Unformatted text preview: Vector Algebra
A set of vectors S {u1 , u 2 , , u n } is said to linearly dependent if we can find scalars
a1 , a 2 , , a n , not all of them zero, such that
a1 u1 a 2 u 2 a n u n 0. If a1 a 2 a n 0, then the set is linearly independent.
Geometrically, if three vectors are not in the same plane, they are independent and if three
vectors u1 , u 2 , u 3 are in the same plane, they are dependent.
A sequence of vectors is either dependent or independent. They can be combined to give the zero
vector. We begin with some small examples in R 2 :
(a) The vectors (1,0) and (0,1) are independent.
(b) The vectors (1,0) and (1,0.00001) are independent.
(c) The vectors (1,1) and (1, 1) are dependent.
(d) The vectors (1,1) and (0,0) are dependent because of the zero vector.
(e) In R 2 , any three vectors (a, b), and (c, d) and (e, f) are dependent.
Any set of n vectors in R m must be linearly dependent if n > m.
Geometrically, since the points (1,1) and (1,1) are on the same line through the origin, so that
they are dependent.
Let u be a vector and a scalar. Then the product u is defined to be a vector parallel to u
and whose length is times the length of u . In this case, the vector u and u are linearly
dependent.
Remark: Two vectors u and v are linearly dependent if one of them is a multiple of the other. Theorem: If u , v, w be position vectors of the points A,B,C, respectively and if there are
nonzero scalars x , y , z for which xu y v z w 0 with x y z 0, then the points A,B,C are
collinear.
Let there be four vectors u1 , u 2 , u 3 , u 4 in R 3 which are linearly dependent. Then any one of
them, say, u 4 can be expressed as a linear combination of u1 , u 2 , u 3 :
u 4 x1 u1 x 2 u 2 x3 u 3 , where x1 , x 2 , x3 are scalars (not all of them zero).
Problems
1. Show that the three vector u 2iˆ ˆj 3kˆ, v iˆ 4kˆ and w 4iˆ 3 ˆj kˆ are linearly
dependent. Determine a relation among them and show that the terminal points are
collinear.
Solution: Write the given vectors in the form as u ( 2,1, 3) , v (1,0, 4) and
w ( 4,3, 1) .
Now write xu y v z w (0,0,0) for scalars x , y, z . x( 2,1, 3) y (1,0, 4) z ( 4,3, 1) (0,0,0) So that 2 x y 4 z 0, x 3 z 0 and 3 x 4 y z 0 .
Now reduce the system by Gaussian elimination: 2
2 x y 4 z 0
x 3 z 0 3 x 4 y z 0 x 3 z 0
2 x y 4 z 0 3 x 4 y z 0 x 3 z 0
y 2 z 0 4 y 8 z 0 x 3 z 0
y 2 z 0 In the reduced system (last system), the leading variables are x,y while z is the nonleading (free)
variable. So the system has infinitely many nonzero solutions.
Setting z 1, we get y 2 and x 3 .
Since we now have xu y v z w (0,0,0) and x y z 0, the terminal points are collinear.
Finally, write w u v for nonzero scalars , . Then we have ( 4,3, 1) ( 2,1, 3) (1,0, 4)
and 3 4 1 Solving them we get
Hence, we can write 2 4 , 3, 3, and 2 .
w 3u 2v . 2. Write down the geometrical interpretation of dot product and cross product of vectors.
3. Consider the position vectors OA 4i 5 j k , OB ( j k ), OC 3i 9 j 4k and
OD 4i 4 j 4k . Determine whether the terminal points of the vectors are coplanar or
not.
Or, Show that the four points 4i 5 j k , ( j k ), 3i 9 j 4k and 4i 4 j 4k are
coplanar.
4. For the vectors a , b , c prove that a (b c) ( a .c)b (a .b)c. Hence calculate
( a b) c . 5. For the vectors a,b,c . b c ) (c a ) ( a .b c) 2 .
prove that (a b) ( Problems on Limits, Continuity and Differentiability
1. A function f (x ) is defined as follows: 6 4 x, f ( x ) 6 4 x, 6 4 x, 3 x 0
2
3
0 x 2
3
x .
2
3
2 Discuss the continuity at x and differentiability at x 0. Also sketch the graph of f (x).
2. A function f (x ) is defined by 2x 1, x 1
f ( x ) 2 x x 1, x 1.
Examine the continuity and differentiability of f (x) at x 1. Also sketch the graph of
f (x ). 2 3
3. A function f (x ) is defined by 1 x, x, f ( x ) 2 x, 2 x x 2 , x 0
0 x 1
1 x 2
x 2. Examine the continuity and differentiability of f (x) at x 1 and x 2. Also sketch the
graph
of f (x).
4. A function f (x ) is defined by x 3 , x 3
f ( x) x 3 0,
x 3. f ( x ), if it exists.
Find lim
x 3
5. A function f (x ) is defined by x,
0 x 1 f ( x ) 2 x , 1 x 2 1 x x 2 , x 2. 2
Examine the continuity and differentiability of f (x) at x 2. Also sketch the graph of
f (x ). 6. A function f (x ) is defined by x 2 1,
0 x 1 2
f ( x ) 4 x 3 x, 1 x 2 3x 4, x 2. Discuss the continuity of f (x) at x 1 and the existence f ' ( x) at x 2. Also sketch the
graph of f (x ).
7. A function f (x ) is defined by x 2 1, x 0 f ( x) 2 x 1, 0 x 1 x 2 x 1, x 1. Test the continuity and differentiability of f (x) at x 0 and at x 1. Also sketch the
graph of f (x).
3 4
8. A function f (x ) is defined by 2 1 x sin , x 0
f ( x ) x x 2 , x 0. Discuss the continuity and differentiability of f (x) at x 0.
9. Consider the function 2x 5, 3 x 0 f ( x) 3, x 0 5x 2 , x 0. Discuss the continuity and differentiability of f (x) at x 0.
10. Consider the function x 2 , x 0 f ( x ) x ,
0 x 1
1 , x 1.
x
Discuss the continuity and differentiability of f (x) at x 0 and x 1.
Differentiation Explicit and Implicit Functions
a bx 1. If f ( x) ln a bx a bx a bx , find for what values of x, 1 0.
f ' ( x) ( x 3 2 x 5) 4 3 x 2 3x 2
dy
y .
, where
2. Find
1
2
dx
x tan x
2 ax bx 3. If f ( x) a b 2 x a b2 a2
, show that f ' (0) 2 log b
ab Problems on L’ Hospital’s rule
Evaluate the following:
1
x sin 1 x
tan x x (i) lim (ii) lim x 0
x 0
sin 3 x x x (v) lim a x
x 0 1
x [Hints: ln ln(1 x 2 )
(vi) lim
x 0
ln ln cos x (iii) lim sin x x
2 d
( a x ) a x log e a, where
dx (vii) lim
x 4 a b . tan x (iv) lim
x 0 x sin x
tan 3 x a is a positive constant] sec 2 x 2 tan x
1 cos 4 x 4 a b 5
1 (viii) lim sin x x
x 0 x 1 2 1 (xvii) lim ( x ) (xv) lim
x 0 (x) lim
x 0 1 1 x sin x xe x sin x
3x 2
1
1 x 2 (xii) lim
x 0 x 0 x 0 x (xi) lim (1 x) x
(xiv) lim (ix) lim tan x x
x 0 x sin x
x3 (e x 1) tan 2 x
x3
x 1 ln x x 1 (xiii) lim
x 0 (xvi) lim
x 0 sin x ln(e x cos x)
x sin x . x 0 Problems on Expansion of Functions
1. State Taylor’s theorem and Maclaurin’s theorem. Use Maclaurin’s theorem to expand e 2 x sin 3 x
in a power series. 2. Expand tan x in a series of ascending powers of x up to fourth powers of x. 4 3. State Rolle’s theorem. Give the geometrical interpretation of Rolle’s theorem and Mean value
theorem.
4. Verify Rolle’s theorem for f ( x) e x sin x in the interval (0, ). 5. Give the statement of Taylor’s theorem. Expand cos 2 x in powers of x .
2 6. Give the statement of Maclaurin’s theorem.
x
Expand x
in a series of ascending powers of x.
e 1
7. State Lagrange’s Mean value theorem. Find the value of x 0 in the Mean value theorem for the
function f ( x) x 3 5 x 2 12 x 20 in the interval (0,4).
8. Use the Mean Value theorem to show
x ln(1 x) x for 1 x 0 and for x 0.
1 x
5
h2
f ' ' ( x h), 0 1 , where f ( x) ( x a ) 2 . Show that
9. Given f ( x h) f ( x) hf ' ( x) 2!
64 for x = a.
225 10. Write the Maclaurin’s series for the following functions:
(i) y ln(1 sin x)
(ii) y ln(1 sin x) (iii) y ln(1 cos x)
Problems on Successive Differentiation
2x 3
, then find y n .
1. If y 2
x 3x 2
2. If y x2
, then find y n .
( x 1) 2 ( x 2) 3. Find the nth derivative of the following:
(i) y cos x cos 2 x cos 3x (ii) y e ax sin(bx c)
(iv) y e ax cos 2 x sin x
(v) y cos 4 x
6
(vii) y sin x .
5 (iii) y e ax cos bx
(vi) y cos 5 x 6 ln y . By applying Leibnitz’s theorem prove that m (1 x 2 ) y n 2 ( 2n 1) xy n1 ( n 2 m 2 ) y n 0. 4. Given x sin n y x
5. If cos 1 ln then prove that
b n 2 x y n 2 ( 2n 1) xy n 1 2n 2 y n 0.
1 sin 1 y , then by applying Leibnitz’s theorem prove that
m (1 x 2 ) y n 2 ( 2n 1) xy n1 (m 2 n 2 ) y n 0. 6. If x sin 7. If y A cos{m sin 1 (ax b)}, then show that
{1 (ax b) 2 } y n 2 (2n 1)a (ax b) y n 1 (m 2 n 2 )a 2 y n 0. 8. If y ( x 2 1) n , then show that
( x 2 1) y n 2 2 xy n 1 n(n 1) y n 0. 9. If y [ x 1 x 2 ] m , then show that 2 (1 x ) y n 2 (2n 1) xy n 1 (n 2 m 2 ) y n 0.
1 1 10. If y m y m 2 x, then show that
( x 2 1) y n 2 ( 2n 1) xy n 1 ( n 2 m 2 ) y n 0. 11. If y sin( m sin 1 x), then by applying Leibnitz’s theorem show that
(1 x 2 ) y n 2 (2n 1) xy n 1 (m 2 n 2 ) y n 0. 12. If y e m cos 1 x , then by applying Leibnitz’s theorem show that (1 x 2 ) y n 2 ( 2n 1) xy n1 ( m 2 n 2 ) y n 0.
13. If y a cos(ln x) b sin(ln x ), then by applying Leibnitz’s theorem show that (i) x 2 y 2 xy1 y 0.
(ii) x 2 y n 2 (2n 1) xy n 1 (n 2 1) y n 0.
Maxima and Minima
1. Consider the following functions:
1
3 1
2 3
2
i. y x x 6 x 8 ii. y x 4 2 x 3 3x 2 4 x 4 iii. y 3 x 4 10 x 3 12 x 2 12 x 7
iv. y 3 x 4 10 x 3 12 x 2 12 x 7
4
3
2
v. y x 6 x 12 x 8 x.
Find the following:
(i)
the critical points
(ii)
the intervals on which y is increasing and decreasing
(iii)
the maximum and minimum values of y
(iv)
the points of inflection
(v)
the equations of the tangents at the points of inflection
Finally, sketch the graph of f (x).
2. Find the critical numbers of f if f ( x) ( x 5) 2 3 x 4 .
3. Given f ( x) x 3 x 2 5 x 5.
(i) Find the intervals on which f is increasing and the intervals on which f is decreasing.
(ii) Find the intervals on which the graph of f is concave upward and concave downward.
(iii) Sketch the graph of f .
6 7
4. If f ( x) 12 2 x 2 x 4 , find the maximum and minimum values of f . Discuss concavity, find
the points of inflection and sketch the graph of f .
5. Find the equations of the tangents at the points of inflection of
y f ( x) x 4 6 x 3 12 x 2 8 x. Sketch the graph of the function.
6. Find the maximum volume of a right circular cylinder that can be inscribed in a cone of
altitude 12 cm and base radius 4 cm, if the axes of the cylinder and cone coincide.
7. A cylindrical container with circular base is to hold 64 cubic inches. Find the dimensions so
that the amount of metal required is a minimum.
8. A circular cylindrical container, open at the top and having a capacity of 24 cubic inches, is
to be manufactured. If the cost of the material used for the bottom of the container is three
times that used for the curved part and if there is no waste of material, find the dimensions
which will minimize the cost.
9. Find the slopes of the lines that are parallel to xz and yzplane and tangent to the surface
z 3 x 2 4 y 2 6 at P(1,3,2).
Partial Differentiation
1
1. State Euler’s theorem for homogeneous functions. If u tan u u 5 x 4 3 xy 3 7 y 2 z 2
then show
x 2 2 y 2 3 yz u that x x y y z z sin 2u.
2. If u tan 1 u
u
u
1
x2 y2 z2
then show that x x y y z z 2 sin 2u.
xyz
1 3. If u ( x, y, z ) ( x 2 y 2 z 2 ) 2 , then show that u satisfies the Laplace’s equation.
4. If u ( x, y, z ) 3e 2 x cos 2 y x 3 3xy 2 5 z 7, then show that u satisfies the Laplace’s
equation.
u
u 1 x 1 y 5. If u sin tan , then find the value of x x y y . x y
m 2v 2v 2v 6. If v ( x 2 y 2 z 2 ) 2 , then find the value of
.
x 2 y 2 z 2 7. If v f ( x, y ) with x r cos and y r sin , then express v
v
and y in polar
x coordinates. y x z y
w
w
w y2 z2
0.
, then prove that x 2
, x y
z
xy
yz 8. If w f 9. If u f ( x, y ), where x r cos , y r sin , prove that 2 u 2 u 2 u 1 u
1 2u .
x 2 y 2
r 2 r r r 2 2 Solution: Given u f ( x, y ) and x r cos , y r sin . So that
Therefore, we get
7 r x2 y2 and tan 1 y
.
x 8
r
2x
x r cos cos ,
2
2
x 2 x y
r
r
r
2y
y r sin sin ,
2
2
y 2 x y
r
r y 2 y
r sin sin x 2 2 ,
2
2
x
r
y
x y
r
1 2
x
1
2 x
cos x 2 2 .
2
y
r
y
x y
1 2
x
v v r v v sin v cos Hence,
x r x x
r
r and (2) v
v r
v v cos v sin y
r y y
r
r (3) sin cos x
r
r cos sin y
r
r 2 v v sin v sin v ( ) 2 cos cos x x
r
r r
r x 2 v sin 2 v
sin v cos cos 2 2 r r
r
r We can write (4) and (5) So sin 2v
v cos v sin 2 v sin cos r r
r
r r 2 2 v 2 sin cos 2 v sin 2 2 v sin 2 v 2 sin cos v
cos 2 2 r
r
r r r
r 2 2
r2 (6) Similarly, we can show that
2v 2 v 2 sin cos 2 v
cos 2 2 v cos 2 v 2 sin cos v
sin 2 2 2
r
r
r
r y
r
r 2 2
r2 Thus 2 v 2 v 2 v 1 v 1 2 v 0
x 2 y 2 r 2 r r r 2 2 which is the Laplace equation in polar coordinates. 8 (7) (8) ...
View
Full Document
 Winter '13
 Calculus, Derivative, Vector Space, Sin

Click to edit the document details