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CHAP. 5] 1 5.76. Evaluate Ans. (a) (a)
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INTEGRALS dx ; 1 x3 =2
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1 dx p . x x2 1
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sin 2x dx; sin x 4=3
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b 3 c does not exist
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Evaluate lim
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=2 ex2 = e=4 x esin t dt . x!=2 1 cos 2x d dx x3
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Ans.
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e=2 d dx x2
Prove:
t2 t 1 dt 3x3 x5 2x3 3x2 2x;
cos t2 dt 2x cos x4 cos x2 .
5.79. 5.80.
=2  p p p dx 2 ln 2 1 . 1 sin x dx 4; b sin x cos x 0 1 0 1 dx dy I, using the transformation x 1=y. Hence I 0. Explain the fallacy: I 2 2 1 1 x 1 1 y But I tan 1 1 tan 1 1 =4 =4 =2. Thus =2 0. Prove that (a) cos x 1 1 p dx @ tan 1 . 4 2 1 x2 (p p p ) n 1 n 2 2n 1 Evaluate lim . n!1 n3=2
1=2
Prove that
Ans.
2 3 2
p 2 1
& 5.83. Prove that f x
1 if x is irrational is not Riemann integrable in 0; 1 . 0 if x is rational
[Hint: In (2), Page 91, let k , k 1; 2; 3; . . . ; n be rst rational and then irrational points of subdivision and examine the lower and upper sums of Problem 5.31.] 5.84. Prove the result (3) of Problem 5.31. subdivision.] " In Problem 5.31, prove that s @ S. [Hint: First consider the e ect of only one additional point of
5.85. 5.86.
[Hint: Assume the contrary and obtain a contradiction.] b If f x is sectionally continuous in a; b , prove that f x dx exists. [Hint: Enclose each point of discontia
nuity in an interval, noting that the sum of the lengths of such intervals can be made arbitrarily small. Then consider the di erence between the upper and lower sums. 8 2 0<x<1 < 2x If f x 3 x 1 , nd f x dx. Interpret the result graphically. Ans. 9 : 0 6x 1 1 < x < 2 3 Evaluate fx x 1g dx where x denotes the greatest integer less than or equal to x. Interpret the result 2
5.87. 5.88.
graphically. 5.89.
Ans. 3 =2 sinm x  for all real values of m. dx sinm x cosm x 4
(a) Prove that 2 (b) Prove that
dx . 1 tan4 x
=2 5.90. 5.91. Prove that
0 0:5
Show that
sin x dx exists. x 1 tan x dx 0:4872 approximately. x
 5.92. Show that
x dx 2 p : 1 cos2 x 2 2
Partial Derivatives
FUNCTIONS OF TWO OR MORE VARIABLES The de nition of a function was given in 3 (page 39). For us the distinction for functions of two or more variables is that the domain is a set of n-tuples of numbers. The range remains one dimensional and is referred to an interval of numbers. If n 2, the domain is pictured as a twodimensional region. The region is referred to a rectangular Cartesian coordinate system described through number pairs x; y , and the range variable is usually denoted by z. The domain variables are independent while the range variable is dependent. We use the notation f x; y , F x; y , etc., to denote the value of the function at x; y and write z f x; y , z F x; y , etc. We shall also sometimes use the notation z z x; y although it should be understood that in this case z is used in two senses, namely as a function and as a variable.
EXAMPLE. If f x; y x2 2y3 , then f 3; 1 3 2 2 1 3 7:
The concept is easily extended. Thus w F x; y; z denotes the value of a function at x; y; z [a point in three-dimensional space], etc.
p EXAMPLE. If z 1 x2 y2 , the domain for which z is real consists of the set of points x; y such that x2 y2 @ 1, i.e., the set of points inside and on a circle in the xy plane having center at 0; 0 and radius 1.
THREE-DIMENSIONAL RECTANGULAR COORDINATE SYSTEMS A three-dimensional rectangular coordinate system, as referred to in the previous paragraph, obtained by constructing three mutually perpendicular axes (the x-, y-, and z-axes) intersecting in point O (the origin). It forms a natural extension of the usual xy plane for representing functions of two variables graphically. A point in three dimensions is represented by the triplet x; y; z called coordinates of the point. In this coordinate system z f x; y [or F x; y; z 0] represents a surface, in general.
EXAMPLE. The set of points x; y; z such that z 1 and center at 0; 0; 0 . p 1 x2 y2 comprises the surface of a hemisphere of radius
For functions of more than two variables such geometric interpretation fails, although the terminology is still employed. For example, x; y; z; w is a point in four-dimensional space, and w f x; y; z [or F x; y; z; w 0] represents a hypersurface in four dimensions; thus x2 y2 z2 p w2 a2 represents a hypersphere in four dimensions with radius a > 0 and center at 0; 0; 0; 0 . w a2 x2 y2 z2 , x2 y2 z2 @ a2 describes a function generated from the hypersphere. 116
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