Download ASSIGNMENT 1

MTE-01
Assignment Booklet
Bachelor’s Degree Programme
Calculus
School of Sciences
Indira Gandhi National Open University
New Delhi
2004
1
Dear Student,
Please read the section on assignments in the Programme Guide for Elective Courses that we sent you
after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for
continuous evaluation, which would consist of two tutor-marked assignments for this course. The
assignments are in this booklet.
A sample exam paper is also given at the end of this booklet for your information.
Instructions for Formating Your Assignments
Before attempting the assignments please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following
format :
ROLL NO:………………………….
NAME:………………………….
ADDRESS:………………………….
…………………………..
…………………………..
…………………………..
COURSE CODE:………………………….
COURSE TITLE:………………………….
ASSIGNMENT NO:………………………
STUDY CENTRE:………………………...
DATE:…………………….…….
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND
TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) The assignments are to be submitted to your Study Centre.
Your answer sheets will only be accepted within 2003, not later.
Please keep a copy of your answer sheets.
Wish you good luck.
2
ASSIGNMENT 1
(To be done after studying Blocks 1 and 2.)
Course Code : MTE-01
Assignment Code : MTE-01/AST-1/2004
Maximum Marks : 100
1.
Which of the following statements are true? Give reasons for your answer. The
reasons should be presented as a short proof or an example to make your point. (For
example, if you say (i) below is true, then give a short proof for showing it is true. If
you say (i) is false, then given an example for which it is not true.)
i)
If y = f(x) such that
dy
exists at every point of [a, b], then y must have a
dx
vii)
critical point in [a, b].
A function that is continuous everywhere has no point of inflection.
If f : [a, b]  R is differentiable over [a, b], then f(a) = f(b).
x 5
The domain of the function f given by f(x) =
is R  {–5}.
x5
Any proper subset of R has an infimum.
If f : [a, b]  R is a function such that x = y  f(x) = f(y)  x, y  [a, b], then
f is 1–1.
lim f ( x )  L if f ( x )  f (a )  L  x  ] a – h, a + h [ , where h > 0.
viii)
ix)
x)
If y = f (x) is such that yn = yn – 1 for some n  N, then f(x) = c, a constant.
y2 – x2 = 3x has oblique asymptotes.
If f is concave on [a, b], then it is convex on [ – b, – a ] .
(20)
ii)
iii)
iv)
v)
vi)
x a
2. (a) Express the following without using the absolute value signs.
i) 1  2 1  2 ,
ii ) x  1



(b) Express the following as an interval or union of intervals.
i)
x R x  2  3 ,
ii )
x R

x  2 2 ,
iii )
xR
(2)

x 2  6x  10  0 .
(5)
(c) Draw the graph of a 1–1 function which is not onto. Justify your choice of function.
(3)
3. (a) Examine the continuity of the following function f. (This means that you must give
all the points at which the function is continuous, and all the points at which it is
discontinuous, along with the justification.)
1  x 2 , x  0

(4)
f ( x )  5x  4 , 0  x 1

2
1  x , x 1.
3
 1.
x 1 x  1
Further, define a function f : [ 0, 1 ]  R such that lim f (x)  1 .
(b) Using the - definition, show that lim
x 1
3
(6)
4. (a) Find the slope of the tangent at each point of the curve y 
2x
.
x 2
2
(b) If y is a function of x such that xy . yx = 1 for x  0, find y1.
(2)
(2)
(c) Is f given by f(x) = x5 + 2x + 1 invertible? If so, check if f –1 is differentiable. If f is
not invertible, define another function g which is invertible and for which g–1 is
derivable.
d 1
d 1
(f ) or
(g ) , whichever exists.
Also find
(5)
dx
dx
(d) A spherical balloon is leaking air so that its surface area is decreasing at the rate of
1 sq. cm./sec. At the moment that the radius is 10 cm., how fast is the volume
decreasing?
(3)
(e) Find
d

(coth 1 (cos ec x )), 0  x  .
dx
2
(3)
5. (a) Sketch the curve r = 1 – 2cos . Clearly state the properties you use for doing so. (7)
   
,
(b) Draw the graph of tan x and its derivative for x  
in the same diagram,
 2 2 
giving the features you used for doing so.
(6)
x 2 , x  0
(c) Let F : R  R be defined by F(x) = 
 x , x  0
Find the intervals over which F is concave upwards. Also find all the inflection points
of F.
(5)
(d) Give an example of a function with two inflection points.
example.
Justify your choice of
(2)
6. (a) A woman on an island wishes to go to a small town on the mainland, the coast of
which is 1 km. away from the island. From the point on the coast closest to the
island it is 4 km. along the straight shore line to the town. The woman can row her
boat at a speed of 2 km. per hour and walk at a speed of 4 km. per hour along the
beach. Where should she land the boat in order to get to town as quickly as possible?
(6)
(b) Let f(x) = x2 + 4x + 3. Find c so that f  (c) equals the average slope of f(x) on [ a, b ].
(2)
(c) Using Rolle’s theorem, show that   such that  1 and sin 2 = – 43.
(5)
(d) Find all the critical points of the function f defined by f(x) = x3(x2 – 1)2  x [ –2, 2 ].
Classify these critical points as local extrema, absolute extrema and those which are
not extrema.
(7)
7. (a) Find the Taylor polynomial of degree 4 around zero for f(x) = sinh x. Hence estimate
2
sinh   .
(3)
 15 
(b) Find a and b if we know that the Maclaurin’s expansion of degree at most 4 for
4
f(x) = ax + b sin x is x –
x3
.
6
(2)
5
ASSIGNMENT 2
(To be done after studying Blocks 3 and 4.)
Course Code : MTE-01
Assignment Code : MTE-01/AST-2/2004
Maximum Marks : 100
1.
Which of the following statements are true, and which are false? Give reasons for
your answers.
i)
ii)
iii)
iv)
v)
vi)
 
The function f, defined by f(x) = sin x + cos 2x, is monotonic in 0,  .
 4
1
d sin x 3 3 
2
1 3
1 3
2
3
3
 sin ( t ) dt  3 sin (sin x ) cos (sin x )  sin (8x ) cos (8x )
dx  2 x



For a function f defined on [a,b] and a partition P ={x 0, x1, , xn} of [a,b],
L(P,f)  U(P,f) only if x1 – x0 = x2 –x1 =  = xn – xn – 1.
If f is integrable on [a,b], then it is continuous on [a,b].
If f is defined and differentiable on [a,b], then it is integrable on [a,b].
g  fdx  f  gdx
f 


, where f and g are two integrable functions and
dx

  g 
(  gdx) 2
g(x)  0  x.
a
vii)
If f(x) = f(a – x)  x  [0,a], then
 f (x)dx  0.
0
viii)
ix)
x)
The area under the curve y = f(x) for x[a,b] is U(P,f) – L(P,f), where P is a
partition of [a,b].
The Maclaurin’s series for a function is a reduction formula for the integral of
the function.
We can find the integral of any function by Simpson’s rule.
(20)
2. (a) Find the upper and lower integrals of the function g, defined on [0,1] by
x , x  0
g(x) = 
.
0, x  0
Hence conclude whether g is integrable on [0, 1] or not.
(6)
(b) Prove Th.6, Sec. 10.4, by proving the following steps.
(i)
Let m = min {f(x)x  [a,b]} and M = max {f(x)x  [a,b]}. Show that
b
m(b – a) 
 f (x)dx  M (b – a).
a
(ii)
Next, use the intermediate value theorem (Block 1) to find x  [a,b] so that
1
f( x ) =
b  a 
b
 f (x)dx
(4)
a
6
3. (a) Find constants b, c, d so that the function g defined by g(x) = bx2 + cx + d satisfies
1
0

g ( x ) dx  1,
1
2
 g(x) dx  1,  g(x) dx  9.
0
(4)
1
(b) Find the intervals in which G is increasing or decreasing, where
x
G(x) =
 (u  1) e
3 u du  x   1.
(3)
1
a
(c)
Suppose a  0 and f is an odd function. Evaluate
 f (x)dx.
Also choose an odd
a
function and verify your answer for it.
(3)
4. Evaluate the following integrals :
i)
x
dx
2
ii )
4
x
x 2  2x  3
2
5. (a) Find a reduction formula for
 4x  8

dx
iii )
 (x  2) (x  1) dx
9x
2
(2+4+4)

2
x n cos x dx, n  2. Hence evaluate

x 4 cos x dx.
(4)
0
(b) Find a reduction formula for  tan m x sec n x dx, m odd and n  0. Use this to find
 tan
3
x sec 4 x dx.
(6)


 tan 1   tan 1  
if 0    .
2
1 
1 2
 1
4  3
Deduce that   tan 1   .
4 6
3 4 25
6. (a) Prove that
(7)
(b) Find an approximate value of ln (0.9) upto 3 decimal places.
(3)
7. (a) The velocity of a train which starts from rest is given by the following table. The time
is recorded in minutes and the speed in kilometres per hour.
Minute
Km./hr.
2
15
4
20
6
27
8
33
10
26
12
24
14
18
Use the Trapezoidal rule and Simpson’s rule to estimate approximately (upto 2
decimal places) the total distance travelled in 12 minutes.
(4)
(b) Find the area of the region enclosed by the curves x2 = y and y =
1 4
(x + x).
2
(2½)
(c) Give an example to show that the region between the curves f(x) and g(x) over [a,b]
may not have the same area as the region between f (x) and g(x) over [a, b].
(2)
7
(d) Let y = x and x2 + y2 = 1 intersect in A in the 1st quadrant. Find the area of the
surface obtained by revolving the portion OAB of the curves about the x-axis, where
O is the origin and B is the point (1,0).
(2½)
(e) Consider two single-cell organisms. One is spherical, and the other has the shape of
the solid of revolution obtained by revolving x2 + 2y2 = 1 around the x-axis. If both
have the same volume, do they have the same requirements for nutrients? Note that
the amount of nutrients absorbed is proportional to the surface area.
(4)
8. (a) Find the length of the arc of the curve y = ex between x = ln 3 and x = ln
8.
(2)
(b) The half-life of uranium 238U is 4.5 billion years. If the quantity of this isotope in a
rock sample has diminished to 90% of its original amount, how old is the rock
sample? (1 billion = 109.)
(4)
(c) Find the volume of the solid obtained by revolving about the y-axis the planar region
bounded by the y-axis, y = x and y = 2x2 – 1.
(4)
9. (a) Find the arc length of the curve r(t) = (1 + t2, 3 – 2t3) for t [0,1].
(2½)
(b) Use Simpson’s rule with n = 4 to estimate the arc length of r(t) = (cos2t, sin3t),
t [0,].
(2½)
8
SAMPLE PAPER
MTE-1 : CALCULUS
Time : 2 hours
Maximum Marks : 50
(Weightage 70%)
___________________________________________________________________________
Note : Q. No. 1 is compulsory.
1.
Which of the following statements are true? Justify your answers with a short proof
or a counterexample.
10
(i)
f(x) = x3 has no critical point.
(ii)
The function f(x) = x 2  5 is differentiable in  2, 4 .
(iii)
(iv)
(v)
2.
Do any four questions from Q.Nos. 2 to 7.
(a)
(b)
 
The function f(x) = sin x + cos 2x is not monotonic in  0,  .
 4

1
sin x

d 

3 3
sin ( t )dt   3 sin 2 (sin 1 x ) 3 cos(sin 1 x ) 3  sin 2 (8x 3 ) cos (8x 3 )

dx
 2x

x 3
is the int erval  3, 3 .
The domain of the function f given by f(x) =
x 3


Examine the continuity of the function f defined by
 x 2
if x  0

if 0  x 1
f(x) = 5x  4
 2
4 x  3x if x 1
over R.
4
The velocity of a train which starts from rest is given by the following table,
the time being recorded in minutes from the start and the speed in kilometres
per hour.
Minute
Km./hr.
2
15
4
20
6
27
8
33
10
26
12
24
14
18
Estimate approximately (upto 2 decimal places) the total distance travelled in
12 minutes.
3
(c)
3.
(a)
Let f : [0, 1]  R be a function defined by f(x) = x2 – 3 and let
 1 1 3 
P = 0 , , , , 1 be a partition of [0, 1]. Show that L(P, f)  U(P, f).
 4 2 4 
3
1
If y = e m sin x , prove that
(1 – x2)yn+2 – (2n + 1)xyn+1 – (n2 + m2)yn = 0.
9
4

(b)
4.
(a)
(b)
3
 1.
x 1 x  1
3
Further, for what value of a is lim
 1?
x a x  1
Use the - definition to show that lim
Using Lagrange’s mean value theorem, prove that


 tan 1   tan 1  
if 0    .
2
1 
1 2
 3
4  1
Deduce that 
 tan 1   .
4 25
3 4 6
Evaluate the following integrals :
(i)
5.
(a)
(b)
(c)
6.
x
 (x  1)
xe
2
dx ,
(ii)

6
4
(tan
1
x)
1 x2
2
dx .
Find an approximate value of (1.01)5/2 using Maclaurin’s series, upto 3
decimal places.
3
Show that (2, 0) is a singular point of the curve y2 = (x – 1) (x – 2)2.
Also determine the type of singular point it is.
3
Find the length of the arc of the spiral r = a, for any , the arc being measured
from the pole.
4
Trace the curve x(x2 + y2) = a(x2 – y2), clearly stating the necessary properties
required for tracing it.
10
2
7.
6
(a)
If I n 

0

x n sin ( x ) dx (n  1), then prove that I n  n (n 1) I n  2  n  
 2
n 1
.
2
Hence find the value of I n 
x
3
sin x dx .
5
0
x
 cos 1 x  (  1) for  1 x 1.
1 x
(b)
Show that
(c)
Give an example, with justification, of a real-valued function which is
differentiable on R but is not one-one.
2
10
3