StudyNp | CSIT First Semester – 2077 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: 2077, syllabus wise question

Applications of Derivatives

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? Sketch the curve

y = \frac{1}{x-3}

[5+5]

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

f(x) = x^3 - x^2 - 6x + 2

in [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 derivatives of

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

and find the unit tangent vector at

t=0

. [5]

Function of One Variable

f(x) = x^2

then find

\frac{f(2+h)-f(2)}{h}

Dry air is moving upward. If the ground temperature is

20^{\circ}C

and the temperature at a height of 1km is

10^{\circ}C

, express the temperature T in

^{\circ}C

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). [2.5+5+5]

f(x) = x^2 - 1

g(x) = 2x + 1

, find

fog

and

gof

and domain of

fog

. [5]

Find the third approximation

x_3

to the root of the equation

f(x) = x^3 - 2x - 7

, setting

x_1 = 2

. [5]

Infinite Sequence and Series

Show that the following integrals converge and diverge respectively.

\int_1^{\infty} \frac{1}{x^2} dx \text{ and } \int_1^{\infty} \frac{1}{x} dx

f(x, y) = xy/(x^2 + y^2)

, does

f(x, y)

exist as

(x, y) \to (0, 0)

? A particle moves in a straight line and has acceleration given by

a(t) = 6t^2 + t

. Its initial velocity is 4m/sec and its initial displacement is

s(0) = 5

cm. Find its position function

s(t)

. [2+3+5]

Show that the series converges.

\sum_{n=0}^{\infty} \frac{1}{1+n^2}

[5]

Limits and Continuity

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]

. [5]

Ordinary Differential Equations

Solve:

y'' + y = 0

y(0) = 5

y(\pi/4) = 3

. [5]

Partial Derivatives and Multiple Integrals

Evaluate

\int_{3}^{2} \int_{0}^{\frac{\pi}{2}} \left( y + y^{2} \cos x \right) \, dx \, dy

Find the Maclaurin's series for

\cos x

and prove that it represents

\cos x

for all x. [5+5]

Find the partial derivative of

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

at (2,1). [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]