so that its velocity v at time t is

v = t^2 – t + 6

1. Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
2. Find the distance travelled during this time period.

Using trapezoidal rule, approximate ${\int }_1^2\frac{1}{x}dx$ with n = 5

Solve the initial value problem: y" + y' – 6y = 0, y(0) = 1, y'(0) = 0

Recent studies indicates that the average surface temperature of the earth has been rising rapidly. 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.

1. What do the slope and T-intercept represent?
2. Use the equation to predict the average global surface temperature in 2100

Find the equation of tangent at (1, 2) to the curve y = 2x^2

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

Use Newton's method to find ^6√2 correct five decimal places.

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' + 2xy – 1 = 0

What is sequence? Is the sequence

$a_n=\frac{n}{\sqrt{5+n}}$

convergent?

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^2 + 2x^3y^2 – 3y^2 + x + y at (1. 2)

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