StudyNp | CSIT First Semester – 2078 Question | Mathematics I (Syllabus Wise)

Bachelors Level/First Year/First Semester/Science csit/first semester/mathematics i/syllabus wise questions

B.Sc Computer Science and Information Technology

Institute of Science and Technology, TU

Mathematics I (MTH117)

Year Asked: 2078, syllabus wise question

Applications of Derivatives

Using rectangles, estimate the area under the parabola

y = x^2

from 0 to 1. A particle moves along a line so that its velocity v at time t is

v = t^2 + t + 6

. (i) find the displacement of the particle during the time period

1 \leq t \leq 4

. (ii) find the distance traveled during this time period. [5+5+0]

Find the area of the region bounded by

y = x^2

and

y = 2x - x^2

. Using trapezoidal rule, approximate

\int_1^2 \frac{1}{x} dx \text{ with n=5.}

[5+5]

Recent studies indicate that the average surface temperature of the earth has been rising steadily. 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. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. [5]

Find the equation of tangent at (1,2) to the curve

y = 2x^3

. [5]

State Rolle's theorem and verify the Rolle's theorem for

f(x) = x^2 - 3x + 2

[0, 3]

. [5]

Find the volume of the solid obtained by rotating about the y-axis the region between

y = x

and

y = x^2

. [5]

Find the local maximum and minimum values, saddle points of

f(x, y) = x^4 + y^4 - 4xy + 1

. [5]

Derivatives

Find the derivative of

r(t) = (1 + t^2)\hat{i} - t e^{-t} \hat{j} + \sin 2t \hat{k}

and find the unit tangent vector at

t=0

. [5]

Function of One Variable

f(x) = \sqrt{x}

and

g(x) = \sqrt{3-x}

, then find

fog

and its domain and range. 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. [5+5]

Use Newton's method to find

\sqrt[6]{2}

, correct to five decimal places. [5]

Infinite Sequence and Series

What is a sequence? Is the sequence

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

convergent? [5]

Ordinary Differential Equations

Solve:

y' = \frac{x^2}{y^2}

y(0) = 2

. Solve the initial value problem:

y'' + y' - 6y = 0

y(0) = 0

y(0)' = 1

. [5+5]

Solve

y' + 2xy - 1 = 0

. [5]

Partial Derivatives and Multiple Integrals

Find the partial derivatives of

f(x, y) = x^2 + 2x^3y^2 - 3y^2 + x + y

at (1,2). [5]

Plane and Space Vectors

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)

. [5]