StudyNp Notes | Notes of Mathematics I(MTH117), First Semester CSIT

Asked in 2081Long Question5+5 Marks

1.

Where the function

f(x) = |x|

is differentiable? Discuss. A farmer has 1200 m. of fencing and wants to fence off a rectangular field that borders a straight river. He needs to fence along the river. What are the dimensions of the field that has the largest area? [5+5]

Asked in 2081Long Question5+5 Marks

2.

Sketch the graph of

f(x) = x^2

. Find its domain and range. Evaluate

\lim_{x \to 1^-} \sin^{-1} \left( \frac{1-\sqrt{x}}{1-x} \right)

[5+5]

Asked in 2079Short Question5 Marks

3.

Starting with

x_1 = 1

, find the third approximate

x_3

to the root of the equation

x^3 - x - 5 = 0

. [5]

Asked in 2079Long Question10+0 Marks

4.

Sketch the curve

y = x^2 + 1

with the guidelines of sketching. If

z = xy^2 + y^3

,

x = \sin t

,

y = \cos t

, find

\frac{dz}{dt}

at

t = 0

. [10+0]

Asked in 2079Long Question10+0 Marks

5.

If a function is defined by

f(x) = \begin{cases} 1 + x, & x \leq -1 \\ x^2, & x > -1 \end{cases}

, evaluate

f(-3)

,

f(-1)

and

f(0)

and sketch the graph. Prove that

\lim_{x \to 0} \frac{|x|}{x}

does not exist. [10+0]

Asked in 2078Short Question5 Marks

6.

Use Newton's method to find

\sqrt[6]{2}

, correct to five decimal places. [5]

Asked in 2078Long Question5+5 Marks

7.

If

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]

Asked in 2077Short Question5 Marks

8.

Find the third approximation

x_3

to the root of the equation

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

, setting

x_1 = 2

. [5]

Asked in 2077Short Question5 Marks

9.

If

f(x) = x^2 - 1

,

g(x) = 2x + 1

, find

fog

and

gof

and domain of

fog

. [5]

Asked in 2077Long Question2.5+5+5 Marks

10.

If

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]

Asked in 2075Short Question5 Marks

11.

Starting with

x_1 = 2

, find the third approximation

x_3

to the root of the equation

x^3 - 2x - 5 = 0

. [5]

Asked in 2075Short Question5 Marks

12.

If

f(x) = \sqrt{2-x}

and

g(x) = \sqrt{x}

, find

fof

and

fog

. [5]

Asked in 2075Long Question3+2 Marks

13.

Find the domain and sketch the graph of the function

f(x) = x^2 - 6x

Estimate the area between the curve

y = x^2

and the line y = 1 and y = 2. [3+2]

Asked in 2075Long Question5+0+5 Marks

14.

A function is defined by

f(x) = |x|

Calculate f(-3), f(4), and sketch the graph. Prove that the limit does not exist.

\lim_{x \to 2} \frac{|x-2|}{x-2}

[5+0+5]

Asked in 2074Short Question5 Marks

15.

Define limit of a function.

\lim_{x \to \infty} \left({x - \sqrt{x}} \right)

[5]

Asked in 2074Short Question5 Marks

16.

If

f(x) = \sqrt{x}

and

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

, find

gof

and

fog

. [5]

Asked in 2074Long Question5+0+5 Marks

17.

A function is defined by

f(x) = \begin{cases} x+2, & x<0 \\ 1-x, & x>0 \end{cases}

Calculate f(-1), f(3), and sketch the graph. Prove that the limit does not exist.

\lim_{x \to 0} \frac{|x|}{x}

[5+0+5]

Function of One Variable

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