Module-IV(Vector Calculus)
Note 1:
The directional derivative of a scalar eld (real-valued function) f at a point P
in the direction of a unit vector u is denoted by (Du f )P and (Du f )P = (f )P .u, where
f is the gradient of f.
Note 2:
If f (x, y, z) =
1. Find the volume of the ellipsoid
2
x2
+ y2 =1
a2
b
=2c
=2c
2
b 1 x2
=4c
=4c
4c
=
b
where
aa
a
0
b
a
1
z2
c2
= 1 by triple integration.
dz dy dx
b
x2
a2
y2
dy dx
b2
1
x2 y 2
2 dy dx
a2
b
1
x2 y 2
2 dy dx (the integrand being even function of y)
a2
Module-I
Solutions of Linear equations
1.
Verify whether the following equations
x + 2y z = 3
3x y + 2z = 1
2x 2y + 3z = 2
x y + z = 1,
are consistent. If they are consistent nd a solution.
Solution:
The system of equations is consistent if they possess a
Chapter 2: Graphs, Lines, and Inequalities
Section 2.1 Graphs
1. (1, 2) lies in quadrant IV
( 2,1) lies in quadrant II
( 3, 4 ) lies in quadrant I
( 5, 6) lies in quadrant III
7. Continued
2. ( ,2) lies in quadrant I
3 y 12
lies in quadrant IV
y4
(4,0) li
Chapter 3: Functions and Graphs
Section 3.1 Functions
1.
x3210
y9410
1
1
2
4
3
9
This rule defines y as a function of x because each
value of x determines one and only one value of
y.
2.
x9410
y3210
1
1
4
2
9
3
The rule does not define y as a function of
Chapter 4: Exponential and Logarithmic Functions
Section 4.1 Exponential Functions
1.
f ( x ) = 6x
This function is exponential, because the variable
is in the exponent and the base is a positive
constant.
8. g( x ) = 4 - x
x
a.
This function is linear be
Chapter 11: Differential Calculus
Section 11.1 Limits
1. a.
By reading the graph, as x gets closer to
3 from the left or the right, f(x) gets closer
to 3, so
lim f ( x ) 3.
x
b.
5. a.
3
x
b.
As x gets closer to 1.5 from the left or right,
f(x) gets closer
Chapter 1: Algebra and Equations
Section 1.1 The Real Numbers
13.
1. True. This statement is true, since every integer
can be written as the ratio of the integer and 1.
5
.
For example, 5
1
2. False. For example, 5 is a real number, and
10
5
which is not