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SET 1

Indian Institute of Technology Department of Mathematics

Autumn Mid Semester Examination-2012 Subject Name: Mathematics I

Subject No: MA10001 No. of students:1300 Time: 2 hrs F.M. 30

Instructions: Answer ALL qu~stions. Numerals in right margin indicate marks. Answer each question in a NEW page and all the parts of the same question TOGETHER. Write question SET NUMBER on the top of your answer script.

1. (a) Let f be a function continuous on (a, b] and differentiable on (a, b). Using Lagrange'~ mean value theorem prove that

(i) f is increasing in (a, b) if f'(x) 2 0, for all x E. (a, b)

(ii) f is decreasing in (a, b) if f'(x) S 0, for all x E (a, b) [2]

· (b) Write Taylor's expansion for the function f(x) =-IX about 1, up to six terms including the remainder term in Lagrange's form. [3]

2. (a) Let

{

xy(2x2 - 3y2) f(x,y)= x2+y2 '

0,

Find fyx(O, 0) and fxy(O, 0).

( ) 82u (b) Let u = esin xyz . Find Oy Oz.

(x,y) f. (0, 0)

(x, y) = (0, 0)

[3]

[2]

3. (a) Using E-8 method find limit ofxy sin(x+y) when (x,y) --t (7r/2,0).[2]

(b) Test the differentiability of the function

f(x,y) = {

at (0, 0).

x2 + y2' 0,

1

(x, y) f. (0, 0)

(x, y) = (0, 0)

[3]

I

4. (a) C .d h X+ 1 F. d . onsi er t e curve y = 2

• In the intervals where the curve IS X + 1

concave upwards (convex downwards) or concave downwards (convex upwards). Also find the point(s) of inflection. [3]

x 2 - 3x (b) Find the asymptotes of the curve y =

x-1 [2]

5. (a) State and prove Euler's theorem for homogeneous functions of two variables. [2]

(b) Determine Taylor's series expansion of f(x + 1, y + 1r /3) in ascending powers of x and y 'when f(x,y) =sin xy neglecting terms of degree higher than two. [3]

6. (a) Investigate for local maxima and minima of

(3]

(b) Find the general solution of

(y2exy2

+ 4x3 )dx + (2xyexy2

- 3y2 )dy = 0

[2]

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